Special relativity

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standard_configuration"> <div class="vector-toc-text"> 3.2 Standard configuration </div> </a> <ul id="toc-Standard_configuration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lack_of_an_absolute_reference_frame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lack_of_an_absolute_reference_frame"> <div class="vector-toc-text"> 3.3 Lack of an absolute reference frame </div> </a> <ul id="toc-Lack_of_an_absolute_reference_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativity_without_the_second_postulate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativity_without_the_second_postulate"> <div class="vector-toc-text"> 3.4 Relativity without the second postulate </div> </a> <ul id="toc-Relativity_without_the_second_postulate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lorentz_invariance_as_the_essential_core_of_special_relativity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lorentz_invariance_as_the_essential_core_of_special_relativity"> <div class="vector-toc-text"> 4 Lorentz invariance as the essential core of special relativity </div> </a> <button aria-controls="toc-Lorentz_invariance_as_the_essential_core_of_special_relativity-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Lorentz invariance as the essential core of special relativity subsection </button> <ul id="toc-Lorentz_invariance_as_the_essential_core_of_special_relativity-sublist" class="vector-toc-list"> <li id="toc-Alternative_approaches_to_special_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_approaches_to_special_relativity"> <div class="vector-toc-text"> 4.1 Alternative approaches to special relativity </div> </a> <ul id="toc-Alternative_approaches_to_special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_transformation_and_its_inverse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_transformation_and_its_inverse"> <div class="vector-toc-text"> 4.2 Lorentz transformation and its inverse </div> </a> <ul id="toc-Lorentz_transformation_and_its_inverse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graphical_representation_of_the_Lorentz_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graphical_representation_of_the_Lorentz_transformation"> <div class="vector-toc-text"> 4.3 Graphical representation of the Lorentz transformation </div> </a> <ul id="toc-Graphical_representation_of_the_Lorentz_transformation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Consequences_derived_from_the_Lorentz_transformation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Consequences_derived_from_the_Lorentz_transformation"> <div class="vector-toc-text"> 5 Consequences derived from the Lorentz transformation </div> </a> <button aria-controls="toc-Consequences_derived_from_the_Lorentz_transformation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Consequences derived from the Lorentz transformation subsection </button> <ul id="toc-Consequences_derived_from_the_Lorentz_transformation-sublist" class="vector-toc-list"> <li id="toc-Invariant_interval" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invariant_interval"> <div class="vector-toc-text"> 5.1 Invariant interval </div> </a> <ul id="toc-Invariant_interval-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativity_of_simultaneity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativity_of_simultaneity"> <div class="vector-toc-text"> 5.2 Relativity of simultaneity </div> </a> <ul id="toc-Relativity_of_simultaneity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_dilation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_dilation"> <div class="vector-toc-text"> 5.3 Time dilation </div> </a> <ul id="toc-Time_dilation-sublist" class="vector-toc-list"> <li id="toc-Langevin's_light-clock" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Langevin's_light-clock"> <div class="vector-toc-text"> 5.3.1 Langevin's light-clock </div> </a> <ul id="toc-Langevin's_light-clock-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocal_time_dilation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reciprocal_time_dilation"> <div class="vector-toc-text"> 5.3.2 Reciprocal time dilation </div> </a> <ul id="toc-Reciprocal_time_dilation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Twin_paradox" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Twin_paradox"> <div class="vector-toc-text"> 5.4 Twin paradox </div> </a> <ul id="toc-Twin_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Length_contraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Length_contraction"> <div class="vector-toc-text"> 5.5 Length contraction </div> </a> <ul id="toc-Length_contraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_transformation_of_velocities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_transformation_of_velocities"> <div class="vector-toc-text"> 5.6 Lorentz transformation of velocities </div> </a> <ul id="toc-Lorentz_transformation_of_velocities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thomas_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thomas_rotation"> <div class="vector-toc-text"> 5.7 Thomas rotation </div> </a> <ul id="toc-Thomas_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Causality_and_prohibition_of_motion_faster_than_light" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Causality_and_prohibition_of_motion_faster_than_light"> <div class="vector-toc-text"> 5.8 Causality and prohibition of motion faster than light </div> </a> <ul id="toc-Causality_and_prohibition_of_motion_faster_than_light-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Optical_effects" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Optical_effects"> <div class="vector-toc-text"> 6 Optical effects </div> </a> <button aria-controls="toc-Optical_effects-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Optical effects subsection </button> <ul id="toc-Optical_effects-sublist" class="vector-toc-list"> <li id="toc-Dragging_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dragging_effects"> <div class="vector-toc-text"> 6.1 Dragging effects </div> </a> <ul id="toc-Dragging_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_aberration_of_light" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_aberration_of_light"> <div class="vector-toc-text"> 6.2 Relativistic aberration of light </div> </a> <ul id="toc-Relativistic_aberration_of_light-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_Doppler_effect" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_Doppler_effect"> <div class="vector-toc-text"> 6.3 Relativistic Doppler effect </div> </a> <ul id="toc-Relativistic_Doppler_effect-sublist" class="vector-toc-list"> <li id="toc-Relativistic_longitudinal_Doppler_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relativistic_longitudinal_Doppler_effect"> <div class="vector-toc-text"> 6.3.1 Relativistic longitudinal Doppler effect </div> </a> <ul id="toc-Relativistic_longitudinal_Doppler_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transverse_Doppler_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transverse_Doppler_effect"> <div class="vector-toc-text"> 6.3.2 Transverse Doppler effect </div> </a> <ul id="toc-Transverse_Doppler_effect-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measurement_versus_visual_appearance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_versus_visual_appearance"> <div class="vector-toc-text"> 6.4 Measurement versus visual appearance </div> </a> <ul id="toc-Measurement_versus_visual_appearance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dynamics"> <div class="vector-toc-text"> 7 Dynamics </div> </a> <button aria-controls="toc-Dynamics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Dynamics subsection </button> <ul id="toc-Dynamics-sublist" class="vector-toc-list"> <li id="toc-Equivalence_of_mass_and_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence_of_mass_and_energy"> <div class="vector-toc-text"> 7.1 Equivalence of mass and energy </div> </a> <ul id="toc-Equivalence_of_mass_and_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einstein's_1905_demonstration_of_E_=_mc2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein's_1905_demonstration_of_E_=_mc2"> <div class="vector-toc-text"> 7.2 Einstein's 1905 demonstration of E = mc2 </div> </a> <ul id="toc-Einstein's_1905_demonstration_of_E_=_mc2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-How_far_can_you_travel_from_the_Earth?" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#How_far_can_you_travel_from_the_Earth?"> <div class="vector-toc-text"> 7.3 How far can you travel from the Earth? </div> </a> <ul id="toc-How_far_can_you_travel_from_the_Earth?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elastic_collisions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elastic_collisions"> <div class="vector-toc-text"> 7.4 Elastic collisions </div> </a> <ul id="toc-Elastic_collisions-sublist" class="vector-toc-list"> <li id="toc-Newtonian_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Newtonian_analysis"> <div class="vector-toc-text"> 7.4.1 Newtonian analysis </div> </a> <ul id="toc-Newtonian_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relativistic_analysis"> <div class="vector-toc-text"> 7.4.2 Relativistic analysis </div> </a> <ul id="toc-Relativistic_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Beyond_the_basics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Beyond_the_basics"> <div class="vector-toc-text"> 8 Beyond the basics </div> </a> <button aria-controls="toc-Beyond_the_basics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Beyond the basics subsection </button> <ul id="toc-Beyond_the_basics-sublist" class="vector-toc-list"> <li id="toc-Rapidity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rapidity"> <div class="vector-toc-text"> 8.1 Rapidity </div> </a> <ul id="toc-Rapidity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4‑vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#4‑vectors"> <div class="vector-toc-text"> 8.2 4‑vectors </div> </a> <ul id="toc-4‑vectors-sublist" class="vector-toc-list"> <li id="toc-Definition_of_4-vectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definition_of_4-vectors"> <div class="vector-toc-text"> 8.2.1 Definition of 4-vectors </div> </a> <ul id="toc-Definition_of_4-vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_4-vectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties_of_4-vectors"> <div class="vector-toc-text"> 8.2.2 Properties of 4-vectors </div> </a> <ul id="toc-Properties_of_4-vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_4-vectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_of_4-vectors"> <div class="vector-toc-text"> 8.2.3 Examples of 4-vectors </div> </a> <ul id="toc-Examples_of_4-vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4-vectors_and_physical_law" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#4-vectors_and_physical_law"> <div class="vector-toc-text"> 8.2.4 4-vectors and physical law </div> </a> <ul id="toc-4-vectors_and_physical_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Acceleration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acceleration"> <div class="vector-toc-text"> 8.3 Acceleration </div> </a> <ul id="toc-Acceleration-sublist" class="vector-toc-list"> <li id="toc-Dewan–Beran–Bell_spaceship_paradox" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dewan–Beran–Bell_spaceship_paradox"> <div class="vector-toc-text"> 8.3.1 Dewan–Beran–Bell spaceship paradox </div> </a> <ul id="toc-Dewan–Beran–Bell_spaceship_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Accelerated_observer_with_horizon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Accelerated_observer_with_horizon"> <div class="vector-toc-text"> 8.3.2 Accelerated observer with horizon </div> </a> <ul id="toc-Accelerated_observer_with_horizon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Relativity_and_unifying_electromagnetism" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relativity_and_unifying_electromagnetism"> <div class="vector-toc-text"> 9 Relativity and unifying electromagnetism </div> </a> <ul id="toc-Relativity_and_unifying_electromagnetism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theories_of_relativity_and_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Theories_of_relativity_and_quantum_mechanics"> <div class="vector-toc-text"> 10 Theories of relativity and quantum mechanics </div> </a> <ul id="toc-Theories_of_relativity_and_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Status" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Status"> <div class="vector-toc-text"> 11 Status </div> </a> <ul id="toc-Status-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Technical_discussion_of_spacetime" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Technical_discussion_of_spacetime"> <div class="vector-toc-text"> 12 Technical discussion of spacetime </div> </a> <button aria-controls="toc-Technical_discussion_of_spacetime-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Technical discussion of spacetime subsection </button> <ul id="toc-Technical_discussion_of_spacetime-sublist" class="vector-toc-list"> <li id="toc-Geometry_of_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_of_spacetime"> <div class="vector-toc-text"> 12.1 Geometry of spacetime </div> </a> <ul id="toc-Geometry_of_spacetime-sublist" class="vector-toc-list"> <li id="toc-Comparison_between_flat_Euclidean_space_and_Minkowski_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Comparison_between_flat_Euclidean_space_and_Minkowski_space"> <div class="vector-toc-text"> 12.1.1 Comparison between flat Euclidean space and Minkowski space </div> </a> <ul id="toc-Comparison_between_flat_Euclidean_space_and_Minkowski_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3D_spacetime" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#3D_spacetime"> <div class="vector-toc-text"> 12.1.2 3D spacetime </div> </a> <ul id="toc-3D_spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4D_spacetime" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#4D_spacetime"> <div class="vector-toc-text"> 12.1.3 4D spacetime </div> </a> <ul id="toc-4D_spacetime-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physics_in_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics_in_spacetime"> <div class="vector-toc-text"> 12.2 Physics in spacetime </div> </a> <ul id="toc-Physics_in_spacetime-sublist" class="vector-toc-list"> <li id="toc-Transformations_of_physical_quantities_between_reference_frames" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transformations_of_physical_quantities_between_reference_frames"> <div class="vector-toc-text"> 12.2.1 Transformations of physical quantities between reference frames </div> </a> <ul id="toc-Transformations_of_physical_quantities_between_reference_frames-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Metric"> <div class="vector-toc-text"> 12.2.2 Metric </div> </a> <ul id="toc-Metric-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_kinematics_and_invariance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relativistic_kinematics_and_invariance"> <div class="vector-toc-text"> 12.2.3 Relativistic kinematics and invariance </div> </a> <ul id="toc-Relativistic_kinematics_and_invariance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_dynamics_and_invariance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relativistic_dynamics_and_invariance"> <div class="vector-toc-text"> 12.2.4 Relativistic dynamics and invariance </div> </a> <ul id="toc-Relativistic_dynamics_and_invariance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 13 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 14 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primary_sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Primary_sources"> <div class="vector-toc-text"> 15 Primary sources </div> </a> <ul id="toc-Primary_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 16 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 17 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Texts_by_Einstein_and_text_about_history_of_special_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Texts_by_Einstein_and_text_about_history_of_special_relativity"> <div class="vector-toc-text"> 17.1 Texts by Einstein and text about history of special relativity </div> </a> <ul id="toc-Texts_by_Einstein_and_text_about_history_of_special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks"> <div class="vector-toc-text"> 17.2 Textbooks </div> </a> <ul id="toc-Textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Journal_articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Journal_articles"> <div class="vector-toc-text"> 17.3 Journal articles </div> </a> <ul id="toc-Journal_articles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 18 External links </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle External links subsection </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Original_works" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Original_works"> <div class="vector-toc-text"> 18.1 Original works </div> </a> <ul id="toc-Original_works-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_relativity_for_a_general_audience_(no_mathematical_knowledge_required)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_relativity_for_a_general_audience_(no_mathematical_knowledge_required)"> <div class="vector-toc-text"> 18.2 Special relativity for a general audience (no mathematical knowledge required) </div> </a> <ul id="toc-Special_relativity_for_a_general_audience_(no_mathematical_knowledge_required)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_relativity_explained_(using_simple_or_more_advanced_mathematics)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_relativity_explained_(using_simple_or_more_advanced_mathematics)"> <div class="vector-toc-text"> 18.3 Special relativity explained (using simple or more advanced mathematics) </div> </a> <ul id="toc-Special_relativity_explained_(using_simple_or_more_advanced_mathematics)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Visualization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visualization"> <div class="vector-toc-text"> 18.4 Visualization </div> </a> <ul id="toc-Visualization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Special relativity</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 110 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-110" aria-hidden="true" > 110 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Spesiale_relatiwiteit" title="Spesiale relatiwiteit – Afrikaans" lang="af" hreflang="af" data-title="Spesiale relatiwiteit" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie – Alemannic" lang="gsw" hreflang="gsw" data-title="Spezielle Relativitätstheorie" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%8D%E1%8B%A9_%E1%8A%A0%E1%8A%95%E1%8C%BB%E1%88%AB%E1%8B%8A%E1%8A%90%E1%89%B5" title="ልዩ አንጻራዊነት – Amharic" lang="am" hreflang="am" data-title="ልዩ አንጻራዊነት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%86%D8%B3%D8%A8%D9%8A%D8%A9_%D8%A7%D9%84%D8%AE%D8%A7%D8%B5%D8%A9" title="النسبية الخاصة – Arabic" lang="ar" hreflang="ar" data-title="النسبية الخاصة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Relatividat_especial" title="Relatividat especial – Aragonese" lang="an" hreflang="an" data-title="Relatividat especial" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%87%E0%A6%B7_%E0%A6%86%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%BF%E0%A6%95%E0%A6%A4%E0%A6%BE%E0%A6%AC%E0%A6%BE%E0%A6%A6_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="বিশেষ আপেক্ষিকতাবাদ তত্ত্ব – Assamese" lang="as" hreflang="as" data-title="বিশেষ আপেক্ষিকতাবাদ তত্ত্ব" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_la_relativid%C3%A1_especial" title="Teoría de la relatividá especial – Asturian" lang="ast" hreflang="ast" data-title="Teoría de la relatividá especial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Mba%27ekuaar%C3%A3_joguerahavi%C3%A1rava_ijap%C3%BDva" title="Mba'ekuaarã joguerahaviárava ijapýva – Guarani" lang="gn" hreflang="gn" data-title="Mba'ekuaarã joguerahaviárava ijapýva" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C3%BCsusi_nisbilik_n%C9%99z%C9%99riyy%C9%99si" title="Xüsusi nisbilik nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Xüsusi nisbilik nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D8%A4%D8%B2%D9%84_%D9%86%DB%8C%D8%B3%D8%A8%DB%8C%D8%AA" title="اؤزل نیسبیت – South Azerbaijani" lang="azb" hreflang="azb" data-title="اؤزل نیسبیت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/R%C3%A9lativitas_khusus" title="Rélativitas khusus – Balinese" lang="ban" hreflang="ban" data-title="Rélativitas khusus" data-language-autonym="Basa Bali" data-language-local-name="Balinese" class="interlanguage-link-target">Basa Bali</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%87%E0%A6%B7_%E0%A6%86%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%BF%E0%A6%95%E0%A6%A4%E0%A6%BE" title="বিশেষ আপেক্ষিকতা – Bangla" lang="bn" hreflang="bn" data-title="বিশেষ আপেক্ষিকতা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D1%85%D1%81%D1%83%D1%81_%D1%81%D0%B0%D2%93%D1%8B%D1%88%D1%82%D1%8B%D1%80%D0%BC%D0%B0%D0%BB%D1%8B%D2%A1_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Махсус сағыштырмалыҡ теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Махсус сағыштырмалыҡ теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%8B%D1%8F%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B0%D0%B4%D0%BD%D0%BE%D1%81%D0%BD%D0%B0%D1%81%D1%86%D1%96" title="Спецыяльная тэорыя адноснасці – Belarusian" lang="be" hreflang="be" data-title="Спецыяльная тэорыя адноснасці" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D0%BF%D1%8D%D1%86%D1%8B%D1%8F%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B0%D0%B4%D0%BD%D0%BE%D1%81%D0%BD%D0%B0%D1%81%D1%8C%D1%86%D1%96" title="Спэцыяльная тэорыя адноснасьці – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Спэцыяльная тэорыя адноснасьці" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AC%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B8_%E0%A4%B8%E0%A4%BE%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A4%E0%A4%BE" title="बिशेस सापेक्षता – Bhojpuri" lang="bh" hreflang="bh" data-title="बिशेस सापेक्षता" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D0%B8%D1%82%D0%B5%D0%BB%D0%BD%D0%BE%D1%81%D1%82%D1%82%D0%B0" title="Специална теория на относителността – Bulgarian" lang="bg" hreflang="bg" data-title="Специална теория на относителността" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Spezieje_Relativitetstheorie" title="Spezieje Relativitetstheorie – Bavarian" lang="bar" hreflang="bar" data-title="Spezieje Relativitetstheorie" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Posebna_teorija_relativnosti" title="Posebna teorija relativnosti – Bosnian" lang="bs" hreflang="bs" data-title="Posebna teorija relativnosti" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%B8%D1%81%D0%B0%D0%BD%D0%B3%D1%8B_%D0%B1%D0%B0%D0%B9%D0%B4%D0%B0%D0%BB%D0%B0%D0%B9_%D1%82%D1%83%D1%81%D1%85%D0%B0%D0%B9_%D0%BE%D0%BD%D0%BE%D0%BB" title="Харисангы байдалай тусхай онол – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Харисангы байдалай тусхай онол" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Relativitat_especial" title="Relativitat especial – Catalan" lang="ca" hreflang="ca" data-title="Relativitat especial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BD%D0%BB%D0%B0%D1%88%D1%82%D0%B0%D1%80%D1%83%D0%BB%C4%83%D1%85%C4%83%D0%BD_%D1%8F%D1%82%D0%B0%D1%80%D0%BB%C4%83_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Танлаштарулăхăн ятарлă теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Танлаштарулăхăн ятарлă теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Speci%C3%A1ln%C3%AD_teorie_relativity" title="Speciální teorie relativity – Czech" lang="cs" hreflang="cs" data-title="Speciální teorie relativity" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Perthnasedd_arbennig" title="Perthnasedd arbennig – Welsh" lang="cy" hreflang="cy" data-title="Perthnasedd arbennig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Speciel_relativitetsteori" title="Speciel relativitetsteori – Danish" lang="da" hreflang="da" data-title="Speciel relativitetsteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie – German" lang="de" hreflang="de" data-title="Spezielle Relativitätstheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Erirelatiivsusteooria" title="Erirelatiivsusteooria – Estonian" lang="et" hreflang="et" data-title="Erirelatiivsusteooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CE%AE_%CF%83%CF%87%CE%B5%CF%84%CE%B9%CE%BA%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Ειδική σχετικότητα – Greek" lang="el" hreflang="el" data-title="Ειδική σχετικότητα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_la_relatividad_especial" title="Teoría de la relatividad especial – Spanish" lang="es" hreflang="es" data-title="Teoría de la relatividad especial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Speciala_teorio_de_relativeco" title="Speciala teorio de relativeco – Esperanto" lang="eo" hreflang="eo" data-title="Speciala teorio de relativeco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erlatibitate_berezia" title="Erlatibitate berezia – Basque" lang="eu" hreflang="eu" data-title="Erlatibitate berezia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%DB%8C%D8%AA_%D8%AE%D8%A7%D8%B5" title="نسبیت خاص – Persian" lang="fa" hreflang="fa" data-title="نسبیت خاص" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Special_relativity" title="Special relativity – Fiji Hindi" lang="hif" hreflang="hif" data-title="Special relativity" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Relativit%C3%A9_restreinte" title="Relativité restreinte – French" lang="fr" hreflang="fr" data-title="Relativité restreinte" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Te%C3%B2irig_sh%C3%B2nraichte_na_d%C3%A0imheachd" title="Teòirig shònraichte na dàimheachd – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Teòirig shònraichte na dàimheachd" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Relatividade_especial" title="Relatividade especial – Galician" lang="gl" hreflang="gl" data-title="Relatividade especial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8A%B9%EC%88%98_%EC%83%81%EB%8C%80%EC%84%B1%EC%9D%B4%EB%A1%A0" title="특수 상대성이론 – Korean" lang="ko" hreflang="ko" data-title="특수 상대성이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D6%80%D5%A1%D5%A2%D5%A5%D6%80%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%B0%D5%A1%D5%BF%D5%B8%D6%82%D5%AF_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հարաբերականության հատուկ տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Հարաբերականության հատուկ տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A4%BF%E0%A4%B7%E0%A5%8D%E0%A4%9F_%E0%A4%86%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BF%E0%A4%95%E0%A4%A4%E0%A4%BE" title="विशिष्ट आपेक्षिकता – Hindi" lang="hi" hreflang="hi" data-title="विशिष्ट आपेक्षिकता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://hr.wikipedia.org/wiki/Posebna_teorija_relativnosti" title="Posebna teorija relativnosti – Croatian" lang="hr" hreflang="hr" data-title="Posebna teorija relativnosti" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Specala_relativeso" title="Specala relativeso – Ido" lang="io" hreflang="io" data-title="Specala relativeso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Relativitas_khusus" title="Relativitas khusus – Indonesian" lang="id" hreflang="id" data-title="Relativitas khusus" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Relativitate_special" title="Relativitate special – Interlingua" lang="ia" hreflang="ia" data-title="Relativitate special" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Takmarka%C3%B0a_afst%C3%A6%C3%B0iskenningin" title="Takmarkaða afstæðiskenningin – Icelandic" lang="is" hreflang="is" data-title="Takmarkaða afstæðiskenningin" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Relativit%C3%A0_ristretta" title="Relatività ristretta – Italian" lang="it" hreflang="it" data-title="Relatività ristretta" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%99%D7%97%D7%A1%D7%95%D7%AA_%D7%94%D7%A4%D7%A8%D7%98%D7%99%D7%AA" title="תורת היחסות הפרטית – Hebrew" lang="he" hreflang="he" data-title="תורת היחסות הפרטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%90%E1%83%A0%E1%83%93%E1%83%9D%E1%83%91%E1%83%98%E1%83%97%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%A1%E1%83%9E%E1%83%94%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ფარდობითობის სპეციალური თეორია – Georgian" lang="ka" hreflang="ka" data-title="ფარდობითობის სპეციალური თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%BD%D0%B0%D0%B9%D1%8B_%D1%81%D0%B0%D0%BB%D1%8B%D1%81%D1%82%D1%8B%D1%80%D0%BC%D0%B0%D0%BB%D1%8B%D0%BB%D1%8B%D2%9B_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Арнайы салыстырмалылық теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Арнайы салыстырмалылық теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uhusianifu_maalumu" title="Uhusianifu maalumu – Swahili" lang="sw" hreflang="sw" data-title="Uhusianifu maalumu" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%82%D0%B0%D0%B9%D1%8B%D0%BD_%D1%81%D0%B0%D0%BB%D1%8B%D1%88%D1%82%D1%8B%D1%80%D0%BC%D0%B0%D0%BB%D1%83%D1%83%D0%BB%D1%83%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Атайын салыштырмалуулук теориясы – Kyrgyz" lang="ky" hreflang="ky" data-title="Атайын салыштырмалуулук теориясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Relativitas_specialis" title="Relativitas specialis – Latin" lang="la" hreflang="la" data-title="Relativitas specialis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Speci%C4%81l%C4%81_relativit%C4%81tes_teorija" title="Speciālā relativitātes teorija – Latvian" lang="lv" hreflang="lv" data-title="Speciālā relativitātes teorija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Specialioji_reliatyvumo_teorija" title="Specialioji reliatyvumo teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Specialioji reliatyvumo teorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Speci%C3%A1lis_relativit%C3%A1selm%C3%A9let" title="Speciális relativitáselmélet – Hungarian" lang="hu" hreflang="hu" data-title="Speciális relativitáselmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B7%D0%B0_%D1%80%D0%B5%D0%BB%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D0%B0" title="Специјална теорија за релативноста – Macedonian" lang="mk" hreflang="mk" data-title="Специјална теорија за релативноста" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B4%BF%E0%B4%B7%E0%B5%8D%E0%B4%9F_%E0%B4%86%E0%B4%AA%E0%B5%87%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B4%BF%E0%B4%95%E0%B4%A4%E0%B4%BE_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വിശിഷ്ട ആപേക്ഷികതാ സിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="വിശിഷ്ട ആപേക്ഷികതാ സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Relattivit%C3%A0_ristretta" title="Relattività ristretta – Maltese" lang="mt" hreflang="mt" data-title="Relattività ristretta" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7_%E0%A4%B8%E0%A4%BE%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A4%E0%A4%BE" title="विशेष सापेक्षता – Marathi" lang="mr" hreflang="mr" data-title="विशेष सापेक्षता" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%D9%8A%D9%87_%D8%AE%D8%A7%D8%B5%D9%87" title="نسبيه خاصه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="نسبيه خاصه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kerelatifan_khas" title="Kerelatifan khas – Malay" lang="ms" hreflang="ms" data-title="Kerelatifan khas" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D1%8C%D1%86%D0%B0%D0%BD%D0%B3%D1%83%D0%B9%D0%BD_%D1%82%D1%83%D1%81%D0%B3%D0%B0%D0%B9_%D0%BE%D0%BD%D0%BE%D0%BB" title="Харьцангуйн тусгай онол – Mongolian" lang="mn" hreflang="mn" data-title="Харьцангуйн тусгай онол" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%91%E1%80%B0%E1%80%B8%E1%80%94%E1%80%BE%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9B%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="အထူးနှိုင်းရသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="အထူးနှိုင်းရသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Speciale_relativiteitstheorie" title="Speciale relativiteitstheorie – Dutch" lang="nl" hreflang="nl" data-title="Speciale relativiteitstheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E6%80%A7%E7%90%86%E8%AB%96" title="特殊相対性理論 – Japanese" lang="ja" hreflang="ja" data-title="特殊相対性理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Den_spesielle_relativitetsteorien" title="Den spesielle relativitetsteorien – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Den spesielle relativitetsteorien" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Den_spesielle_relativitetsteorien" title="Den spesielle relativitetsteorien – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Den spesielle relativitetsteorien" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Relativitat_especiala" title="Relativitat especiala – Occitan" lang="oc" hreflang="oc" data-title="Relativitat especiala" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AC%BF%E0%AC%B6%E0%AD%87%E0%AC%B7_%E0%AC%86%E0%AC%AA%E0%AD%87%E0%AC%95%E0%AD%8D%E0%AC%B7%E0%AC%BF%E0%AC%95_%E0%AC%A4%E0%AC%A4%E0%AD%8D%E0%AC%A4%E0%AD%8D%E0%AD%B1" title="ବିଶେଷ ଆପେକ୍ଷିକ ତତ୍ତ୍ୱ – Odia" lang="or" hreflang="or" data-title="ବିଶେଷ ଆପେକ୍ଷିକ ତତ୍ତ୍ୱ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Maxsus_nisbiylik_nazariyasi" title="Maxsus nisbiylik nazariyasi – Uzbek" lang="uz" hreflang="uz" data-title="Maxsus nisbiylik nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BF%E0%A8%B8%E0%A8%BC%E0%A9%87%E0%A8%B8%E0%A8%BC_%E0%A8%B8%E0%A8%BE%E0%A8%AA%E0%A9%87%E0%A8%96%E0%A8%A4%E0%A8%BE" title="ਵਿਸ਼ੇਸ਼ ਸਾਪੇਖਤਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਵਿਸ਼ੇਸ਼ ਸਾਪੇਖਤਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B3%D9%BE%DB%8C%D8%B4%D9%84_%D8%B1%DB%8C%D9%84%DB%8C%D9%B9%DB%8C%D9%88%D9%B9%DB%8C" title="سپیشل ریلیٹیوٹی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="سپیشل ریلیٹیوٹی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%81%D8%A7%D9%86%DA%AB%DA%93%DB%8C_%D9%86%D8%B3%D8%A8%D9%8A%D8%AA" title="ځانګړی نسبيت – Pashto" lang="ps" hreflang="ps" data-title="ځانګړی نسبيت" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teor%C3%ACa_dla_relativit%C3%A0_limit%C3%A0" title="Teorìa dla relatività limità – Piedmontese" lang="pms" hreflang="pms" data-title="Teorìa dla relatività limità" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Spetschale_Relativit%C3%A4tstheorie" title="Spetschale Relativitätstheorie – Low German" lang="nds" hreflang="nds" data-title="Spetschale Relativitätstheorie" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szczeg%C3%B3lna_teoria_wzgl%C4%99dno%C5%9Bci" title="Szczególna teoria względności – Polish" lang="pl" hreflang="pl" data-title="Szczególna teoria względności" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Relatividade_restrita" title="Relatividade restrita – Portuguese" lang="pt" hreflang="pt" data-title="Relatividade restrita" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_relativit%C4%83%C8%9Bii_restr%C3%A2nse" title="Teoria relativității restrânse – Romanian" lang="ro" hreflang="ro" data-title="Teoria relativității restrânse" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Специальная теория относительности – Russian" lang="ru" hreflang="ru" data-title="Специальная теория относительности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Special_relativity" title="Special relativity – Scots" lang="sco" hreflang="sco" data-title="Special relativity" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_speciale_e_relativitetit" title="Teoria speciale e relativitetit – Albanian" lang="sq" hreflang="sq" data-title="Teoria speciale e relativitetit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiur%C3%ACa_di_la_rilativitati_spiciali" title="Tiurìa di la rilativitati spiciali – Sicilian" lang="scn" hreflang="scn" data-title="Tiurìa di la rilativitati spiciali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%80%E0%B7%92%E0%B7%81%E0%B7%9A%E0%B7%82_%E0%B7%83%E0%B7%8F%E0%B6%B4%E0%B7%9A%E0%B6%9A%E0%B7%8A%E0%B7%82%E0%B6%AD%E0%B7%8F%E0%B7%80%E0%B7%8F%E0%B6%AF%E0%B6%BA" title="විශේෂ සාපේක්ෂතාවාදය – Sinhala" lang="si" hreflang="si" data-title="විශේෂ සාපේක්ෂතාවාදය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Special_relativity" title="Special relativity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Special relativity" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5_%D9%86%D8%B3%D8%A8%D8%AA_%D8%AC%D9%88_%D9%86%D8%B8%D8%B1%D9%8A%D9%88" title="خاص نسبت جو نظريو – Sindhi" lang="sd" hreflang="sd" data-title="خاص نسبت جو نظريو" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://sk.wikipedia.org/wiki/%C5%A0peci%C3%A1lna_te%C3%B3ria_relativity" title="Špeciálna teória relativity – Slovak" lang="sk" hreflang="sk" data-title="Špeciálna teória relativity" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Posebna_teorija_relativnosti" title="Posebna teorija relativnosti – Slovenian" lang="sl" hreflang="sl" data-title="Posebna teorija relativnosti" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%DB%8E%DA%98%DB%95%DB%8C%DB%8C%DB%8C_%D8%AA%D8%A7%DB%8C%D8%A8%DB%95%D8%AA" title="ڕێژەییی تایبەت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕێژەییی تایبەت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Specijalna_teorija_relativnosti" title="Specijalna teorija relativnosti – Serbian" lang="sr" hreflang="sr" data-title="Specijalna teorija relativnosti" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Specijalna_teorija_relativnosti" title="Specijalna teorija relativnosti – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Specijalna teorija relativnosti" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Teori_Relativitas_Khusus" title="Teori Relativitas Khusus – Sundanese" lang="su" hreflang="su" data-title="Teori Relativitas Khusus" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Erityinen_suhteellisuusteoria" title="Erityinen suhteellisuusteoria – Finnish" lang="fi" hreflang="fi" data-title="Erityinen suhteellisuusteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Speciella_relativitetsteorin" title="Speciella relativitetsteorin – Swedish" lang="sv" hreflang="sv" data-title="Speciella relativitetsteorin" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_natatanging_relatibidad" title="Teorya ng natatanging relatibidad – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng natatanging relatibidad" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BF%E0%AE%B1%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="சிறப்புச் சார்புக் கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="சிறப்புச் சார்புக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://tt.wikipedia.org/wiki/Maxsus_%C3%A7a%C4%9F%C4%B1%C5%9Ft%C4%B1rmal%C4%B1l%C4%B1q_teori%C3%A4se" title="Maxsus çağıştırmalılıq teoriäse – Tatar" lang="tt" hreflang="tt" data-title="Maxsus çağıştırmalılıq teoriäse" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9E%E0%B8%B1%E0%B8%97%E0%B8%98%E0%B8%A0%E0%B8%B2%E0%B8%9E%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9" title="ทฤษฎีสัมพัทธภาพพิเศษ – Thai" lang="th" hreflang="th" data-title="ทฤษฎีสัมพัทธภาพพิเศษ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96zel_g%C3%B6relilik" title="Özel görelilik – Turkish" lang="tr" hreflang="tr" data-title="Özel görelilik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B2%D1%96%D0%B4%D0%BD%D0%BE%D1%81%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Спеціальна теорія відносності – Ukrainian" lang="uk" hreflang="uk" data-title="Спеціальна теорія відносності" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%B6%D8%A7%D9%81%DB%8C%D8%AA_%D9%85%D8%AE%D8%B5%D9%88%D8%B5%DB%81" title="اضافیت مخصوصہ – Urdu" lang="ur" hreflang="ur" data-title="اضافیت مخصوصہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Specialine_rel%C3%A4tivi%C5%BEusen_teorii" title="Specialine relätivižusen teorii – Veps" lang="vep" hreflang="vep" data-title="Specialine relätivižusen teorii" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Thuy%E1%BA%BFt_t%C6%B0%C6%A1ng_%C4%91%E1%BB%91i_h%E1%BA%B9p" title="Thuyết tương đối hẹp – Vietnamese" lang="vi" hreflang="vi" data-title="Thuyết tương đối hẹp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Erirelatiivsusteooria" title="Erirelatiivsusteooria – Võro" lang="vro" hreflang="vro" data-title="Erirelatiivsusteooria" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%8B%B9%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96" title="狹義相對論 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="狹義相對論" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pinaurog_nga_relatibidad" title="Pinaurog nga relatibidad – Waray" lang="war" hreflang="war" data-title="Pinaurog nga relatibidad" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8B%AD%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" title="狭义相对论 – Wu" lang="wuu" hreflang="wuu" data-title="狭义相对论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A1%D7%A4%D7%A2%D7%A6%D7%99%D7%A2%D7%9C%D7%A2_%D7%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2_%D7%A4%D7%95%D7%9F_%D7%A8%D7%A2%D7%9C%D7%90%D7%98%D7%99%D7%95%D7%95%D7%99%D7%98%D7%A2%D7%98" title="ספעציעלע טעאריע פון רעלאטיוויטעט – Yiddish" lang="yi" hreflang="yi" data-title="ספעציעלע טעאריע פון רעלאטיוויטעט" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%8B%B9%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96" title="狹義相對論 – Cantonese" lang="yue" hreflang="yue" data-title="狹義相對論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Teoriya_Relatifiya_X%C4%B1susiye" title="Teoriya Relatifiya Xısusiye – Zazaki" lang="diq" hreflang="diq" data-title="Teoriya Relatifiya Xısusiye" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target">Zazaki</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Spec%C4%93liuoj%C4%97_rel%C4%93t%C4%ABvoma_teuor%C4%97j%C4%97" title="Specēliuojė relētīvoma teuorėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Specēliuojė relētīvoma teuorėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Special relativity</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:World_line.svg" class="mw-file-description" title="The world line: a diagrammatic representation of spacetime"><img alt="The world line: a diagrammatic representation of spacetime" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/200px-World_line.svg.png" decoding="async" width="200" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/300px-World_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/400px-World_line.svg.png 2x" data-file-width="481" data-file-height="491" /></a></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Formulations_of_special_relativity" title="Formulations of special relativity">Formulations</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c">Foundations</div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Postulates_of_special_relativity" title="Postulates of special relativity">Einstein's postulates</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c">Consequences</div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">Relativistic disk</a></li> <li><a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a></li> <li><a href="/wiki/Ehrenfest_paradox" title="Ehrenfest paradox">Ehrenfest paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Spacetime diagram</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">Dynamics</a></div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Invariant_mass" title="Invariant mass">Proper mass</a></li> <li><a href="/wiki/Four-momentum" title="Four-momentum">Four-momentum</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist" style="font-size:110%;"><ul><li><a href="/wiki/History_of_special_relativity" title="History of special relativity">History</a></li><li>Precursors</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a></li> <li><a href="/wiki/Aether_theories" title="Aether theories">Aether theories</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)"><div class="sidebar-list-title-c">People</div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/Albert_A._Michelson" title="Albert A. Michelson">Michelson</a></li> <li><a href="/wiki/Edward_W._Morley" title="Edward W. Morley">Morley</a></li> <li><a href="/wiki/George_Francis_FitzGerald" title="George Francis FitzGerald">FitzGerald</a></li> <li><a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Herglotz</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Max_Abraham" title="Max Abraham">Abraham</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Richard_C._Tolman" title="Richard C. 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In <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s 1905 paper, <a href="/wiki/Annus_Mirabilis_papers#Special_relativity" class="mw-redirect" title="Annus Mirabilis papers">On the Electrodynamics of Moving Bodies</a>, the theory is presented as being based on just <a href="/wiki/Postulates_of_special_relativity" title="Postulates of special relativity">two postulates</a>:<a href="#cite_note-electro-1">[p 1]</a><a href="#cite_note-Griffiths-2013-2">[1]</a><a href="#cite_note-Jackson-1999-3">[2]</a> <ol><li>The <a href="/wiki/Laws_of_physics" class="mw-redirect" title="Laws of physics">laws of physics</a> are <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> (identical) in all <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frames of reference</a> (that is, <a href="/wiki/Frame_of_reference" title="Frame of reference">frames of reference</a> with no <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>). This is known as the <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a>.</li> <li>The <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> is the same for all observers, regardless of the motion of light source or observer. This is known as the principle of light constancy, or the principle of light speed invariance.</li></ol> The first postulate was first formulated by <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> (see <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a>). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origins_and_significance">Origins and significance</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=1" title="Edit section: Origins and significance">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_special_relativity" title="History of special relativity">History of special relativity</a></div> Special relativity was described by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".<a href="#cite_note-electro-1">[p 1]</a> <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> appeared to be incompatible with <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, and the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> failed to detect the Earth's motion against the hypothesized <a href="/wiki/Luminiferous_aether" title="Luminiferous aether">luminiferous aether</a>. These led to the development of the <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a>, by <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a>, which adjust distances and times for moving objects. Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>relativistic velocities). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible.<a href="#cite_note-4">[3]</a><a href="#cite_note-Lanczos-1970-5">[4]</a> Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth. Special relativity has a wide range of consequences that have been experimentally verified.<a href="#cite_note-6">[5]</a> These include the <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>, <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a>, <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a>, the relativistic velocity addition formula, the relativistic <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a>, <a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">relativistic mass</a>, <a href="/wiki/Speed_of_light#Upper_limit_on_speeds" title="Speed of light">a universal speed limit</a>, <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>, the speed of causality and the <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a>.<a href="#cite_note-Griffiths-2013-2">[1]</a><a href="#cite_note-Jackson-1999-3">[2]</a> It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and <a href="/wiki/Space" title="Space">spatial</a> position. Rather than an invariant time interval between two events, there is an invariant <a href="/wiki/Spacetime_interval" class="mw-redirect" title="Spacetime interval">spacetime interval</a>. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Energy" title="Energy">energy</a>, as expressed in the <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a> formula ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=mc^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f73dbd37a0cac34406ee89057fa1b36a1e6a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}">⁠, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum.<a href="#cite_note-relativity-7">[6]</a><a href="#cite_note-Feynman-8">[7]</a> It also explains how the phenomena of electricity and magnetism are related.<a href="#cite_note-Griffiths-2013-2">[1]</a><a href="#cite_note-Jackson-1999-3">[2]</a> A defining feature of special relativity is the replacement of the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformations</a> of Newtonian mechanics with the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into <a href="/wiki/Spacetime" title="Spacetime">a single continuum known as "spacetime"</a>. Events that occur at the same time for one observer can occur at different times for another. Until several years later when Einstein developed <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, which introduced a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case".<a href="#cite_note-9">[p 2]</a><a href="#cite_note-10">[p 3]</a><a href="#cite_note-11">[p 4]</a><a href="#cite_note-12">[note 1]</a> Some of the work of Albert Einstein in special relativity is built on the earlier work by <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a> and <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>. The theory became essentially complete in 1907, with <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>'s papers on spacetime.<a href="#cite_note-Lanczos-1970-5">[4]</a> The theory is "special" in that it only applies in the <a href="/wiki/Special_case" title="Special case">special case</a> where the spacetime is "flat", that is, where the <a href="/wiki/Curvature_of_spacetime" class="mw-redirect" title="Curvature of spacetime">curvature of spacetime</a> (a consequence of the <a href="/wiki/Energy%E2%80%93momentum_tensor" class="mw-redirect" title="Energy–momentum tensor">energy–momentum tensor</a> and representing <a href="/wiki/Gravity" title="Gravity">gravity</a>) is negligible.<a href="#cite_note-13">[8]</a><a href="#cite_note-14">[note 2]</a> To correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate <a href="/wiki/Acceleration_(special_relativity)" title="Acceleration (special relativity)">accelerations</a> as well as <a href="/wiki/Rindler_coordinates" title="Rindler coordinates">accelerating frames of reference</a>.<a href="#cite_note-15">[9]</a><a href="#cite_note-16">[10]</a> Just as <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a> is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational fields</a>, that is, at a sufficiently small scale (e.g., when <a href="/wiki/Tidal_force" title="Tidal force">tidal forces</a> are negligible) and in conditions of <a href="/wiki/Free_fall" title="Free fall">free fall</a>. But general relativity incorporates <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. As long as the universe can be modeled as a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this <a href="/wiki/Curved_spacetime" title="Curved spacetime">curved spacetime</a>. <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> had already postulated that there is no absolute and well-defined state of rest (no <a href="/wiki/Preferred_frame" title="Preferred frame">privileged reference frames</a>), a principle now called <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galileo's principle of relativity</a>. Einstein extended this principle so that it accounted for the constant speed of light,<a href="#cite_note-Taylor_1992-17">[11]</a> a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the <a href="/wiki/Laws_of_physics" class="mw-redirect" title="Laws of physics">laws of physics</a>, including both the laws of mechanics and of <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>.<a href="#cite_note-Rindler0-18">[12]</a> <div class="mw-heading mw-heading2"><h2 id="Traditional_"two_postulates"_approach_to_special_relativity">Traditional "two postulates" approach to special relativity </h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=2" title="Edit section: Traditional "two postulates" approach to special relativity">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote floatright" style="width:40%; ;"> <blockquote class="quotebox-quote left-aligned" style=""> "Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results ... How, then, could such a universal principle be found?" </blockquote> <cite class="left-aligned" style="">Albert Einstein: Autobiographical Notes<a href="#cite_note-autogenerated1-19">[p 5]</a></cite> </div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Postulates_of_special_relativity" title="Postulates of special relativity">Postulates of special relativity</a></div> Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:<a href="#cite_note-electro-1">[p 1]</a> <ul><li>The <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a> – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.<a href="#cite_note-electro-1">[p 1]</a></li> <li>The principle of invariant light speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).<a href="#cite_note-electro-1">[p 1]</a> That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.</li></ul> The constancy of the speed of light was motivated by <a href="/wiki/Maxwell%27s_theory_of_electromagnetism" class="mw-redirect" title="Maxwell's theory of electromagnetism">Maxwell's theory of electromagnetism</a><a href="#cite_note-20">[13]</a> and the lack of evidence for the <a href="/wiki/Luminiferous_ether" class="mw-redirect" title="Luminiferous ether">luminiferous ether</a>.<a href="#cite_note-21">[14]</a> There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.<a href="#cite_note-22">[15]</a><a href="#cite_note-mM1905-23">[16]</a> In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance. The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (<a href="/wiki/Duhem%E2%80%93Quine_thesis" title="Duhem–Quine thesis">made in almost all theories of physics</a>), including the <a href="/wiki/Isotropy" title="Isotropy">isotropy</a> and <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneity</a> of space and the independence of measuring rods and clocks from their past history.<a href="#cite_note-24">[p 6]</a> Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.<a href="#cite_note-25">[17]</a> But the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively to K.<a href="#cite_note-Einstein-26">[18]</a></blockquote> <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> provided the mathematical framework for relativity theory by proving that <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a> are a subset of his <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> of symmetry transformations. Einstein later derived these transformations from his axioms. Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.<a href="#cite_note-27">[p 7]</a> <div class="mw-heading mw-heading2"><h2 id="Principle_of_relativity">Principle of relativity</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=3" title="Edit section: Principle of relativity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a></div> <div class="mw-heading mw-heading3"><h3 id="Reference_frames_and_relative_motion">Reference frames and relative motion</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=4" title="Edit section: Reference frames and relative motion">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Frames_of_reference_in_relative_motion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Frames_of_reference_in_relative_motion.svg/300px-Frames_of_reference_in_relative_motion.svg.png" decoding="async" width="300" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Frames_of_reference_in_relative_motion.svg/450px-Frames_of_reference_in_relative_motion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Frames_of_reference_in_relative_motion.svg/600px-Frames_of_reference_in_relative_motion.svg.png 2x" data-file-width="478" data-file-height="283" /></a><figcaption>Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a>, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −v. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.</figcaption></figure> <a href="/wiki/Frame_of_reference" title="Frame of reference">Reference frames</a> play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a "clock" (any reference device with uniform periodicity). An <a href="/wiki/Event_(relativity)" title="Event (relativity)">event</a> is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. Since the speed of light is constant in relativity irrespective of the reference frame, pulses of light can be used to unambiguously measure distances and refer back to the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S. In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called transformation equations. <div class="mw-heading mw-heading3"><h3 id="Standard_configuration">Standard configuration</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=5" title="Edit section: Standard configuration">edit</a>]</div> To gain insight into how the spacetime coordinates measured by observers in different <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">reference frames</a> compare with each other, it is useful to work with a simplified setup with frames in a standard configuration.<a href="#cite_note-Collier-28">[19]</a>: 107  With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two <a href="/wiki/Galilean_reference_frame" class="mw-redirect" title="Galilean reference frame">Galilean reference frames</a> (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′. <ul><li>The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.</li> <li>Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.</li> <li>The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.</li></ul> Since there is no absolute reference frame in relativity theory, a concept of "moving" does not strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving. <div class="mw-heading mw-heading3"><h3 id="Lack_of_an_absolute_reference_frame">Lack of an absolute reference frame</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=6" title="Edit section: Lack of an absolute reference frame">edit</a>]</div> The <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a>, which states that physical laws have the same form in each <a href="/wiki/Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">inertial reference frame</a>, dates back to <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a>, and was incorporated into Newtonian physics. But in the late 19th century the existence of <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic waves</a> led some physicists to suggest that the universe was filled with a substance they called "<a href="/wiki/Luminiferous_aether" title="Luminiferous aether">aether</a>", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an <a href="/wiki/Preferred_frame" title="Preferred frame">absolute reference frame</a> against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelson–Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist.<a href="#cite_note-29">[20]</a> Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities. <div class="mw-heading mw-heading3"><h3 id="Relativity_without_the_second_postulate">Relativity without the second postulate</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=7" title="Edit section: Relativity without the second postulate">edit</a>]</div> From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity) <a href="/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates" title="Derivations of the Lorentz transformations">it can be shown</a> that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.<a href="#cite_note-Friedman-30">[p 8]</a><a href="#cite_note-Morin2007-31">[21]</a> <div class="mw-heading mw-heading2"><h2 id="Lorentz_invariance_as_the_essential_core_of_special_relativity">Lorentz invariance as the essential core of special relativity </h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=8" title="Edit section: Lorentz invariance as the essential core of special relativity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></div> <div class="mw-heading mw-heading3"><h3 id="Alternative_approaches_to_special_relativity">Alternative approaches to special relativity</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=9" title="Edit section: Alternative approaches to special relativity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Derivations_of_the_Lorentz_transformations" title="Derivations of the Lorentz transformations">Derivations of the Lorentz transformations</a></div> Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of: relativity and invariance of the speed of light. He wrote: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events ... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws ...<a href="#cite_note-autogenerated1-19">[p 5]</a></blockquote> Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>.<a href="#cite_note-32">[p 9]</a><a href="#cite_note-33">[p 10]</a> Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations.<a href="#cite_note-Miller2009-34">[22]</a> Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler<a href="#cite_note-Taylor_1992-17">[11]</a> and by Callahan.<a href="#cite_note-Callahan-35">[23]</a> This is also the approach followed by the Wikipedia articles <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a> and <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a>. <div class="mw-heading mw-heading3"><h3 id="Lorentz_transformation_and_its_inverse">Lorentz transformation and its inverse</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=10" title="Edit section: Lorentz transformation and its inverse">edit</a>]</div> Define an <a href="/wiki/Spacetime#Basic_concepts" title="Spacetime">event</a> to have spacetime coordinates (t, x, y, z) in system S and (t′, x′, y′, z′) in a reference frame moving at a velocity v on the x-axis with respect to that frame, S′. Then the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> specifies that these coordinates are related in the following way: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>v</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4f6066009b1535e4856fecbd58921b8de48ccc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:18.877ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}t'&=\gamma \ (t-vx/c^{2})\\x'&=\gamma \ (x-vt)\\y'&=y\\z'&=z,\end{aligned}}}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b130ec5e5e9586833b7888f7cbe2433f1e295e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.929ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}"> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> and c is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum, and the velocity v of S′, relative to S, is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a <a href="/wiki/One-parameter_group" title="One-parameter group">one-parameter group</a> of <a href="/wiki/Linear_mapping" class="mw-redirect" title="Linear mapping">linear mappings</a>, that parameter being called <a href="/wiki/Rapidity" title="Rapidity">rapidity</a>. Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d661443f4fe56eab9d5b9209391d5481deb2ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:18.982ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}"> This shows that the unprimed frame is moving with the velocity −v, as measured in the primed frame.<a href="#cite_note-36">[24]</a> There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction parallel to the motion (which are warped by the γ factor) and perpendicular; see the article <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> for details. A quantity that is invariant under <a href="/wiki/Lorentz_transformations" class="mw-redirect" title="Lorentz transformations">Lorentz transformations</a> is known as a <a href="/wiki/Lorentz_scalar" title="Lorentz scalar">Lorentz scalar</a>. Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x1, t1) and (x′1, t′1), another event has coordinates (x2, t2) and (x′2, t′2), and the differences are defined as <ul><li>Eq. 1:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568223f2885721b0a8f0442e723780462d305533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.266ex; height:3.176ex;" alt="{\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}"></li> <li>Eq. 2:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3c81cd5e1a3c738c63e1b55e085dcba51f72ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.478ex; height:2.509ex;" alt="{\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}"></li></ul> we get <ul><li>Eq. 3:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70bf86b3cdce212f0e8d3227bdde155a2a65bd2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.873ex; height:3.009ex;" alt="{\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>−</mo> <mi>v</mi> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8129c910ae372135cf81cf89e982907c1d2b1f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.346ex; height:3.343ex;" alt="{\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}"></li> <li>Eq. 4:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fbbbce3c4c6d0e32a6cdca0a7018bddd95757c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.977ex; height:3.009ex;" alt="{\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e67e882d66376bd3170df90945ffe4cf4603c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.031ex; height:3.343ex;" alt="{\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}"></li></ul> If we take differentials instead of taking differences, we get <ul><li>Eq. 5:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69aec35a89bc0eab2766e33b188d649b6ace16ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.713ex; height:3.009ex;" alt="{\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mi>v</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e103e5c426a1349efd51885e7a1d96fe57f7a3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.186ex; height:3.343ex;" alt="{\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}"></li> <li>Eq. 6:    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed2f958a99ff20bb81deb94d6151f73c37ad1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.817ex; height:3.009ex;" alt="{\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mtext> </mtext> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f65ee47b4da221c155e9ff3f66fe985df1d6baf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.871ex; height:3.343ex;" alt="{\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graphical_representation_of_the_Lorentz_transformation">Graphical representation of the Lorentz transformation</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=11" title="Edit section: Graphical representation of the Lorentz transformation">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:192px;overflow:hidden"><a href="/wiki/File:Spacetime_diagram_development_A.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Spacetime_diagram_development_A.svg/192px-Spacetime_diagram_development_A.svg.png" decoding="async" width="192" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Spacetime_diagram_development_A.svg/288px-Spacetime_diagram_development_A.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Spacetime_diagram_development_A.svg/384px-Spacetime_diagram_development_A.svg.png 2x" data-file-width="535" data-file-height="535" /></a></div></div><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:192px;overflow:hidden"><a href="/wiki/File:Spacetime_diagram_development_B.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Spacetime_diagram_development_B.svg/192px-Spacetime_diagram_development_B.svg.png" decoding="async" width="192" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Spacetime_diagram_development_B.svg/288px-Spacetime_diagram_development_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Spacetime_diagram_development_B.svg/384px-Spacetime_diagram_development_B.svg.png 2x" data-file-width="535" data-file-height="535" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:192px;overflow:hidden"><a href="/wiki/File:Spacetime_diagram_development_C.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Spacetime_diagram_development_C.svg/192px-Spacetime_diagram_development_C.svg.png" decoding="async" width="192" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Spacetime_diagram_development_C.svg/288px-Spacetime_diagram_development_C.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Spacetime_diagram_development_C.svg/384px-Spacetime_diagram_development_C.svg.png 2x" data-file-width="535" data-file-height="535" /></a></div></div><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:192px;overflow:hidden"><a href="/wiki/File:Spacetime_diagram_development_D.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Spacetime_diagram_development_D.svg/192px-Spacetime_diagram_development_D.svg.png" decoding="async" width="192" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Spacetime_diagram_development_D.svg/288px-Spacetime_diagram_development_D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Spacetime_diagram_development_D.svg/384px-Spacetime_diagram_development_D.svg.png 2x" data-file-width="535" data-file-height="535" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.</div></div></div></div> Spacetime diagrams (<a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagrams</a>) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.<a href="#cite_note-Morin2007-31">[21]</a> To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig. 2-1.<a href="#cite_note-Morin2007-31">[21]</a><a href="#cite_note-Mermin1968-37">[25]</a>: 155–199  Fig. 3-1a. Draw the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> axes of frame S. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> axis is horizontal and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> (actually <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72479bb6f1dc1b592b57dd9fed06d5f50030a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.846ex; height:2.009ex;" alt="{\displaystyle ct}">) axis is vertical, which is the opposite of the usual convention in kinematics. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72479bb6f1dc1b592b57dd9fed06d5f50030a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.846ex; height:2.009ex;" alt="{\displaystyle ct}"> axis is scaled by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9248d91021260015d75d2b7540612616bbb36b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.747ex; height:2.176ex;" alt="{\displaystyle t=0.}"> The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{B}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb2ddddccead55f997b42d41c75bc6d4e525781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle {\text{B}},}"> have been plotted on this graph so that their coordinates may be compared in the S and S' frames. Fig. 3-1b. Draw the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> axes of frame S'. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=c/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=c/2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4730c201ece9c4cdda275cb7a3ae623f7b53cfb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle v=c/2.}"> Both the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"> axes are tilted from the unprimed axes by an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\tan ^{-1}(\beta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\tan ^{-1}(\beta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295c2e4be7b7b821a6a2b161fb994939e278b029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.067ex; height:3.176ex;" alt="{\displaystyle \alpha =\tan ^{-1}(\beta ),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =v/c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =v/c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358f21f49108a1ee981d4a38e963c02c4d28e8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:2.843ex;" alt="{\displaystyle \beta =v/c.}"> The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t'=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t'=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad4572232dcdaadc4a1d2cc438996ec27ca6224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.432ex; height:2.509ex;" alt="{\displaystyle t'=0.}"> Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',ct')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',ct')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3007fde756a005e6784e4d609189b1faf174af9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.389ex; height:3.009ex;" alt="{\displaystyle (x',ct')}"> coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"> in the primed coordinate system transform to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\beta \gamma ,\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>β</mi> <mi>γ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\beta \gamma ,\gamma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c330f35cf90e5c7b710d67814bab8c5bc365f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.7ex; height:2.843ex;" alt="{\displaystyle (\beta \gamma ,\gamma )}"> in the unprimed coordinate system. Likewise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',ct')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',ct')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3007fde756a005e6784e4d609189b1faf174af9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.389ex; height:3.009ex;" alt="{\displaystyle (x',ct')}"> coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53cc1773694affcc1d4d6c2c778d43156a1206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}"> in the primed coordinate system transform to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma ,\beta \gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>β</mi> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma ,\beta \gamma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e574b0e57933bb22498e9d6aba5e559986a5c606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.7ex; height:2.843ex;" alt="{\displaystyle (\gamma ,\beta \gamma )}"> in the unprimed system. Draw gridlines parallel with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> axis through points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\gamma ,k\beta \gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>γ</mi> <mo>,</mo> <mi>k</mi> <mi>β</mi> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k\gamma ,k\beta \gamma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d533b78efc7460c32ca80517ca6c3057e25ade89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.122ex; height:2.843ex;" alt="{\displaystyle (k\gamma ,k\beta \gamma )}"> as measured in the unprimed frame, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is an integer. Likewise, draw gridlines parallel with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"> axis through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\beta \gamma ,k\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>β</mi> <mi>γ</mi> <mo>,</mo> <mi>k</mi> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k\beta \gamma ,k\gamma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d49b28ad7efea06285f6188b8a5ffd4b7634809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.122ex; height:2.843ex;" alt="{\displaystyle (k\beta \gamma ,k\gamma )}"> as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> units equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee2f7395cfed9371f1fad8cf7852583e3f78d04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.893ex; height:3.343ex;" alt="{\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}}"> times the spacing between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72479bb6f1dc1b592b57dd9fed06d5f50030a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.846ex; height:2.009ex;" alt="{\displaystyle ct}"> units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \to 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">→</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \to 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75626015c0357be299fd7300820d9783bee5a3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.755ex; height:2.509ex;" alt="{\displaystyle \beta \to 1.}"> Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t'=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/485223d4a356b062533d406614af224efa771628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.785ex; height:2.509ex;" alt="{\displaystyle t'=0}"> still plot as 45° diagonal lines. The primed coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"> are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space. While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>, but the frames are actually equivalent. <div class="mw-heading mw-heading2"><h2 id="Consequences_derived_from_the_Lorentz_transformation">Consequences derived from the Lorentz transformation</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=12" title="Edit section: Consequences derived from the Lorentz transformation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a> and <a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a></div> The consequences of special relativity can be derived from the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> equations.<a href="#cite_note-38">[26]</a> These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially <a href="/wiki/Counterintuitive" class="mw-redirect" title="Counterintuitive">counterintuitive</a>. <div class="mw-heading mw-heading3"><h3 id="Invariant_interval">Invariant interval</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=13" title="Edit section: Invariant interval">edit</a>]</div> In Galilean relativity, an object's length (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4f36cb1759b5dbc5003f91e305d51f58205200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.985ex; height:2.176ex;" alt="{\displaystyle \Delta r}">⁠)<a href="#cite_note-39">[note 3]</a> and the temporal separation between two events (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}">⁠) are independent invariants, the values of which do not change when observed from different frames of reference.<a href="#cite_note-43">[note 4]</a><a href="#cite_note-47">[note 5]</a> In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d76e6fc5f1353a997222afbf3f50f1e57731a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.081ex; height:2.676ex;" alt="{\displaystyle \Delta s^{2}}">⁠:<a href="#cite_note-48">[note 6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mspace width="thickmathspace" /> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53a989fbfd98b36341494b61fe4d14164d9c4af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.29ex; height:3.843ex;" alt="{\displaystyle \Delta s^{2}\;{\overset {\text{def}}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}"> The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames. The form of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d76e6fc5f1353a997222afbf3f50f1e57731a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.081ex; height:2.676ex;" alt="{\displaystyle \Delta s^{2}}">⁠, being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances.<a href="#cite_note-49">[note 7]</a> The invariance of this interval is a property of the general Lorentz transform (also called the <a href="/wiki/Poincar%C3%A9_transformation" class="mw-redirect" title="Poincaré transformation">Poincaré transformation</a>), making it an <a href="/wiki/Isometry" title="Isometry">isometry</a> of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boosts</a>, in the x-direction) with all other <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>, and <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> between any Cartesian inertial frame.<a href="#cite_note-Rindler1977-50">[30]</a>: 33–34  In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d2843330f7657b08e59cc4420cc96bcbf74368" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.004ex; height:2.843ex;" alt="{\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}}"> Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:<a href="#cite_note-Morin2007-31">[21]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcc8d4f5514d1d42a9f138446511f1acec7a978" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:88.289ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}c^{2}\Delta t^{2}-\Delta x^{2}&=c^{2}\gamma ^{2}\left(\Delta t'+{\dfrac {v\Delta x'}{c^{2}}}\right)^{2}-\gamma ^{2}\ (\Delta x'+v\Delta t')^{2}\\&=\gamma ^{2}\left(c^{2}\Delta t'^{\,2}+2v\Delta x'\Delta t'+{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\right)-\gamma ^{2}\ (\Delta x'^{\,2}+2v\Delta x'\Delta t'+v^{2}\Delta t'^{\,2})\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}-\gamma ^{2}v^{2}\Delta t'^{\,2}-\gamma ^{2}\Delta x'^{\,2}+\gamma ^{2}{\dfrac {v^{2}\Delta x'^{\,2}}{c^{2}}}\\&=\gamma ^{2}c^{2}\Delta t'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)-\gamma ^{2}\Delta x'^{\,2}\left(1-{\dfrac {v^{2}}{c^{2}}}\right)\\&=c^{2}\Delta t'^{\,2}-\Delta x'^{\,2}\end{aligned}}}"> The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d76e6fc5f1353a997222afbf3f50f1e57731a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.081ex; height:2.676ex;" alt="{\displaystyle \Delta s^{2}}"> is hence independent of the frame in which it is measured. In considering the physical significance of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d76e6fc5f1353a997222afbf3f50f1e57731a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.081ex; height:2.676ex;" alt="{\displaystyle \Delta s^{2}}">⁠, there are three cases to note:<a href="#cite_note-Morin2007-31">[21]</a><a href="#cite_note-Taylor1966-51">[31]</a>: 25–39  <ul><li>Δs2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert \Delta x/\Delta t\vert <c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \Delta x/\Delta t\vert <c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b3527a4b04b18b725fa0fb6d08d66dc784a452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.602ex; height:2.843ex;" alt="{\displaystyle \vert \Delta x/\Delta t\vert <c}">⁠, and given the Lorentz transformation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x'=\gamma \ (\Delta x-v\ \Delta t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x'=\gamma \ (\Delta x-v\ \Delta t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe78c2bf9bafe8dd192b53f66859fcc982bff76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.291ex; height:3.009ex;" alt="{\displaystyle \Delta x'=\gamma \ (\Delta x-v\ \Delta t)}">⁠, it is evident that there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x'=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13efc239af4739c8857438b3aebf273a2edd4266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.211ex; height:2.509ex;" alt="{\displaystyle \Delta x'=0}"> (in particular, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\Delta x/\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\Delta x/\Delta t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dced72fdf712754acc3ee84c950f882909bdb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.43ex; height:2.843ex;" alt="{\displaystyle v=\Delta x/\Delta t}">⁠). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s/c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd532bc5b666192d6cac502a6c694a82e4101e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.196ex; height:2.843ex;" alt="{\displaystyle \Delta s/c}">⁠, is called the proper time.</li> <li>Δs2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert \Delta x/\Delta t\vert >c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \Delta x/\Delta t\vert >c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c9ec708a1135ea0d2e023be2d76c0862b4a3f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.602ex; height:2.843ex;" alt="{\displaystyle \vert \Delta x/\Delta t\vert >c}">⁠, and given the Lorentz transformation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>−</mo> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d76202856d3cd6ef93aa1c2cebf6e8dc08ef9c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.444ex; height:3.176ex;" alt="{\displaystyle \Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2})}">⁠, there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79312af2caef81493091b785626154e3b97c172c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.721ex; height:2.509ex;" alt="{\displaystyle \Delta t'=0}"> (in particular, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=c^{2}\Delta t/\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=c^{2}\Delta t/\Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e18d875265386f590cf5016a44358c43590d3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.491ex; height:3.176ex;" alt="{\displaystyle v=c^{2}\Delta t/\Delta x}">⁠). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sqrt {-\Delta s^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {-\Delta s^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d33552c15c81e40910239dd328666fe67023d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.212ex; height:3.509ex;" alt="{\displaystyle \textstyle {\sqrt {-\Delta s^{2}}}}">⁠, is called the proper distance, or proper length. For values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> greater than and less than ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}\Delta t/\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}\Delta t/\Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f9edf99cb4c2d855f683f5d86ce5a21d01930c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.265ex; height:3.176ex;" alt="{\displaystyle c^{2}\Delta t/\Delta x}">⁠, the sign of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ea24905e052383d75e10e87e33a3d805d39b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:2.509ex;" alt="{\displaystyle \Delta t'}"> changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> could be greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}\Delta t/\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}\Delta t/\Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f9edf99cb4c2d855f683f5d86ce5a21d01930c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.265ex; height:3.176ex;" alt="{\displaystyle c^{2}\Delta t/\Delta x}"> would be if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v>c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>></mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v>c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd68cc1d60b0f5aadc893dcc3fae7bb50f0c9eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.233ex; height:1.843ex;" alt="{\displaystyle v>c}">⁠.</li> <li>Δs2 = 0: In this case, the two events are said to be lightlike separated. This implies that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert \Delta x/\Delta t\vert =c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \Delta x/\Delta t\vert =c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db88008a09a68b061c82868c0e666e192a938c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.602ex; height:2.843ex;" alt="{\displaystyle \vert \Delta x/\Delta t\vert =c}">⁠, and this relationship is frame independent due to the invariance of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="{\displaystyle s^{2}}">⁠. From this, we observe that the speed of light is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Relativity_of_simultaneity">Relativity of simultaneity</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=14" title="Edit section: Relativity of simultaneity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a> and <a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relativity_of_Simultaneity_Animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Relativity_of_Simultaneity_Animation.gif/220px-Relativity_of_Simultaneity_Animation.gif" decoding="async" width="220" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Relativity_of_Simultaneity_Animation.gif/330px-Relativity_of_Simultaneity_Animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/7/78/Relativity_of_Simultaneity_Animation.gif 2x" data-file-width="400" data-file-height="430" /></a><figcaption>Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer O. In a reference frame moving at v = 0.3c, as measured by O, the events occur in the order C, B, A. In a reference frame moving at v = −0.5c with respect to O, the events occur in the order A, B, C. The white lines, the lines of simultaneity, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the <a href="/wiki/Light_cone" title="Light cone">light cone</a> with respect to the origin of all considered frames.</figcaption></figure> Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of <a href="/wiki/Absolute_simultaneity" class="mw-redirect" title="Absolute simultaneity">absolute simultaneity</a>). From <a href="#math_3">Equation 3</a> (the forward Lorentz transformation in terms of coordinate differences) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5082c4895196aa92bc26edc9bb606b7669fb1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.862ex; height:6.176ex;" alt="{\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)}"> It is clear that the two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′. The <a href="/wiki/Sagnac_effect" title="Sagnac effect">Sagnac effect</a> can be considered a manifestation of the relativity of simultaneity.<a href="#cite_note-Ashby2003-52">[32]</a> Since relativity of simultaneity is a first order effect in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">⁠,<a href="#cite_note-Morin2007-31">[21]</a> instruments based on the Sagnac effect for their operation, such as <a href="/wiki/Ring_laser_gyroscope" title="Ring laser gyroscope">ring laser gyroscopes</a> and <a href="/wiki/Fiber_optic_gyroscope" class="mw-redirect" title="Fiber optic gyroscope">fiber optic gyroscopes</a>, are capable of extreme levels of sensitivity.<a href="#cite_note-Lin1979-53">[p 14]</a> <div class="mw-heading mw-heading3"><h3 id="Time_dilation">Time dilation</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=15" title="Edit section: Time dilation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></div> The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames. Suppose a <a href="/wiki/Clock" title="Clock">clock</a> is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, <a href="#math_3">Equation 3</a> can be used to find: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=\gamma \,\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=\gamma \,\Delta t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f216b0be7c75fbdddecc96761aa6bb08ec6f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.984ex; height:3.009ex;" alt="{\displaystyle \Delta t'=\gamma \,\Delta t}"> for events satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcaea5f99fa8bf00b7aff6f5a919e63d085a97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.754ex; height:2.176ex;" alt="{\displaystyle \Delta x=0\ .}"></dd></dl> This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed <a href="/wiki/Muon" title="Muon">muons</a> created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.<a href="#cite_note-54">[33]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Observer_in_special_relativity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Observer_in_special_relativity.svg/220px-Observer_in_special_relativity.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Observer_in_special_relativity.svg/330px-Observer_in_special_relativity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Observer_in_special_relativity.svg/440px-Observer_in_special_relativity.svg.png 2x" data-file-width="605" data-file-height="605" /></a><figcaption>Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame</figcaption></figure> Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.<a href="#cite_note-French_1968-55">[34]</a>: 149–152  The measurements that we would get if we actually looked at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a>. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.<a href="#cite_note-French_1968-55">[34]</a>: 149–152  <div class="mw-heading mw-heading4"><h4 id="Langevin's_light-clock">Langevin's light-clock</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=16" title="Edit section: Langevin's light-clock">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Langevin_Light_Clock.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Langevin_Light_Clock.gif/320px-Langevin_Light_Clock.gif" decoding="async" width="320" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Langevin_Light_Clock.gif/480px-Langevin_Light_Clock.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/ae/Langevin_Light_Clock.gif 2x" data-file-width="510" data-file-height="302" /></a><figcaption>Figure 4–3. Thought experiment using a light-clock to explain time dilation</figcaption></figure> <a href="/wiki/Paul_Langevin" title="Paul Langevin">Paul Langevin</a>, an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909<a href="#cite_note-Lewis_Tolman_1909-56">[35]</a>), which he used to perform a novel derivation of the Lorentz transformation.<a href="#cite_note-Cuvaj_1971-57">[36]</a> A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.<a href="#cite_note-58">[37]</a><a href="#cite_note-59">[38]</a><a href="#cite_note-60">[39]</a><a href="#cite_note-Feynman_Lectures_1-61">[40]</a> (Langevin himself made use of a light-clock oriented parallel to its line of motion.<a href="#cite_note-Cuvaj_1971-57">[36]</a>) Consider the scenario illustrated in Fig. 4-3A. Observer A holds a light-clock of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\text{A}}=2L/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{A}}=2L/c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16010dfcc3b3036fb58e341793f713d4916cbcc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.317ex; height:2.843ex;" alt="{\displaystyle t_{\text{A}}=2L/c}">⁠. A time interval measured using a single clock that is motionless in a particular reference frame is called a <a href="/wiki/Proper_time_interval" class="mw-redirect" title="Proper time interval">proper time interval</a>.<a href="#cite_note-Halliday_1988-62">[41]</a> Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">⁠. Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pm {\sqrt {c^{2}-v^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pm {\sqrt {c^{2}-v^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b995584110eb04dc3dfe15c594927952430e514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.862ex; height:3.509ex;" alt="{\textstyle \pm {\sqrt {c^{2}-v^{2}}},}"> so that the total round-trip time of the pulses is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t_{\text{B}}=2L{\big /}{\sqrt {c^{2}-v^{2}}}={}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t_{\text{B}}=2L{\big /}{\sqrt {c^{2}-v^{2}}}={}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a549a9d2b1e23a53419f82beac5c749dc762d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.928ex; height:3.676ex;" alt="{\textstyle t_{\text{B}}=2L{\big /}{\sqrt {c^{2}-v^{2}}}={}}">⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle t_{\text{A}}{\big /}{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle t_{\text{A}}{\big /}{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7219111c066a3702ca1b3cab14deeb58b58e2c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.38ex; height:3.343ex;" alt="{\displaystyle \textstyle t_{\text{A}}{\big /}{\sqrt {1-v^{2}/c^{2}}}}">⁠. Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore not a proper time interval because he did not measure it with a single resting clock.<a href="#cite_note-Halliday_1988-62">[41]</a> <div class="mw-heading mw-heading4"><h4 id="Reciprocal_time_dilation">Reciprocal time dilation</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=17" title="Edit section: Reciprocal time dilation">edit</a>]</div> In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> to the left, in which case observer A would perceive B's clock as running slower than her local clock. There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with both A's and B's measurement of time.<a href="#cite_note-63">[42]</a> <div class="mw-heading mw-heading3"><h3 id="Twin_paradox">Twin paradox</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=18" title="Edit section: Twin paradox">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></div> The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called <a href="/wiki/Twin_paradox" title="Twin paradox">twin paradox</a>, articulated in its present form by Langevin in 1911.<a href="#cite_note-Langevin_1911-64">[43]</a> Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older. This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in order for the two observers to perform side-by-side comparisons of their elapsed proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.<a href="#cite_note-Debs_Redhead-65">[44]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Twin_paradox_Doppler_analysis.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Twin_paradox_Doppler_analysis.svg/220px-Twin_paradox_Doppler_analysis.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Twin_paradox_Doppler_analysis.svg/330px-Twin_paradox_Doppler_analysis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Twin_paradox_Doppler_analysis.svg/440px-Twin_paradox_Doppler_analysis.svg.png 2x" data-file-width="870" data-file-height="840" /></a><figcaption>Figure 4-4. Doppler analysis of twin paradox</figcaption></figure> Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the <a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a> article. We will examine in the following one such solution to the paradox. Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences. Fig 4-4 illustrates a scenario where the traveling twin flies at 0.6 c to and from a star 3 ly distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f'=f{\sqrt {(1-\beta )/(1+\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f'=f{\sqrt {(1-\beta )/(1+\beta )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d956c649f7e2ab654a1c31694f163ebfbaa86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.157ex; height:3.343ex;" alt="{\displaystyle \textstyle f'=f{\sqrt {(1-\beta )/(1+\beta )}}}">⁠. Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f''=f{\sqrt {(1+\beta )/(1-\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo>=</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f''=f{\sqrt {(1+\beta )/(1-\beta )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43967e02e29d320a5070d0f25d94bfbc2374305e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.609ex; height:3.343ex;" alt="{\displaystyle \textstyle f''=f{\sqrt {(1+\beta )/(1-\beta )}}}">⁠. The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.<a href="#cite_note-French_1968-55">[34]</a>: 152–159  <table class="wikitable"> <tbody><tr> <th>Item</th> <th>Measured by the stay-at-home</th> <th>Fig 4-4</th> <th>Measured by the traveler</th> <th>Fig 4-4 </th></tr> <tr> <td>Total time of trip </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {2L}{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {2L}{v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa027081ef36e4c3ba0519ce0e8e4a818c51e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.316ex; height:5.176ex;" alt="{\displaystyle T={\frac {2L}{v}}}"> </td> <td>10 yr </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'={\frac {2L}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'={\frac {2L}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990d9233674c0064ae9c6f122084c4c0487b50b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.085ex; height:5.676ex;" alt="{\displaystyle T'={\frac {2L}{\gamma v}}}"> </td> <td>8 yr </td></tr> <tr> <td>Total number of pulses sent </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fT={\frac {2fL}{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fT={\frac {2fL}{v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e887faf8a4a46e8d218081912de74dccaf6679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.873ex; height:5.343ex;" alt="{\displaystyle fT={\frac {2fL}{v}}}"> </td> <td>10 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fT'={\frac {2fL}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>f</mi> <mi>L</mi> </mrow> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fT'={\frac {2fL}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef54fafd6d4dcb67739439ecc256a7f1b34322ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.642ex; height:5.843ex;" alt="{\displaystyle fT'={\frac {2fL}{\gamma v}}}"> </td> <td>8 </td></tr> <tr> <td>Time when traveler's turnaround is detected </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>v</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f5d79091d28c376cf88936745fab274ab22afa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.671ex; height:5.176ex;" alt="{\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}}"> </td> <td>8 yr </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}'={\frac {L}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}'={\frac {L}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ebb090072e6115de6b0520dce6c0f3a0b437e47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.218ex; height:5.676ex;" alt="{\displaystyle t_{1}'={\frac {L}{\gamma v}}}"> </td> <td>4 yr </td></tr> <tr> <td>Number of pulses received at initial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"> rate </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f't_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f't_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ec22055c2c9f49b2532b60ea49b3bbf90850e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.899ex; height:2.843ex;" alt="{\displaystyle f't_{1}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117525a822c99df5a523361df56417e7ab7962b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.585ex; height:6.676ex;" alt="{\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c5e0c902c4a3a75bb4b4ba8908a75a829ee710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.052ex; height:5.343ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}"> </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f't_{1}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f't_{1}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50bf4dae052d97e4fbd0356bd5f59acfa9d26cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.899ex; height:3.176ex;" alt="{\displaystyle f't_{1}'}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ac03531f28e65d3f5e762fb9ce43a7bd908a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.343ex; height:6.676ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f1ed616f9660f5ea85ff5055826d844239475d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.295ex; height:5.343ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta )}"> </td> <td>2 </td></tr> <tr> <td>Time for remainder of trip </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>v</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69524cbdae74c2e1f9faea3e5d7fe6a4b481405b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.671ex; height:5.176ex;" alt="{\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}}"> </td> <td>2 yr </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}'={\frac {L}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}'={\frac {L}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a133848607084f8024f8b581db7384476cfa84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.218ex; height:5.676ex;" alt="{\displaystyle t_{2}'={\frac {L}{\gamma v}}}"> </td> <td>4 yr </td></tr> <tr> <td>Number of signals received at final <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbdf186092f4353b7630fa8dda903e493cbbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle f''}"> rate </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''t_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5b8514126102f9782992c0b7852019d664c466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.352ex; height:2.843ex;" alt="{\displaystyle f''t_{2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c188b7c14b08f3fc6ef967f9e45e707d8a128e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.585ex; height:6.676ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c5e0c902c4a3a75bb4b4ba8908a75a829ee710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.052ex; height:5.343ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}"> </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''t_{2}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''t_{2}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0996d7cc735a333bf7118cc1450d086677283acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.352ex; height:3.176ex;" alt="{\displaystyle f''t_{2}'}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41edd4caf8a03c886ee83537192d5f5fafaf7500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.343ex; height:6.676ex;" alt="{\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {fL}{v}}(1+\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {fL}{v}}(1+\beta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb1a554ab6c1a173b6ee22370bf87df4d7acbe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.295ex; height:5.343ex;" alt="{\displaystyle ={\frac {fL}{v}}(1+\beta )}"> </td> <td>8 </td></tr> <tr> <td>Total number of received pulses </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d46601c273574f9ed809c7ef755009b46040eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.762ex; height:5.343ex;" alt="{\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {2fL}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>f</mi> <mi>L</mi> </mrow> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {2fL}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a80319eb3a9b346bcdcba858e2061e962f0a94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.313ex; height:5.843ex;" alt="{\displaystyle ={\frac {2fL}{\gamma v}}}"> </td> <td>8 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2fL}{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>f</mi> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2fL}{v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7099a8000e545c5e75448c622abc907b9efe7b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.86ex; height:5.343ex;" alt="{\displaystyle {\frac {2fL}{v}}}"> </td> <td>10 </td></tr> <tr> <td>Twin's calculation as to how much the other twin should have aged </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'={\frac {2L}{\gamma v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mrow> <mi>γ</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'={\frac {2L}{\gamma v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990d9233674c0064ae9c6f122084c4c0487b50b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.085ex; height:5.676ex;" alt="{\displaystyle T'={\frac {2L}{\gamma v}}}"> </td> <td>8 yr </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {2L}{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mi>v</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {2L}{v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa027081ef36e4c3ba0519ce0e8e4a818c51e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.316ex; height:5.176ex;" alt="{\displaystyle T={\frac {2L}{v}}}"> </td> <td>10 yr </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Length_contraction">Length contraction</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=19" title="Edit section: Length contraction">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Lorentz_contraction" class="mw-redirect" title="Lorentz contraction">Lorentz contraction</a></div> The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the <a href="/wiki/Ladder_paradox" title="Ladder paradox">ladder paradox</a> involves a long ladder traveling near the speed of light and being contained within a smaller garage). Similarly, suppose a <a href="/wiki/Measuring_rod" title="Measuring rod">measuring rod</a> is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with <a href="#math_4">Equation 4</a> to find the relation between the lengths Δx and Δx′: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mi>γ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8187d1893fb6b7efb0d681c4b47531b4707341e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.15ex; height:5.843ex;" alt="{\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}}">  for events satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t'=0\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=0\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0826d5835ba3c1661c6d5d607755e074ab01d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.949ex; height:2.509ex;" alt="{\displaystyle \Delta t'=0\ .}"></dd></dl> This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S). Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. <div class="mw-heading mw-heading3"><h3 id="Lorentz_transformation_of_velocities">Lorentz transformation of velocities</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=20" title="Edit section: Lorentz transformation of velocities">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Velocity-addition_formula" title="Velocity-addition formula">Velocity-addition formula</a></div> Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }">⁠. What is its velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">u</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u'} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1517e7ccf782a3519acbc06b4b883a840b4d0459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.509ex;" alt="{\displaystyle \mathbf {u'} }"> in frame S′? We can write <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {|u|} =u=dx/dt\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">|</mo> </mrow> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">|</mo> </mrow> </mrow> <mo>=</mo> <mi>u</mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {|u|} =u=dx/dt\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c72dd41de286498dfe7b26ec75da9d077e75e42" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.293ex; height:2.843ex;" alt="{\displaystyle \mathbf {|u|} =u=dx/dt\,.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(7)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="bold">u</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">|</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6165b6151c53a183922284310dce6f32dcfb65cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.032ex; height:3.009ex;" alt="{\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(8)</td></tr></tbody></table> Substituting expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ddc78263ecbad98bf1f9b504531d0de37d614e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle dx'}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffce1e90f72ed15a6afcd4be6fe28ac13fdc88e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.509ex;" alt="{\displaystyle dt'}"> from <a href="#math_5">Equation 5</a> into <a href="#math_8">Equation 8</a>, followed by straightforward mathematical manipulations and back-substitution from <a href="#math_7">Equation 7</a> yields the Lorentz transformation of the speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee645f272e64333868e2baa275419eca4ee0718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle u'}">⁠: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>u</mi> <mi>v</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bf35c6151c71c5eb19029ce3262c5f28da70c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:54.458ex; height:12.176ex;" alt="{\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(9)</td></tr></tbody></table> The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d12e513906523af26c5372b10aee063aa11926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.936ex; height:2.176ex;" alt="{\displaystyle -v}">⁠. <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd11fda494417d40d6492961920b6ace1ab0f94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.28ex; height:6.343ex;" alt="{\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(10)</td></tr></tbody></table> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> not aligned along the x-axis, we write:<a href="#cite_note-Rindler0-18">[12]</a>: 47–49  <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f893791837973ee622cc0c8a97a3fa9b812bb7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.013ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(11)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">u</mi> <mo>′</mo> </msup> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <msup> <mi>z</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6395f7b466f525c52a6829ce22deaa606ebfded6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.813ex; height:3.176ex;" alt="{\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(12)</td></tr></tbody></table> The forward and inverse transformations for this case are: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f97f9f38602360f5cf95cd61315e79dc36529e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.834ex; height:5.843ex;" alt="{\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(13)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="2em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dbcd650693f03df7d21c552f94f2d4b7f16fef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:71.834ex; height:6.676ex;" alt="{\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(14)</td></tr></tbody></table> <a href="#math_10">Equation 10</a> and <a href="#math_14">Equation 14</a> can be interpreted as giving the resultant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> of the two velocities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">u</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u'} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1517e7ccf782a3519acbc06b4b883a840b4d0459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.509ex;" alt="{\displaystyle \mathbf {u'} }">⁠, and they replace the formula ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u=u'+v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> <mo mathvariant="bold">=</mo> <msup> <mi mathvariant="bold">u</mi> <mo>′</mo> </msup> <mo mathvariant="bold">+</mo> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u=u'+v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9160bf66391ca30a404c05c7cd565fd2f4151fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.544ex; height:2.676ex;" alt="{\displaystyle \mathbf {u=u'+v} }">⁠. which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned with each other (although not necessarily in standard configuration).<a href="#cite_note-Rindler0-18">[12]</a>: 47–49  We note the following points: <ul><li>If an object (e.g., a <a href="/wiki/Photon" title="Photon">photon</a>) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±c), then it would also be moving at the speed of light in any other frame, moving at |v| < c.</li> <li>The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.</li> <li>If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g., |<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠u/c⁠| ≪ 1), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity</li> <li>Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.</li></ul> There is nothing special about the x direction in the standard configuration. The above <a href="/wiki/Formalism_(mathematics)" class="mw-redirect" title="Formalism (mathematics)">formalism</a> applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See <a href="/wiki/Velocity-addition_formula" title="Velocity-addition formula">Velocity-addition formula</a> for details. <div class="mw-heading mw-heading3"><h3 id="Thomas_rotation">Thomas rotation</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=21" title="Edit section: Thomas rotation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Thomas_rotation" class="mw-redirect" title="Thomas rotation">Thomas rotation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:224px;max-width:224px"><div class="trow"><div class="tsingle" style="width:222px;max-width:222px"><div class="thumbimage"><a href="/wiki/File:Thomas-Wigner_Rotation_1.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Thomas-Wigner_Rotation_1.svg/220px-Thomas-Wigner_Rotation_1.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Thomas-Wigner_Rotation_1.svg/330px-Thomas-Wigner_Rotation_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Thomas-Wigner_Rotation_1.svg/440px-Thomas-Wigner_Rotation_1.svg.png 2x" data-file-width="485" data-file-height="213" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:222px;max-width:222px"><div class="thumbimage"><a href="/wiki/File:Thomas-Wigner_Rotation_2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Thomas-Wigner_Rotation_2.svg/220px-Thomas-Wigner_Rotation_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Thomas-Wigner_Rotation_2.svg/330px-Thomas-Wigner_Rotation_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Thomas-Wigner_Rotation_2.svg/440px-Thomas-Wigner_Rotation_2.svg.png 2x" data-file-width="485" data-file-height="213" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Figure 4-5. Thomas–Wigner rotation</div></div></div></div> The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. Thomas rotation results from the relativity of simultaneity. In Fig. 4-5a, a rod of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> in its rest frame (i.e., having a <a href="/wiki/Proper_length" title="Proper length">proper length</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">⁠) rises vertically along the y-axis in the ground frame. In Fig. 4-5b, the same rod is observed from the frame of a rocket moving at speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (not <a href="#Measurement_versus_visual_appearance">see</a>) the clock at the right end of the rod as being advanced in time by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Lv/c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Lv/c^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d977d7a03406747cf1d73b4fc0326f5f5a7fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.934ex; height:3.176ex;" alt="{\displaystyle Lv/c^{2}}">⁠, and the rod is correspondingly observed as tilted.<a href="#cite_note-Taylor1966-51">[31]</a>: 98–99  Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the <a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">spin of moving particles</a>, where <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a> is a relativistic correction that applies to the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> of an elementary particle or the rotation of a macroscopic <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a>, relating the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of the spin of a particle following a <a href="/wiki/Curvilinear" class="mw-redirect" title="Curvilinear">curvilinear</a> orbit to the angular velocity of the orbital motion.<a href="#cite_note-Taylor1966-51">[31]</a>: 169–174  Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".<a href="#cite_note-Shaw-66">[p 15]</a><a href="#cite_note-Taylor1966-51">[31]</a>: 98–99  <div class="mw-heading mw-heading3"><h3 id="Causality_and_prohibition_of_motion_faster_than_light">Causality and prohibition of motion faster than light</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=22" title="Edit section: Causality and prohibition of motion faster than light">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Causality_(physics)" title="Causality (physics)">Causality (physics)</a> and <a href="/wiki/Tachyonic_antitelephone" title="Tachyonic antitelephone">Tachyonic antitelephone</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Simple_light_cone_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Simple_light_cone_diagram.svg/220px-Simple_light_cone_diagram.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Simple_light_cone_diagram.svg/330px-Simple_light_cone_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Simple_light_cone_diagram.svg/440px-Simple_light_cone_diagram.svg.png 2x" data-file-width="306" data-file-height="306" /></a><figcaption>Figure 4–6. <a href="/wiki/Light_cone" title="Light cone">Light cone</a></figcaption></figure> In Fig. 4-6, the time interval between the events A (the "cause") and B (the "effect") is 'timelike'; that is, there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect). The interval AC in the diagram is 'spacelike'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. But no frames are accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result. For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).<a href="#cite_note-67">[45]</a><a href="#cite_note-68">[p 16]</a> A variety of causal paradoxes could then be constructed. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:331px;max-width:331px"><div class="trow"><div class="tsingle" style="margin-right:4px;width:162px;max-width:162px"><div class="thumbimage"><a href="/wiki/File:Causality_violation_1.svg" class="mw-file-description"><img alt="Causality violation: Beginning of scenario resulting from use of a fictitious instantaneous communicator" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Causality_violation_1.svg/160px-Causality_violation_1.svg.png" decoding="async" width="160" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Causality_violation_1.svg/240px-Causality_violation_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Causality_violation_1.svg/320px-Causality_violation_1.svg.png 2x" data-file-width="378" data-file-height="390" /></a></div></div><div class="tsingle" style="width:162px;max-width:162px"><div class="thumbimage"><a href="/wiki/File:Causality_violation_2.svg" class="mw-file-description"><img alt="Causality violation: B receives the message before having sent it." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Causality_violation_2.svg/160px-Causality_violation_2.svg.png" decoding="async" width="160" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Causality_violation_2.svg/240px-Causality_violation_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Causality_violation_2.svg/320px-Causality_violation_2.svg.png 2x" data-file-width="378" data-file-height="390" /></a></div></div></div><div class="trow" style="display:flow-root"><div class="thumbcaption" style="text-align:center">Figure 4-7. Causality violation by the use of fictitious "instantaneous communicators"</div></div></div></div> Consider the spacetime diagrams in Fig. 4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The <a href="/wiki/World_lines" class="mw-redirect" title="World lines">world lines</a> of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ct′), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground. <ol><li>Fig. 4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48546807c94ca901131ea91fb0b2e64646d2d8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.823ex; height:2.676ex;" alt="{\displaystyle -x'}"> axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.</li> <li>Fig. 4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2112b13188089a9983fb5b92fd192b21772d0dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle +x}"> axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality.<a href="#cite_note-Takeuchi-69">[46]</a></li></ol> It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"> axis (and the signal from A to B slightly steeper than the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23969b00bf6d4ac97e6b4058b9af2eb87ee3bf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.509ex;" alt="{\displaystyle ct'}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"> axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only slightly faster than the speed of light will result in causality violation.<a href="#cite_note-Morin2017-70">[47]</a> Therefore, if <a href="/wiki/Causality" title="Causality">causality</a> is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel <a href="/wiki/Faster_than_light" class="mw-redirect" title="Faster than light">faster than light</a> in vacuum. This is not to say that all faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.<a href="#cite_note-71">[48]</a> For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).<a href="#cite_note-72">[49]</a><a href="#cite_note-73">[50]</a> <div class="mw-heading mw-heading2"><h2 id="Optical_effects">Optical effects</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=23" title="Edit section: Optical effects">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Dragging_effects">Dragging effects</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=24" title="Edit section: Dragging effects">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Fizeau_experiment_schematic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Fizeau_experiment_schematic.svg/300px-Fizeau_experiment_schematic.svg.png" decoding="async" width="300" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Fizeau_experiment_schematic.svg/450px-Fizeau_experiment_schematic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Fizeau_experiment_schematic.svg/600px-Fizeau_experiment_schematic.svg.png 2x" data-file-width="815" data-file-height="443" /></a><figcaption>Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.</figcaption></figure> In 1850, <a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Hippolyte Fizeau</a> and <a href="/wiki/L%C3%A9on_Foucault" title="Léon Foucault">Léon Foucault</a> independently established that light travels more slowly in water than in air, thus validating a prediction of <a href="/wiki/Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Fresnel's</a> <a href="/wiki/Wave_theory_of_light" class="mw-redirect" title="Wave theory of light">wave theory of light</a> and invalidating the corresponding prediction of Newton's <a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">corpuscular theory</a>.<a href="#cite_note-Lauginie2004-74">[51]</a> The speed of light was measured in still water. What would be the speed of light in flowing water? In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light. According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed through the medium plus the speed of the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'=c/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u'=c/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b3d46bcc8e549a628d05451b79b566b35b386a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.677ex; height:3.009ex;" alt="{\displaystyle u'=c/n}"> is the speed of light in still water, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> is the speed of the water, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\pm }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\pm }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b35ae55d98b0339fccf145688295eef24812dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle u_{\pm }}"> is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>n</mi> </mfrac> </mrow> <mo>±</mo> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600c359886ff9335f50b778121b56c2c1577af9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.849ex; height:6.176ex;" alt="{\displaystyle u_{\pm }={\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)\ .}"> Fizeau's results, although consistent with Fresnel's earlier hypothesis of <a href="/wiki/Aether_drag_hypothesis" title="Aether drag hypothesis">partial aether dragging</a>, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> depends on wavelength, the aether must be capable of sustaining different motions at the same time.<a href="#cite_note-77">[note 8]</a> A variety of theoretical explanations were proposed to explain <a href="/wiki/Fizeau_experiment#Fresnel_drag_coefficient" title="Fizeau experiment">Fresnel's dragging coefficient</a>, that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.<a href="#cite_note-Stachel2005-78">[52]</a> From the point of view of special relativity, Fizeau's result is nothing but an approximation to <a href="#math_10">Equation 10</a>, the relativistic formula for composition of velocities.<a href="#cite_note-Rindler1977-50">[30]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>±</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>±</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223e20e02edd4e5f50666b06d1845b5309ca45aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.597ex; height:6.343ex;" alt="{\displaystyle u_{\pm }={\frac {u'\pm v}{1\pm u'v/c^{2}}}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c/n\pm v}{1\pm v/cn}}\approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>±</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>±</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c/n\pm v}{1\pm v/cn}}\approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1685d33df0cb48b28b8e4093e0a55cbd03314069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.984ex; height:6.509ex;" alt="{\displaystyle {\frac {c/n\pm v}{1\pm v/cn}}\approx }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mrow> <mi>c</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c6891709b6af36fa9ec88a9a4564eaedbbf7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.707ex; height:6.176ex;" alt="{\displaystyle c\left({\frac {1}{n}}\pm {\frac {v}{c}}\right)\left(1\mp {\frac {v}{cn}}\right)\approx }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>n</mi> </mfrac> </mrow> <mo>±</mo> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489bb82f15c4e859f93249175b315300df290b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.295ex; height:6.176ex;" alt="{\displaystyle {\frac {c}{n}}\pm v\left(1-{\frac {1}{n^{2}}}\right)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relativistic_aberration_of_light">Relativistic aberration of light</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=25" title="Edit section: Relativistic aberration of light">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Aberration_of_light" class="mw-redirect" title="Aberration of light">Aberration of light</a> and <a href="/wiki/Light-time_correction" title="Light-time correction">Light-time correction</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Stellar_aberration_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Stellar_aberration_illustration.svg/220px-Stellar_aberration_illustration.svg.png" decoding="async" width="220" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Stellar_aberration_illustration.svg/330px-Stellar_aberration_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Stellar_aberration_illustration.svg/440px-Stellar_aberration_illustration.svg.png 2x" data-file-width="301" data-file-height="337" /></a><figcaption>Figure 5–2. Illustration of stellar aberration</figcaption></figure> Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the <a href="/wiki/Aberration_of_light" class="mw-redirect" title="Aberration of light">aberration of light</a>. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light.<a href="#cite_note-Mould-79">[53]</a> (2) If the source is in motion, the displacement would be the consequence of <a href="/wiki/Light-time_correction" title="Light-time correction">light-time correction</a>. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver.<a href="#cite_note-Seidelmann-80">[54]</a> The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, <a href="/wiki/Fran%C3%A7ois_Arago" title="François Arago">Arago</a> used this expected phenomenon in a failed attempt to measure the speed of light,<a href="#cite_note-Ferraro-81">[55]</a> and in 1870, <a href="/wiki/George_Airy" class="mw-redirect" title="George Airy">George Airy</a> tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.<a href="#cite_note-Dolan-82">[56]</a> A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,<a href="#cite_note-Hollis-83">[57]</a> but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag.<a href="#cite_note-84">[58]</a> Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include<a href="#cite_note-Rindler1977-50">[30]</a>: 57–60  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841d5e8247bcff5b34ec7fe4199135c5c803cdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.293ex; height:6.509ex;" alt="{\displaystyle \cos \theta '={\frac {\cos \theta +v/c}{1+(v/c)\cos \theta }}}">   OR   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta '={\frac {\sin \theta }{\gamma [1+(v/c)\cos \theta ]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>γ</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta '={\frac {\sin \theta }{\gamma [1+(v/c)\cos \theta ]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b1d959a5aa9840f249611011e804cba5079e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.593ex; height:6.176ex;" alt="{\displaystyle \sin \theta '={\frac {\sin \theta }{\gamma [1+(v/c)\cos \theta ]}}}">   OR   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356f6b461b30d528c5a79ad0ab3d0cdc9329043e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.519ex; height:6.676ex;" alt="{\displaystyle \tan {\frac {\theta '}{2}}=\left({\frac {c-v}{c+v}}\right)^{1/2}\tan {\frac {\theta }{2}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relativistic_Doppler_effect">Relativistic Doppler effect</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=26" title="Edit section: Relativistic Doppler effect">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></div> <div class="mw-heading mw-heading4"><h4 id="Relativistic_longitudinal_Doppler_effect">Relativistic longitudinal Doppler effect</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=27" title="Edit section: Relativistic longitudinal Doppler effect">edit</a>]</div> The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> term, and that is the treatment described here.<a href="#cite_note-85">[59]</a><a href="#cite_note-Gill-86">[60]</a> Assume the receiver and the source are moving away from each other with a relative speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> as measured by an observer on the receiver or the source (The sign convention adopted here is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59644b2d90379fa4e587f199754c62a1e31ba50b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.011ex; height:6.176ex;" alt="{\displaystyle f_{r}=\left(1-{\frac {v}{c_{s}}}\right)f_{s}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b83e050da28fa0d83e2aa786963805742ab756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.01ex; height:2.009ex;" alt="{\displaystyle c_{s}}"> is the speed of sound. For light, and with the receiver moving at relativistic speeds, clocks on the receiver are <a href="/wiki/Time_dilation" title="Time dilation">time dilated</a> relative to clocks at the source. The receiver will measure the received frequency to be <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a61ea004a5731152e17e3b2fee73b3da52c289" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.304ex; height:7.509ex;" alt="{\displaystyle f_{r}=\gamma \left(1-\beta \right)f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}"> where <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =v/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =v/c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c09197ac61cf6c55baab7eaaf25cbde57010efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.727ex; height:2.843ex;" alt="{\displaystyle \beta =v/c}">  and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb91924eaa6e5a593ba98cc7e2dfda04f764f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.915ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}"> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>.</li></ul> An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.<a href="#cite_note-Feynman1977-87">[61]</a><a href="#cite_note-Morin2007-31">[21]</a> <div class="mw-heading mw-heading4"><h4 id="Transverse_Doppler_effect">Transverse Doppler effect</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=28" title="Edit section: Transverse Doppler effect">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Transverse_Doppler_effect_scenarios_5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Transverse_Doppler_effect_scenarios_5.svg/300px-Transverse_Doppler_effect_scenarios_5.svg.png" decoding="async" width="300" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Transverse_Doppler_effect_scenarios_5.svg/450px-Transverse_Doppler_effect_scenarios_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Transverse_Doppler_effect_scenarios_5.svg/600px-Transverse_Doppler_effect_scenarios_5.svg.png 2x" data-file-width="536" data-file-height="240" /></a><figcaption>Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.</figcaption></figure> The transverse <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a> is one of the main novel predictions of the special theory of relativity. Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.<a href="#cite_note-Morin2007-31">[21]</a> In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }">⁠. In Fig. 5-3b, the light is redshifted by the same factor. <div class="mw-heading mw-heading3"><h3 id="Measurement_versus_visual_appearance">Measurement versus visual appearance</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=29" title="Edit section: Measurement versus visual appearance">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Animated_Terrell_Rotation_-_Cube.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Animated_Terrell_Rotation_-_Cube.gif/330px-Animated_Terrell_Rotation_-_Cube.gif" decoding="async" width="330" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Animated_Terrell_Rotation_-_Cube.gif/495px-Animated_Terrell_Rotation_-_Cube.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d4/Animated_Terrell_Rotation_-_Cube.gif 2x" data-file-width="640" data-file-height="315" /></a><figcaption>Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.</figcaption></figure> Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of <a href="/wiki/Doppler_shift" class="mw-redirect" title="Doppler shift">Doppler shift</a>, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer. Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Terrell_Rotation_Sphere.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Terrell_Rotation_Sphere.gif/330px-Terrell_Rotation_Sphere.gif" decoding="async" width="330" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Terrell_Rotation_Sphere.gif/495px-Terrell_Rotation_Sphere.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Terrell_Rotation_Sphere.gif/660px-Terrell_Rotation_Sphere.gif 2x" data-file-width="1080" data-file-height="560" /></a><figcaption>Figure 5–5. Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross.</figcaption></figure> For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation.<a href="#cite_note-Terrell-88">[p 19]</a><a href="#cite_note-Penrose-89">[p 20]</a><a href="#cite_note-90">[62]</a><a href="#cite_note-91">[63]</a> A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although the surface of the sphere and the images on it will appear distorted.<a href="#cite_note-92">[64]</a><a href="#cite_note-Boas_1961-93">[65]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:M87_jet_(1).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/M87_jet_%281%29.jpg/220px-M87_jet_%281%29.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/M87_jet_%281%29.jpg/330px-M87_jet_%281%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/M87_jet_%281%29.jpg/440px-M87_jet_%281%29.jpg 2x" data-file-width="1222" data-file-height="916" /></a><figcaption>Figure 5–6. Galaxy <a href="/wiki/Messier_87" title="Messier 87">M87</a> sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.</figcaption></figure> Both Fig. 5-4 and Fig. 5-5 illustrate objects moving transversely to the line of sight. In Fig. 5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig. 5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen.<a href="#cite_note-Muller_2014-94">[66]</a> This illusion has come to be known as <a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a> or the Terrell–Penrose effect.<a href="#cite_note-95">[note 9]</a> Another example where visual appearance is at odds with measurement comes from the observation of apparent <a href="/wiki/Superluminal_motion" title="Superluminal motion">superluminal motion</a> in various <a href="/wiki/Radio_galaxies" class="mw-redirect" title="Radio galaxies">radio galaxies</a>, <a href="/wiki/BL_Lac_objects" class="mw-redirect" title="BL Lac objects">BL Lac objects</a>, <a href="/wiki/Quasars" class="mw-redirect" title="Quasars">quasars</a>, and other astronomical objects that eject <a href="/wiki/Astrophysical_jet" title="Astrophysical jet">relativistic-speed jets</a> of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.<a href="#cite_note-96">[67]</a><a href="#cite_note-97">[68]</a><a href="#cite_note-98">[69]</a> In Fig. 5-6, galaxy <a href="/wiki/Messier_87" title="Messier 87">M87</a> streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5-4 has been stretched out.<a href="#cite_note-99">[70]</a> <div class="mw-heading mw-heading2"><h2 id="Dynamics">Dynamics</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=30" title="Edit section: Dynamics">edit</a>]</div> Section <a href="#Consequences_derived_from_the_Lorentz_transformation">§ Consequences derived from the Lorentz transformation</a> dealt strictly with <a href="/wiki/Kinematics" title="Kinematics">kinematics</a>, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself. <div class="mw-heading mw-heading3"><h3 id="Equivalence_of_mass_and_energy">Equivalence of mass and energy</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=31" title="Edit section: Equivalence of mass and energy">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></div> As an object's speed approaches the speed of light from an observer's point of view, its <a href="/wiki/Relativistic_mass" class="mw-redirect" title="Relativistic mass">relativistic mass</a> increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference. The energy content of an object at rest with mass m equals mc2. Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies. In addition to the papers referenced above – which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> arguments for the equivalence (and transmutability) of mass and energy, for E = mc2. Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0): it has a time component, which is the energy, and three space components, which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c2, 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c2. The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these do not talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.<a href="#cite_note-electro-1">[p 1]</a> The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.<a href="#cite_note-inertia-100">[p 21]</a> Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.<a href="#cite_note-Jammer-101">[71]</a> Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.<a href="#cite_note-Stachel-102">[72]</a> Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.<a href="#cite_note-survey-103">[p 22]</a><a href="#cite_note-104">[note 10]</a> <div class="mw-heading mw-heading3"><h3 id="Einstein's_1905_demonstration_of_E_=_mc2">Einstein's 1905 demonstration of E = mc2</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=32" title="Edit section: Einstein's 1905 demonstration of E = mc2">edit</a>]</div> In his fourth of his 1905 <a href="/wiki/Annus_mirabilis_papers" title="Annus mirabilis papers">Annus mirabilis papers</a>,<a href="#cite_note-inertia-100">[p 21]</a> Einstein presented a heuristic argument for the equivalence of mass and energy. Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time. Einstein took as starting assumptions his recently discovered formula for <a href="/wiki/Relativistic_Doppler_shift" class="mw-redirect" title="Relativistic Doppler shift">relativistic Doppler shift</a>, the laws of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> and <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>, and the relationship between the frequency of light and its energy as implied by <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:225px;max-width:225px"><div class="trow"><div class="tsingle" style="width:223px;max-width:223px"><div class="thumbimage"><a href="/wiki/File:Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg/221px-Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg.png" decoding="async" width="221" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg/332px-Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg/442px-Einstein%27s_derivation_of_E%3Dmc2_Part_1.svg.png 2x" data-file-width="621" data-file-height="359" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:223px;max-width:223px"><div class="thumbimage"><a href="/wiki/File:Einstein%27s_derivation_of_E%3Dmc2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Einstein%27s_derivation_of_E%3Dmc2.svg/221px-Einstein%27s_derivation_of_E%3Dmc2.svg.png" decoding="async" width="221" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Einstein%27s_derivation_of_E%3Dmc2.svg/332px-Einstein%27s_derivation_of_E%3Dmc2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Einstein%27s_derivation_of_E%3Dmc2.svg/442px-Einstein%27s_derivation_of_E%3Dmc2.svg.png 2x" data-file-width="602" data-file-height="354" /></a></div><div class="thumbcaption">Figure 6-1. Einstein's 1905 derivation of E = mc2</div></div></div></div></div> Fig. 6-1 (top). Consider a system of plane waves of light having frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> traveling in direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> relative to the x-axis of reference frame S. The frequency (and hence energy) of the waves as measured in frame S′ that is moving along the x-axis at velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> is given by the relativistic Doppler shift formula that Einstein had developed in his 1905 paper on special relativity:<a href="#cite_note-electro-1">[p 1]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f'}{f}}={\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>f</mi> <mo>′</mo> </msup> <mi>f</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f'}{f}}={\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cc8eb121cf6ef9256b8ae7f482117f3e279fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.156ex; height:7.009ex;" alt="{\displaystyle {\frac {f'}{f}}={\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}}"></dd></dl> Fig. 6-1 (bottom). Consider an arbitrary body that is stationary in reference frame S. Let this body emit a pair of equal-energy light-pulses in opposite directions at angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> with respect to the x-axis. Each pulse has energy ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7038820602ce88d26af4c7080e13d129b552c75e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle L/2}">⁠. Because of conservation of momentum, the body remains stationary in S after emission of the two pulses. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"> be the energy of the body before emission of the two pulses and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"> after their emission. Next, consider the same system observed from frame S′ that is moving along the x-axis at speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> relative to frame S. In this frame, light from the forwards and reverse pulses will be relativistically Doppler-shifted. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}"> be the energy of the body measured in reference frame S′ before emission of the two pulses and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}"> after their emission. We obtain the following relationships:<a href="#cite_note-inertia-100">[p 21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{0}&=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L\\[5mu]H_{0}&=H_{1}+{\tfrac {1}{2}}L{\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\frac {1+(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}=H_{1}+{\frac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{0}&=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L\\[5mu]H_{0}&=H_{1}+{\tfrac {1}{2}}L{\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\frac {1+(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}=H_{1}+{\frac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343ffb26e1d6fe9149832c6e8b3f2ef79fecad0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:73.908ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}E_{0}&=E_{1}+{\tfrac {1}{2}}L+{\tfrac {1}{2}}L=E_{1}+L\\[5mu]H_{0}&=H_{1}+{\tfrac {1}{2}}L{\frac {1-(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}+{\tfrac {1}{2}}L{\frac {1+(v/c)\cos {\phi }}{\sqrt {1-v^{2}/c^{2}}}}=H_{1}+{\frac {L}{\sqrt {1-v^{2}/c^{2}}}}\end{aligned}}}"></dd></dl> From the above equations, we obtain the following: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fec8300f302a4fcab7a132f24185bb2bd5b1924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.616ex; height:7.509ex;" alt="{\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-1)</td></tr></tbody></table> The two differences of form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H-E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>−</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H-E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffaed429ed8ba5adb9afca43994b7dd6d7cca40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.68ex; height:2.343ex;" alt="{\displaystyle H-E}"> seen in the above equation have a straightforward physical interpretation. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> are the energies of the arbitrary body in the moving and stationary frames, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}-E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}-E_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/125a7afbc64e7125307f1128629b4ca3c0aa10e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.595ex; height:2.509ex;" alt="{\displaystyle H_{0}-E_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}-E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}-E_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427c6c94c535e68c0856e4a2abdfe112341c6fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.595ex; height:2.509ex;" alt="{\displaystyle H_{1}-E_{1}}"> represents the kinetic energies of the bodies before and after the emission of light (except for an additive constant that fixes the zero point of energy and is conventionally set to zero). Hence, <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ce2d1cc99f632810f5db717bcf01ce579f9e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.861ex; height:7.509ex;" alt="{\displaystyle \quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-2)</td></tr></tbody></table> Taking a Taylor series expansion and neglecting higher order terms, he obtained <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3283c693d60ef334f3ef866124b203f16c479045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.716ex; height:5.509ex;" alt="{\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-3)</td></tr></tbody></table> Comparing the above expression with the classical expression for kinetic energy, K.E. = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠mv2, Einstein then noted: "If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2." Rindler has observed that Einstein's heuristic argument suggested merely that energy contributes to mass. In 1905, Einstein's cautious expression of the mass–energy relationship allowed for the possibility that "dormant" mass might exist that would remain behind after all the energy of a body was removed. By 1907, however, Einstein was ready to assert that all inertial mass represented a reserve of energy. "To equate all mass with energy required an act of aesthetic faith, very characteristic of Einstein."<a href="#cite_note-Rindler0-18">[12]</a>: 81–84  Einstein's bold hypothesis has been amply confirmed in the years subsequent to his original proposal. For a variety of reasons, Einstein's original derivation is currently seldom taught. Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.<a href="#cite_note-Fernflores_2018-105">[73]</a> <div class="mw-heading mw-heading3"><h3 id="How_far_can_you_travel_from_the_Earth?">How far can you travel from the Earth?</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=33" title="Edit section: How far can you travel from the Earth?">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Space_travel_under_constant_acceleration" title="Space travel under constant acceleration">Space travel under constant acceleration</a></div> Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~ 100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems that exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">1g</a>, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)={\frac {at}{\sqrt {1+a^{2}t^{2}/c^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)={\frac {at}{\sqrt {1+a^{2}t^{2}/c^{2}}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30dbd1d68869634d015a276a37f8e5379a9242f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.086ex; height:6.509ex;" alt="{\displaystyle v(t)={\frac {at}{\sqrt {1+a^{2}t^{2}/c^{2}}}},}"> where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth.<a href="#cite_note-106">[p 23]</a> Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at v = 0.712 c and 0.946 c after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth. During their journey, people on Earth will experience more time than they do – since their clocks (all physical phenomena) would really be ticking 3.1 times faster than those of the spaceship. A 5-year round trip for the traveller will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for them (5 years accelerating, 5 decelerating, twice each) will land them back on Earth having travelled for 335 Earth years and a distance of 331 light years.<a href="#cite_note-gibbskoks-107">[74]</a> A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1 g will take 148000 years and cover about 140000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.<a href="#cite_note-gibbskoks-107">[74]</a> This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its <a href="/wiki/Half-life" title="Half-life">half-life</a> (when at rest).<a href="#cite_note-108">[75]</a> <div class="mw-heading mw-heading3"><h3 id="Elastic_collisions">Elastic collisions</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=34" title="Edit section: Elastic collisions">edit</a>]</div> Examination of the collision products generated by particle accelerators around the world provides scientists evidence of the structure of the subatomic world and the natural laws governing it. Analysis of the collision products, the sum of whose masses may vastly exceed the masses of the incident particles, requires special relativity.<a href="#cite_note-Idema_2022-109">[76]</a> In Newtonian mechanics, analysis of collisions involves use of the <a href="/wiki/Conservation_of_mass" title="Conservation of mass">conservation laws for mass</a>, <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">momentum</a> and <a href="/wiki/Conservation_of_energy" title="Conservation of energy">energy</a>. In relativistic mechanics, mass is not independently conserved, because it has been subsumed into the total relativistic energy. We illustrate the differences that arise between the Newtonian and relativistic treatments of particle collisions by examining the simple case of two perfectly elastic colliding particles of equal mass. (Inelastic collisions are discussed in <a href="/wiki/Spacetime#Conservation_laws" title="Spacetime">Spacetime#Conservation laws</a>. Radioactive decay may be considered a sort of time-reversed inelastic collision.<a href="#cite_note-Idema_2022-109">[76]</a>) Elastic scattering of charged elementary particles deviates from ideality due to the production of <a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a> radiation.<a href="#cite_note-Nakel_1994-110">[77]</a><a href="#cite_note-111">[78]</a> <div class="mw-heading mw-heading4"><h4 id="Newtonian_analysis">Newtonian analysis</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=35" title="Edit section: Newtonian analysis">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg/220px-Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg.png" decoding="async" width="220" height="303" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg/330px-Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg/440px-Elastic_collision_of_moving_particle_with_equal_mass_stationary_particle.svg.png 2x" data-file-width="496" data-file-height="684" /></a><figcaption>Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle</figcaption></figure> Fig. 6-2 provides a demonstration of the result, familiar to billiard players, that if a stationary ball is struck elastically by another one of the same mass (assuming no sidespin, or "English"), then after collision, the diverging paths of the two balls will subtend a right angle. (a) In the stationary frame, an incident sphere traveling at 2v strikes a stationary sphere. (b) In the center of momentum frame, the two spheres approach each other symmetrically at ±v. After elastic collision, the two spheres rebound from each other with equal and opposite velocities ±u. Energy conservation requires that |u| = |v|. (c) Reverting to the stationary frame, the rebound velocities are v ± u. The dot product (v + u) ⋅ (v − u) = v2 − u2 = 0, indicating that the vectors are orthogonal.<a href="#cite_note-Rindler0-18">[12]</a>: 26–27  <div class="mw-heading mw-heading4"><h4 id="Relativistic_analysis">Relativistic analysis</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=36" title="Edit section: Relativistic analysis">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relativistic_elastic_collision_of_equal_mass_particles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Relativistic_elastic_collision_of_equal_mass_particles.svg/220px-Relativistic_elastic_collision_of_equal_mass_particles.svg.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Relativistic_elastic_collision_of_equal_mass_particles.svg/330px-Relativistic_elastic_collision_of_equal_mass_particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Relativistic_elastic_collision_of_equal_mass_particles.svg/440px-Relativistic_elastic_collision_of_equal_mass_particles.svg.png 2x" data-file-width="709" data-file-height="337" /></a><figcaption>Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle</figcaption></figure> Consider the elastic collision scenario in Fig. 6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light: The relativistic momentum and total relativistic energy of a particle are given by <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>E</mi> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fc1824e008290d09f0eedf852c22b9a327394e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.177ex; height:3.176ex;" alt="{\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-4)</td></tr></tbody></table> Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2cb144607317b2c2e3ec806cfbb155f466e56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.455ex; height:3.509ex;" alt="{\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-5)</td></tr></tbody></table> Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc2) equals the sum of the total energies of the emergent particles: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d26a8718dbddd1cde0caf91839d74f0f8b1a446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.606ex; height:3.176ex;" alt="{\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-6)</td></tr></tbody></table> Breaking down (<a href="#math_6-5">6-5</a>) into its components, replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> with the dimensionless ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }">⁠, and factoring out common terms from (<a href="#math_6-5">6-5</a>) and (<a href="#math_6-6">6-6</a>) yields the following:<a href="#cite_note-Champion_1932-112">[p 24]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0954390ac1934bcabb52b221a75b542a1e9cab11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.716ex; height:2.676ex;" alt="{\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-7)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cea69e193495c1c20e5f2c3238ea2571c91c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.736ex; height:2.676ex;" alt="{\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-8)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae3b647d7cc30437fd30c440ee3fb4e94834503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.362ex; height:2.676ex;" alt="{\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-9)</td></tr></tbody></table> From these we obtain the following relationships:<a href="#cite_note-Champion_1932-112">[p 24]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e6db5a9c294cec068283e91d8b67a9a0dc4d82f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.806ex; height:6.509ex;" alt="{\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-10)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239260685a782b104b3cdb3211b2bfaa12c35506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.511ex; height:6.509ex;" alt="{\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-11)</td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd820b1054d2c3aa0a333311be4bef11faa016d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.159ex; height:6.676ex;" alt="{\displaystyle \quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-12)</td></tr></tbody></table> For the symmetrical case in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97060ee8346b6526b71ad2d4d3eeb922633e2d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.574ex; height:2.509ex;" alt="{\displaystyle \phi =\theta }"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}=\beta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}=\beta _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b997bdc2be2de75f7ba6fa02038b31c3ac1bbd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.838ex; height:2.509ex;" alt="{\displaystyle \beta _{2}=\beta _{3}}">⁠, (<a href="#math_6-12">6-12</a>) takes on the simpler form:<a href="#cite_note-Champion_1932-112">[p 24]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05850f06888f7043811c3ace47e8e6fca12c5222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.172ex; height:6.343ex;" alt="{\displaystyle \quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6-13)</td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Beyond_the_basics">Beyond the basics</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=37" title="Edit section: Beyond the basics">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Rapidity">Rapidity</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=38" title="Edit section: Rapidity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:333px;max-width:333px"><div class="trow"><div class="tsingle" style="width:137px;max-width:137px"><div class="thumbimage"><a href="/wiki/File:Trig_functions_(sine_and_cosine).svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trig_functions_%28sine_and_cosine%29.svg/135px-Trig_functions_%28sine_and_cosine%29.svg.png" decoding="async" width="135" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trig_functions_%28sine_and_cosine%29.svg/203px-Trig_functions_%28sine_and_cosine%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Trig_functions_%28sine_and_cosine%29.svg/270px-Trig_functions_%28sine_and_cosine%29.svg.png 2x" data-file-width="465" data-file-height="370" /></a></div><div class="thumbcaption">Figure 7-1a. A ray through the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> x2 + y2 = 1 in the point (cos a, sin a), where a is twice the area between the ray, the circle, and the x-axis.</div></div><div class="tsingle" style="width:192px;max-width:192px"><div class="thumbimage"><a href="/wiki/File:Hyperbolic_functions-2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/190px-Hyperbolic_functions-2.svg.png" decoding="async" width="190" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/285px-Hyperbolic_functions-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/380px-Hyperbolic_functions-2.svg.png 2x" data-file-width="500" data-file-height="437" /></a></div><div class="thumbcaption">Figure 7-1b. A ray through the <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a> x2 − y2 = 1 in the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis.</div></div></div></div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Sinh%2Bcosh%2Btanh.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sinh%2Bcosh%2Btanh.svg/180px-Sinh%2Bcosh%2Btanh.svg.png" decoding="async" width="180" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sinh%2Bcosh%2Btanh.svg/270px-Sinh%2Bcosh%2Btanh.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Sinh%2Bcosh%2Btanh.svg/360px-Sinh%2Bcosh%2Btanh.svg.png 2x" data-file-width="400" data-file-height="500" /></a><figcaption>Figure 7–2. Plot of the three basic <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">Hyperbolic functions</a>: hyperbolic sine (<a href="/wiki/File:Hyperbolic_Sine.svg" title="File:Hyperbolic Sine.svg">sinh</a>), hyperbolic cosine (<a href="/wiki/File:Hyperbolic_Cosine.svg" title="File:Hyperbolic Cosine.svg">cosh</a>) and hyperbolic tangent (<a href="/wiki/File:Hyperbolic_Tangent.svg" title="File:Hyperbolic Tangent.svg">tanh</a>). Sinh is red, cosh is blue and tanh is green.</figcaption></figure> Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters.<a href="#cite_note-Taylor_1992-17">[11]</a>: 47–59  We have previously noted that in an x–ct spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other. The natural functions for expressing these relationships are the <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic analogs of the trigonometric functions</a>. Fig. 7-1a shows a <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the x-axis, but as twice the area of the sector swept out by the ray from the x-axis. Numerically, the angle and 2 × area measures for the unit circle are identical. Fig. 7-1b shows a <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a> with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.<a href="#cite_note-113">[79]</a> Fig. 7-2 presents plots of the sinh, cosh, and tanh functions. For the unit circle, the slope of the ray is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{slope}}=\tan a={\frac {\sin a}{\cos a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>slope</mtext> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>a</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{slope}}=\tan a={\frac {\sin a}{\cos a}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f6609bda2483876ea13240bbbe2561142409a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.435ex; height:5.176ex;" alt="{\displaystyle {\text{slope}}=\tan a={\frac {\sin a}{\cos a}}.}"></dd></dl> In the Cartesian plane, rotation of point (x, y) into point (x', y') by angle θ is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x'\\y'\\\end{pmatrix}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x'\\y'\\\end{pmatrix}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfef0d0bb123d93a39b080b594edda14494e39ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.047ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x'\\y'\\\end{pmatrix}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}.}"></dd></dl> In a spacetime diagram, the velocity parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> is the analog of slope. The rapidity, φ, is defined by<a href="#cite_note-Morin2007-31">[21]</a>: 96–99  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \equiv \tanh \phi \equiv {\frac {v}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≡</mo> <mi>tanh</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \equiv \tanh \phi \equiv {\frac {v}{c}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf4d43a1585b97e5e65068d70c5075552c646da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.564ex; height:4.676ex;" alt="{\displaystyle \beta \equiv \tanh \phi \equiv {\frac {v}{c}},}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh \phi ={\frac {\sinh \phi }{\cosh \phi }}={\frac {e^{\phi }-e^{-\phi }}{e^{\phi }+e^{-\phi }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ϕ</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ϕ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh \phi ={\frac {\sinh \phi }{\cosh \phi }}={\frac {e^{\phi }-e^{-\phi }}{e^{\phi }+e^{-\phi }}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7417028559d0a363e089049293018665cb25b363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.827ex; height:6.176ex;" alt="{\displaystyle \tanh \phi ={\frac {\sinh \phi }{\cosh \phi }}={\frac {e^{\phi }-e^{-\phi }}{e^{\phi }+e^{-\phi }}}.}"></dd></dl> The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;<a href="#cite_note-Taylor_1992-17">[11]</a>: 47–59  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e1d04c777c76f770f7d7630394a32985568a7f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.463ex; height:5.843ex;" alt="{\displaystyle \beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tanh</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>tanh</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>tanh</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>tanh</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b472e444bcc4c8e7731cff0a78ea05154d29bb85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.637ex; height:5.843ex;" alt="{\displaystyle {\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh(\phi _{1}+\phi _{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh(\phi _{1}+\phi _{2}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d64ad79bebdd63cbc635ef9191d648247f579b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.828ex; height:2.843ex;" alt="{\displaystyle \tanh(\phi _{1}+\phi _{2}),}"></dd></dl> or in other words, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\phi _{1}+\phi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\phi _{1}+\phi _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48df15a50be29636192ed28865856453c5d2470c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.204ex; height:2.509ex;" alt="{\displaystyle \phi =\phi _{1}+\phi _{2}}">⁠. The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e7fe3704ea5200e804296d23aff6dde0c16d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:31.655ex; height:8.009ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\cosh \phi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\cosh \phi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581f147045df45196174212022b899d3184a1eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.276ex; height:2.509ex;" alt="{\displaystyle =\cosh \phi ,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tanh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d576400646418d0d23493b1c673b4eb077533cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:32.987ex; height:8.176ex;" alt="{\displaystyle \gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sinh \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sinh \phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f028a8659c371941df6ae7f68c3fdc43efa249b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.021ex; height:2.509ex;" alt="{\displaystyle =\sinh \phi .}"></dd></dl> Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts. Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the x-direction may be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mo>−</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f219d2e3bdb6c0091c27f73a22bb8e8e646187e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.195ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}},}"></dd></dl> and the inverse Lorentz boost in the x-direction may be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}ct\\x\end{pmatrix}}={\begin{pmatrix}\cosh \phi &\sinh \phi \\\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct'\\x'\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}ct\\x\end{pmatrix}}={\begin{pmatrix}\cosh \phi &\sinh \phi \\\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct'\\x'\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e567efa9fbd722ca41aec53f7addb90af802bc0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.316ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}ct\\x\end{pmatrix}}={\begin{pmatrix}\cosh \phi &\sinh \phi \\\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct'\\x'\end{pmatrix}}.}"></dd></dl> In other words, Lorentz boosts represent <a href="/wiki/Hyperbolic_rotation" class="mw-redirect" title="Hyperbolic rotation">hyperbolic rotations</a> in Minkowski spacetime.<a href="#cite_note-Morin2007-31">[21]</a>: 96–99  The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.<a href="#cite_note-Taylor_1992-17">[11]</a><a href="#cite_note-114">[note 11]</a> <div class="mw-heading mw-heading3"><h3 id="4‑vectors">4‑vectors</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=39" title="Edit section: 4‑vectors">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Four-vector" title="Four-vector">Four-vector</a></div> Four‑vectors have been mentioned above in context of the energy–momentum 4‑vector, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, 4‑vectors, and more generally <a href="/wiki/Tensors" class="mw-redirect" title="Tensors">tensors</a>, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the <a href="/wiki/Field_strength_tensor" class="mw-redirect" title="Field strength tensor">field strength tensor</a> formulation.<a href="#cite_note-Post_1962-115">[80]</a> On the other hand, general relativity, from the outset, relies heavily on 4‑vectors, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such 4‑vectors even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime. <div class="mw-heading mw-heading4"><h4 id="Definition_of_4-vectors">Definition of 4-vectors</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=40" title="Edit section: Definition of 4-vectors">edit</a>]</div> A 4-tuple, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6977eff100ca73c0428f61c0687393d1c3a2941a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.942ex; height:2.843ex;" alt="{\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)}">⁠ is a "4-vector" if its component Ai transform between frames according to the Lorentz transformation. If using ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ct,x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ct,x,y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6eac3c6402a2569f6196eae23de908484737e5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.331ex; height:2.843ex;" alt="{\displaystyle (ct,x,y,z)}">⁠ coordinates, A is a 4–vector if it transforms (in the x-direction) according to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A_{0}'&=\gamma \left(A_{0}-(v/c)A_{1}\right)\\A_{1}'&=\gamma \left(A_{1}-(v/c)A_{0}\right)\\A_{2}'&=A_{2}\\A_{3}'&=A_{3}\end{aligned}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{0}'&=\gamma \left(A_{0}-(v/c)A_{1}\right)\\A_{1}'&=\gamma \left(A_{1}-(v/c)A_{0}\right)\\A_{2}'&=A_{2}\\A_{3}'&=A_{3}\end{aligned}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37fcc22c3fefc43d265f24264191d03df0b3fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.646ex; margin-bottom: -0.192ex; width:24.294ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}A_{0}'&=\gamma \left(A_{0}-(v/c)A_{1}\right)\\A_{1}'&=\gamma \left(A_{1}-(v/c)A_{0}\right)\\A_{2}'&=A_{2}\\A_{3}'&=A_{3}\end{aligned}},}"></dd></dl> which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the <a href="#Lorentz_transformations">Lorentz transformation.</a> As usual, when we write x, t, etc. we generally mean Δx, Δt etc. The last three components of a 4–vector must be a standard vector in three-dimensional space. Therefore, a 4–vector must transform like ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c\Delta t,\Delta x,\Delta y,\Delta z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c\Delta t,\Delta x,\Delta y,\Delta z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8632412b69ddb9ff27f0e7b67d9a4da7c59e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.074ex; height:2.843ex;" alt="{\displaystyle (c\Delta t,\Delta x,\Delta y,\Delta z)}">⁠ under Lorentz transformations as well as rotations.<a href="#cite_note-Schutz1985-116">[81]</a>: 36–59  <div class="mw-heading mw-heading4"><h4 id="Properties_of_4-vectors">Properties of 4-vectors</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=41" title="Edit section: Properties of 4-vectors">edit</a>]</div> <ul><li>Closure under linear combination: If A and B are 4-vectors, then ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=aA+aB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>a</mi> <mi>A</mi> <mo>+</mo> <mi>a</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=aA+aB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081a920cedf371e2b6e8e0ff292c0939394aaec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.672ex; height:2.343ex;" alt="{\displaystyle C=aA+aB}">⁠ is also a 4-vector.</li> <li>Inner-product invariance: If A and B are 4-vectors, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 3-vector. In the following, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"> are 3-vectors: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot B\equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B\equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714623f18dc4d4dfdb22fe50222807ecaef8d0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle A\cdot B\equiv }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63734a2b20fd3493a10e2ee165f8f511c5ee40e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.437ex; height:2.509ex;" alt="{\displaystyle A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ee238b44ba58c2d6beea16d2bb14c984699bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.642ex; height:3.343ex;" alt="{\displaystyle A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}}"></dd></dl></li></ul> <dl><dd>In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in 3-space.</dd> <dd>Two vectors are said to be orthogonal if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot B=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5efc0cb2fe9ce4180033bd44b4514b0582e7745c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.447ex; height:2.176ex;" alt="{\displaystyle A\cdot B=0}">⁠. Unlike the case with 3-vectors, orthogonal 4-vectors are not necessarily at right angles to each other. The rule is that two 4-vectors are orthogonal if they are offset by equal and opposite angles from the 45° line, which is the world line of a light ray. This implies that a lightlike 4-vector is orthogonal to itself.</dd></dl> <ul><li>Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a 4-vector with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot A=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot A=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df9f840dad7c9eab62aa2e1e701a6eb4ac01bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.426ex; height:2.176ex;" alt="{\displaystyle A\cdot A=0}">⁠, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}t^{2}-x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}t^{2}-x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1544614c40e69f73aa27510e20ccff546b7295ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.179ex; height:2.843ex;" alt="{\displaystyle c^{2}t^{2}-x^{2}}"> and the invariant length of the relativistic momentum vector ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}-p^{2}c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}-p^{2}c^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5edae8db2fa783bae8dcd1a9dbaaae868effca45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.973ex; height:3.009ex;" alt="{\displaystyle E^{2}-p^{2}c^{2}}">⁠.<a href="#cite_note-Morin2007-31">[21]</a>: 178–181 <a href="#cite_note-Schutz1985-116">[81]</a>: 36–59 </li></ul> <div class="mw-heading mw-heading4"><h4 id="Examples_of_4-vectors">Examples of 4-vectors</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=42" title="Edit section: Examples of 4-vectors">edit</a>]</div> <ul><li>Displacement 4-vector: Otherwise known as the spacetime separation, this is (Δt, Δx, Δy, Δz), or for infinitesimal separations, (dt, dx, dy, dz). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dS\equiv (dt,dx,dy,dz)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>≡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS\equiv (dt,dx,dy,dz)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b3a79e09e4f53baf6731e650bd2d58d872ccb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.001ex; height:2.843ex;" alt="{\displaystyle dS\equiv (dt,dx,dy,dz)}"></dd></dl></li> <li>Velocity 4-vector: This results when the displacement 4-vector is divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00568785317ea373b90759c05c67d795b57b3194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.176ex;" alt="{\displaystyle d\tau }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00568785317ea373b90759c05c67d795b57b3194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.176ex;" alt="{\displaystyle d\tau }"> is the proper time between the two events that yield dt, dx, dy, and dz. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mi>t</mi> <mo>,</mo> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ced520060ffa46a43ff5e545c21bc2e343e488a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.012ex; height:6.509ex;" alt="{\displaystyle V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3faeae5799fe72ffb1a83426b6f4f793b9a02026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.517ex; height:6.176ex;" alt="{\displaystyle \gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma ,\gamma {\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma ,\gamma {\vec {v}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f508c32712542569af0975918323d4bb95d8b221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle (\gamma ,\gamma {\vec {v}})}"></dd></dl></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:308px;max-width:308px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><a href="/wiki/File:Momentarily_Comoving_Reference_Frame.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/150px-Momentarily_Comoving_Reference_Frame.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/225px-Momentarily_Comoving_Reference_Frame.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/300px-Momentarily_Comoving_Reference_Frame.gif 2x" data-file-width="501" data-file-height="501" /></a></div><div class="thumbcaption">Figure 7-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><a href="/wiki/File:Lorentz_transform_of_world_line.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Lorentz_transform_of_world_line.gif/150px-Lorentz_transform_of_world_line.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e4/Lorentz_transform_of_world_line.gif 1.5x" data-file-width="200" data-file-height="200" /></a></div><div class="thumbcaption">Figure 7-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).</div></div></div></div></div> <dl><dd>The 4-velocity is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.</dd> <dd>An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found that is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles.</dd> <dd>Since photons move on null lines, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e31de179716bd64a2eebbc949157606a284867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.679ex; height:2.176ex;" alt="{\displaystyle d\tau =0}"> for a photon, and a 4-velocity cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.</dd></dl> <ul><li>Energy–momentum 4-vector: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea11788251b3afe169700725c8da2a385069b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.643ex; height:3.009ex;" alt="{\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})}"></dd></dl></li></ul> <dl><dd>As indicated before, there are varying treatments for the energy–momentum 4-vector so that one may also see it expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\vec {p}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,{\vec {p}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1445539d4823c2a52fdef0ee636fc353828634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.944ex; height:2.843ex;" alt="{\displaystyle (E,{\vec {p}})}"> or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\vec {p}}c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,{\vec {p}}c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154187fb8caab04245c2594ebba8c99407056fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.951ex; height:2.843ex;" alt="{\displaystyle (E,{\vec {p}}c)}">⁠. The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy–momentum 4-vector is a conserved quantity.</dd></dl> <ul><li>Acceleration 4-vector: This results from taking the derivative of the velocity 4-vector with respect to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">⁠. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\equiv {\frac {dV}{d\tau }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv {\frac {dV}{d\tau }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6190373cbee3695a4a1b9b2d2e4287ad1a6619a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.134ex; height:5.509ex;" alt="{\displaystyle A\equiv {\frac {dV}{d\tau }}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dbdcaa5e12f5b1aa1d610f16633ed36b1dce00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.25ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>γ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03dcac87d0f6bfec3985bde98ae636462d9829a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.718ex; height:6.343ex;" alt="{\displaystyle \gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)}"></dd></dl></li> <li>Force 4-vector: This is the derivative of the momentum 4-vector with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871bb01391136d3551c8ea59059e106be2a403cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\equiv {\frac {dP}{d\tau }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\equiv {\frac {dP}{d\tau }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7a8df77288382a82487f11e4e3487053fbd040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.09ex; height:5.509ex;" alt="{\displaystyle F\equiv {\frac {dP}{d\tau }}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b5a1b00f67805678a7839394cb8b87669f16e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.763ex; height:6.176ex;" alt="{\displaystyle \gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \left({\frac {dE}{dt}},{\vec {f}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \left({\frac {dE}{dt}},{\vec {f}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8cb90349202634bbc78bdea3e4ea8c8a6bc4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.597ex; height:6.176ex;" alt="{\displaystyle \gamma \left({\frac {dE}{dt}},{\vec {f}}\right)}"></dd></dl></li></ul> As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.<a href="#cite_note-Morin2007-31">[21]</a>: 178–181 <a href="#cite_note-Schutz1985-116">[81]</a>: 36–59  <div class="mw-heading mw-heading4"><h4 id="4-vectors_and_physical_law">4-vectors and physical law</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=43" title="Edit section: 4-vectors and physical law">edit</a>]</div> The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum. Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving 4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from 4-vectors.<a href="#cite_note-Morin2007-31">[21]</a>: 186  <div class="mw-heading mw-heading3"><h3 id="Acceleration">Acceleration</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=44" title="Edit section: Acceleration">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Acceleration_(special_relativity)" title="Acceleration (special relativity)">Acceleration (special relativity)</a></div> It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.<a href="#cite_note-PhysicsFAQ-117">[82]</a> Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.<a href="#cite_note-PhysicsFAQ-117">[82]</a> In this section, we analyze several scenarios involving accelerated reference frames. <div class="mw-heading mw-heading4"><h4 id="Dewan–Beran–Bell_spaceship_paradox">Dewan–Beran–Bell spaceship paradox</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=45" title="Edit section: Dewan–Beran–Bell spaceship paradox">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a></div> The Dewan–Beran–Bell spaceship paradox (<a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a>) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png/220px-Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png" decoding="async" width="220" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png/330px-Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png/440px-Bell%27s_spaceship_paradox_-_two_spaceships_-_initial_setup.png 2x" data-file-width="956" data-file-height="308" /></a><figcaption>Figure 7–4. Dewan–Beran–Bell spaceship paradox</figcaption></figure> In Fig. 7-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string that is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.<a href="#cite_note-118">[note 12]</a> Will the string break? When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.<a href="#cite_note-Morin2007-31">[21]</a>: 106, 120–122  <ol><li>To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance L' = γL in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.</li> <li>Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.<a href="#cite_note-Morin2007-31">[21]</a>: 106, 120–122 </li></ol> The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.<a href="#cite_note-Morin2007-31">[21]</a>: 106, 120–122  <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bell_spaceship_paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Bell_spaceship_paradox.svg/220px-Bell_spaceship_paradox.svg.png" decoding="async" width="220" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Bell_spaceship_paradox.svg/330px-Bell_spaceship_paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Bell_spaceship_paradox.svg/440px-Bell_spaceship_paradox.svg.png 2x" data-file-width="641" data-file-height="684" /></a><figcaption>Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.</figcaption></figure> A spacetime diagram (Fig. 7-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> acceleration for proper time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'B''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'B''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3479b3a2b0bffc34034797643226936b0db651da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.329ex; height:2.509ex;" alt="{\displaystyle A'B''}"> turns out to be greater than the length along the line of simultaneity ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}">⁠. The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 7-5, the acceleration is finished, the ships will remain at a constant offset in some frame ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}">⁠. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131920dc49fade5cd528c48af190e33e3c7e0a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.795ex; height:2.009ex;" alt="{\displaystyle x_{A}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{B}=x_{A}+L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{B}=x_{A}+L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ca40362eb910abee23b6229bb4fe4038238f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.125ex; height:2.509ex;" alt="{\displaystyle x_{B}=x_{A}+L}"> are the ships' positions in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">⁠, the positions in frame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"> are:<a href="#cite_note-Franklin-119">[83]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}=\gamma L\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mi>L</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>L</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi>γ</mi> <mi>L</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}=\gamma L\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/496a005e5342d2fd9da69a82534aed90081ba1b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.083ex; margin-bottom: -0.255ex; width:22.143ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}=\gamma L\end{aligned}}}"></dd></dl> The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">⁠. As shown in Fig. 7-5, Bell's example asserts the moving lengths <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'B'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab528dd0aeadceb425686e73e3c71f2de09cf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.877ex; height:2.509ex;" alt="{\displaystyle A'B'}"> measured in frame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> to be fixed, thereby forcing the rest frame length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'B''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'B''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3479b3a2b0bffc34034797643226936b0db651da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.329ex; height:2.509ex;" alt="{\displaystyle A'B''}"> in frame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"> to increase. <div class="mw-heading mw-heading4"><h4 id="Accelerated_observer_with_horizon">Accelerated observer with horizon</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=46" title="Edit section: Accelerated observer with horizon">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Event_horizon#Apparent_horizon_of_an_accelerated_particle" title="Event horizon">Event horizon § Apparent horizon of an accelerated particle</a>, and <a href="/wiki/Rindler_coordinates" title="Rindler coordinates">Rindler coordinates</a></div> Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as <a href="/wiki/Event_horizons" class="mw-redirect" title="Event horizons">event horizons</a>. In the text accompanying <a href="/wiki/Spacetime#Invariant_hyperbola" title="Spacetime">Section "Invariant hyperbola" of the article Spacetime</a>, the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Accelerated_relativistic_observer_with_horizon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Accelerated_relativistic_observer_with_horizon.png/220px-Accelerated_relativistic_observer_with_horizon.png" decoding="async" width="220" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Accelerated_relativistic_observer_with_horizon.png/330px-Accelerated_relativistic_observer_with_horizon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Accelerated_relativistic_observer_with_horizon.png/440px-Accelerated_relativistic_observer_with_horizon.png 2x" data-file-width="640" data-file-height="650" /></a><figcaption>Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed <a href="/wiki/File:ConstantAcceleration02.jpg" title="File:ConstantAcceleration02.jpg">here</a>. </figcaption></figure> Fig. 7-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> approaches a limit of one as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72479bb6f1dc1b592b57dd9fed06d5f50030a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.846ex; height:2.009ex;" alt="{\displaystyle ct}"> increases. Likewise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> approaches infinity. The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows: <ol><li>We remember that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =ct/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =ct/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9028066e4aaa8e00264f13231010cb7e07742d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.769ex; height:2.843ex;" alt="{\displaystyle \beta =ct/x}">⁠.</li> <li>Since ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}t^{2}-x^{2}=s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}t^{2}-x^{2}=s^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f371a4dc1d1a8c82fff3a5de6dc0751d525b8e8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.422ex; height:2.843ex;" alt="{\displaystyle c^{2}t^{2}-x^{2}=s^{2}}">⁠, we conclude that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63d7c189ed809ffaf894f7ade5eff84b9183344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.359ex; height:3.509ex;" alt="{\displaystyle \beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}}">⁠.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1/{\sqrt {1-\beta ^{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1/{\sqrt {1-\beta ^{2}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/441cf3700702a16e7ecbf12fb692635c9eb2790d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:17.857ex; height:4.843ex;" alt="{\displaystyle \gamma =1/{\sqrt {1-\beta ^{2}}}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {c^{2}t^{2}-s^{2}}}/s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {c^{2}t^{2}-s^{2}}}/s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9d06b9f142f43072a75083274073b7fe5a690c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.517ex; height:3.509ex;" alt="{\displaystyle {\sqrt {c^{2}t^{2}-s^{2}}}/s}"></li> <li>From the relativistic force law, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=dp/dt=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>d</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=dp/dt=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678f7e4d2661575eedd51a210a7dfe0bce46a7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.896ex; height:2.843ex;" alt="{\displaystyle F=dp/dt=}">⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mi>γ</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c7ff89052b20b434248374a5bd58f714a1e31f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.28ex; height:3.176ex;" alt="{\displaystyle dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct)}">⁠.</li> <li>Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (ct)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (ct)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61835779ef98de2a8a6e7b236da575b96d573a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.988ex; height:2.843ex;" alt="{\displaystyle \beta (ct)}"> from step 2 and the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> from step 3 yields ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=mc^{2}/s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=mc^{2}/s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b990106c0e01a704134bd43b78a2d48a669c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.194ex; height:3.176ex;" alt="{\displaystyle F=mc^{2}/s}">⁠, which is a constant expression.<a href="#cite_note-Bais-120">[84]</a>: 110–113 </li></ol> Fig. 7-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after 100 hours on his clock (dashed green lines).<a href="#cite_note-Bais-120">[84]</a>: 110–113  After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to receive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an apparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.<a href="#cite_note-Bais-120">[84]</a>: 110–113  <div class="mw-heading mw-heading2"><h2 id="Relativity_and_unifying_electromagnetism">Relativity and unifying electromagnetism</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=47" title="Edit section: Relativity and unifying electromagnetism">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Classical electromagnetism and special relativity</a> and <a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation of classical electromagnetism</a></div> Theoretical investigation in <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical electromagnetism</a> led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the <a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a>, which is a step towards special relativity. The Lorentz transformation of the <a href="/wiki/Electric_field" title="Electric field">electric field</a> of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame. <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a <a href="/wiki/Manifestly_covariant" class="mw-redirect" title="Manifestly covariant">manifestly covariant</a> form, that is, in the language of <a href="/wiki/Tensor" title="Tensor">tensor</a> calculus.<a href="#cite_note-Post_1962-115">[80]</a> <div class="mw-heading mw-heading2"><h2 id="Theories_of_relativity_and_quantum_mechanics">Theories of relativity and quantum mechanics</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=48" title="Edit section: Theories of relativity and quantum mechanics">edit</a>]</div> Special relativity can be combined with <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> to form <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> and <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. How <a href="/wiki/General_relativity" title="General relativity">general relativity</a> and quantum mechanics can be unified is <a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">one of the unsolved problems in physics</a>; <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> and a "<a href="/wiki/Theory_of_everything" title="Theory of everything">theory of everything</a>", which require a unification including general relativity too, are active and ongoing areas in theoretical research. The early <a href="/wiki/Bohr_model#Refinements" title="Bohr model">Bohr–Sommerfeld atomic model</a> explained the <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> of <a href="/wiki/Alkali_metal" title="Alkali metal">alkali metal</a> atoms using both special relativity and the preliminary knowledge on <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> of the time.<a href="#cite_note-121">[85]</a> In 1928, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> constructed an influential <a href="/wiki/Relativistic_wave_equation" class="mw-redirect" title="Relativistic wave equation">relativistic wave equation</a>, now known as the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> in his honour,<a href="#cite_note-Dirac-122">[p 25]</a> that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, it also led to the prediction of the <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a> of the electron (the <a href="/wiki/Positron" title="Positron">positron</a>),<a href="#cite_note-Dirac-122">[p 25]</a><a href="#cite_note-123">[p 26]</a> and <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> could only be fully explained with special relativity. It was the first foundation of <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a>. On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, becomes necessary; in which particles can be <a href="/wiki/Annihilation" title="Annihilation">created and destroyed</a> throughout space and time. <div class="mw-heading mw-heading2"><h2 id="Status">Status</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=49" title="Edit section: Status">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Tests_of_special_relativity" title="Tests of special relativity">Tests of special relativity</a> and <a href="/wiki/Criticism_of_the_theory_of_relativity" title="Criticism of the theory of relativity">Criticism of the theory of relativity</a></div> Special relativity in its <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a> is accurate only when the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a> is much less than c2 in the region of interest.<a href="#cite_note-124">[86]</a> In a strong gravitational field, one must use <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the <a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a> and below, quantum effects must be taken into consideration resulting in <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)<a href="#cite_note-125">[87]</a> and thus accepted by the physics community. Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.<a href="#cite_note-Roberts_2007-126">[88]</a> Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, <a href="/wiki/String_theory" title="String theory">string theory</a>, and general relativity (in the limiting case of negligible gravitational fields). Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See <a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> for a more detailed discussion. Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,<a href="#cite_note-127">[89]</a> and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.<a href="#cite_note-mM1905-23">[16]</a> <ul><li>The <a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a> (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.</li> <li>The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.</li> <li>The <a href="/wiki/Trouton%E2%80%93Noble_experiment" title="Trouton–Noble experiment">Trouton–Noble experiment</a> (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.</li> <li>The <a href="/wiki/Experiments_of_Rayleigh_and_Brace" title="Experiments of Rayleigh and Brace">Experiments of Rayleigh and Brace</a> (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.</li></ul> <a href="/wiki/Particle_accelerator" title="Particle accelerator">Particle accelerators</a> accelerate and measure the properties of particles moving at near the speed of light, where their behavior is consistent with relativity theory and inconsistent with the earlier <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples: <ul><li><a href="/wiki/Tests_of_relativistic_energy_and_momentum" title="Tests of relativistic energy and momentum">Tests of relativistic energy and momentum</a> – testing the limiting speed of particles</li> <li><a href="/wiki/Ives%E2%80%93Stilwell_experiment" title="Ives–Stilwell experiment">Ives–Stilwell experiment</a> – testing relativistic Doppler effect and time dilation</li> <li><a href="/wiki/Experimental_testing_of_time_dilation" title="Experimental testing of time dilation">Experimental testing of time dilation</a> – relativistic effects on a fast-moving particle's half-life</li> <li><a href="/wiki/Kennedy%E2%80%93Thorndike_experiment" title="Kennedy–Thorndike experiment">Kennedy–Thorndike experiment</a> – time dilation in accordance with Lorentz transformations</li> <li><a href="/wiki/Hughes%E2%80%93Drever_experiment" title="Hughes–Drever experiment">Hughes–Drever experiment</a> – testing isotropy of space and mass</li> <li><a href="/wiki/Modern_searches_for_Lorentz_violation" title="Modern searches for Lorentz violation">Modern searches for Lorentz violation</a> – various modern tests</li> <li>Experiments to test <a href="/wiki/Emission_theory_(relativity)" title="Emission theory (relativity)">emission theory</a> demonstrated that the speed of light is independent of the speed of the emitter.</li> <li>Experiments to test the <a href="/wiki/Aether_drag_hypothesis" title="Aether drag hypothesis">aether drag hypothesis</a> – no "aether flow obstruction".</li></ul> <div class="mw-heading mw-heading2"><h2 id="Technical_discussion_of_spacetime">Technical discussion of spacetime</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=50" title="Edit section: Technical discussion of spacetime">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></div> <div class="mw-heading mw-heading3"><h3 id="Geometry_of_spacetime">Geometry of spacetime</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=51" title="Edit section: Geometry of spacetime">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Comparison_between_flat_Euclidean_space_and_Minkowski_space">Comparison between flat Euclidean space and Minkowski space</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=52" title="Edit section: Comparison between flat Euclidean space and Minkowski space">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Line_element" title="Line element">line element</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orthogonality_and_rotation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/350px-Orthogonality_and_rotation.svg.png" decoding="async" width="350" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/525px-Orthogonality_and_rotation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Orthogonality_and_rotation.svg/700px-Orthogonality_and_rotation.svg.png 2x" data-file-width="1154" data-file-height="601" /></a><figcaption>Figure 10–1. Orthogonality and rotation of coordinate systems compared between left: <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> through circular <a href="/wiki/Angle" title="Angle">angle</a> φ, right: in <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a> through <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angle</a> φ (red lines labelled c denote the <a href="/wiki/Worldline" class="mw-redirect" title="Worldline">worldlines</a> of a light signal, a vector is orthogonal to itself if it lies on this line).<a href="#cite_note-128">[90]</a></figcaption></figure> Special relativity uses a "flat" 4-dimensional Minkowski space – an example of a <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. Minkowski spacetime appears to be very similar to the standard 3-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, but there is a crucial difference with respect to time. In 3D space, the <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> of distance (line element) ds is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a5eca9bd44fe2aa0e966708eec74d1170b3e77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.617ex; height:3.343ex;" alt="{\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},}"> where dx = (dx1, dx2, dx3) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X0 derived from time, such that the distance differential fulfills <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873a8c616f9cd684cb8643e2e26b51c10cc0d62c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.503ex; height:3.343ex;" alt="{\displaystyle ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},}"> where dX = (dX0, dX1, dX2, dX3) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1).<a href="#cite_note-129">[91]</a> Just as Euclidean space uses a <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, so spacetime uses a <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>) of Minkowski spacetime. The actual form of ds above depends on the metric and on the choices for the X0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a>: X0 = ict (this is called a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>). According to <a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Misner, Thorne and Wheeler</a> (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X0 = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate. Some authors use X0 = t, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of c±2 are included in the metric tensor.<a href="#cite_note-130">[92]</a> These numerous conventions can be superseded by using <a href="/wiki/Natural_units" title="Natural units">natural units</a> where c = 1. Then space and time have equivalent units, and no factors of c appear anywhere. <div class="mw-heading mw-heading4"><h4 id="3D_spacetime">3D spacetime</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=53" title="Edit section: 3D spacetime">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Special_relativity-_Three_dimensional_dual-cone.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Special_relativity-_Three_dimensional_dual-cone.svg/220px-Special_relativity-_Three_dimensional_dual-cone.svg.png" decoding="async" width="220" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Special_relativity-_Three_dimensional_dual-cone.svg/330px-Special_relativity-_Three_dimensional_dual-cone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Special_relativity-_Three_dimensional_dual-cone.svg/440px-Special_relativity-_Three_dimensional_dual-cone.svg.png 2x" data-file-width="134" data-file-height="112" /></a><figcaption>Figure 10–2. Three-dimensional dual-cone.</figcaption></figure> If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60ef63b9b06797f84052a0ff8dfcb5ec7d44ba6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.157ex; height:3.343ex;" alt="{\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},}"> we see that the <a href="/wiki/Null_geodesic" class="mw-redirect" title="Null geodesic">null</a> <a href="/wiki/Geodesic" title="Geodesic">geodesics</a> lie along a dual-cone (see Fig. 10-2) defined by the equation; <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e43aa213a6206f390257013d99ba7b7c2a33fe2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.771ex; height:3.343ex;" alt="{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}"> or simply <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f136fee2656c3a7d7f608b4732671b36a7a3edc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.956ex; height:3.343ex;" alt="{\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},}"> which is the equation of a circle of radius c dt. <div class="mw-heading mw-heading4"><h4 id="4D_spacetime">4D spacetime</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=54" title="Edit section: 4D spacetime">edit</a>]</div> If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651ec5a5c69df295c86316d566133dfec28b076f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.211ex; height:3.343ex;" alt="{\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}"> so <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b222cb673f2908faef27a39bc8dc0456ac2971" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.396ex; height:3.343ex;" alt="{\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Concentric_Spheres.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Concentric_Spheres.svg/220px-Concentric_Spheres.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Concentric_Spheres.svg/330px-Concentric_Spheres.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Concentric_Spheres.svg/440px-Concentric_Spheres.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Figure 10–3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.</figcaption></figure> As illustrated in Fig. 10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = c dt. This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the <a href="/wiki/Star" title="Star">stars</a> and say "The light from that star that I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad565b128dfdb6fb79ed3ddd90557276826b419a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.471ex; height:4.843ex;" alt="{\textstyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}"> away and a time d/c in the past. For this reason the null dual cone is also known as the "light cone". (The point in the lower left of the Fig. 10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".) The cone in the −t region is the information that the point is "receiving", while the cone in the +t section is the information that the point is "sending". The geometry of Minkowski space can be depicted using <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagrams</a>, which are useful also in understanding many of the <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiments</a> in special relativity. <div class="mw-heading mw-heading3"><h3 id="Physics_in_spacetime">Physics in spacetime</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=55" title="Edit section: Physics in spacetime">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Transformations_of_physical_quantities_between_reference_frames">Transformations of physical quantities between reference frames</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=56" title="Edit section: Transformations of physical quantities between reference frames">edit</a>]</div> Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation. The Lorentz transformation in standard configuration above, that is, for a boost in the x-direction, can be recast into matrix form as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mo>−</mo> <mi>β</mi> <mi>γ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>β</mi> <mi>γ</mi> </mtd> <mtd> <mi>γ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>γ</mi> <mi>β</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mi>γ</mi> <mi>c</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9925a7c7193cc464d3f4f2f08dc9838f3d3578" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:58.466ex; height:12.676ex;" alt="{\displaystyle {\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta x\\\gamma x-\beta \gamma ct\\y\\z\end{pmatrix}}.}"> In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "<a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>", in Minkowski spacetime. The components of vectors are written using <a href="/wiki/Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a>, as this has numerous advantages. The notation makes it clear the equations are <a href="/wiki/Manifestly_covariant" class="mw-redirect" title="Manifestly covariant">manifestly covariant</a> under the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other <a href="/wiki/Physical_quantities" class="mw-redirect" title="Physical quantities">physical quantities</a> as <a href="/wiki/Tensors" class="mw-redirect" title="Tensors">tensors</a> simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used. The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z), in a <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position</a> <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> with components: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf {x} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf {x} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b03fe11c51c95eb4b6374e5c57d32acabed31e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.59ex; height:3.176ex;" alt="{\displaystyle X^{\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf {x} ).}"> where we define X0 = ct so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.<a href="#cite_note-131">[93]</a><a href="#cite_note-132">[94]</a><a href="#cite_note-133">[95]</a> Now the transformation of the contravariant components of the position 4-vector can be compactly written as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\mu '}=\Lambda ^{\mu '}{}_{\nu }X^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\mu '}=\Lambda ^{\mu '}{}_{\nu }X^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d6fd26cf16f7e2e7c5203e697a39db4a7f584b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.422ex; height:3.176ex;" alt="{\displaystyle X^{\mu '}=\Lambda ^{\mu '}{}_{\nu }X^{\nu }}"> where there is an <a href="/wiki/Einstein_notation" title="Einstein notation">implied summation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> from 0 to 3, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{\mu '}{}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ^{\mu '}{}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e3a041aa2487f306dd7bf63846c9ba35a96274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.472ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{\mu '}{}_{\nu }}"> is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. More generally, all contravariant components of a <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293379ba9ad58721f0b5347a8cab23e574fc307e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle T^{\nu }}"> transform from one frame to another frame by a <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\mu '}=\Lambda ^{\mu '}{}_{\nu }T^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\mu '}=\Lambda ^{\mu '}{}_{\nu }T^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2485400dac7d84c23938a74c83a84c8afd124c1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.868ex; height:3.176ex;" alt="{\displaystyle T^{\mu '}=\Lambda ^{\mu '}{}_{\nu }T^{\nu }}"> Examples of other 4-vectors include the <a href="/wiki/Four-velocity" title="Four-velocity">four-velocity</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{\mu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b33064d7e323692b286ae8f79f19d2613f184f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.065ex; height:2.343ex;" alt="{\displaystyle U^{\mu }}">⁠, defined as the derivative of the position 4-vector with respect to <a href="/wiki/Proper_time" title="Proper time">proper time</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f69f85bec48f1ccc29d785ac37175dcb30902c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.463ex; height:5.509ex;" alt="{\displaystyle U^{\mu }={\frac {dX^{\mu }}{d\tau }}=\gamma (v)(c,v_{x},v_{y},v_{z})=\gamma (v)(c,\mathbf {v} ).}"> where the Lorentz factor is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="2em" /> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22946535faa5d862d9139f4faf11a13036593fd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.783ex; height:6.509ex;" alt="{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\qquad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.}"> The <a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">relativistic energy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\gamma (v)mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\gamma (v)mc^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058efaf198edf9b9bd3dbee82cfb069b92e74f7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.175ex; height:3.176ex;" alt="{\displaystyle E=\gamma (v)mc^{2}}"> and <a href="/wiki/Relativistic_momentum" class="mw-redirect" title="Relativistic momentum">relativistic momentum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\gamma (v)m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\gamma (v)m\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4c4610fb0b3233484f6d731c45e969d572ac11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.234ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =\gamma (v)m\mathbf {v} }"> of an object are respectively the timelike and spacelike components of a <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> vector: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\mu }=mU^{\mu }=m\gamma (v)(c,v_{x},v_{y},v_{z})=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)=\left({\frac {E}{c}},\mathbf {p} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{\mu }=mU^{\mu }=m\gamma (v)(c,v_{x},v_{y},v_{z})=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)=\left({\frac {E}{c}},\mathbf {p} \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42bdca8690567fdafd1c240908a5ca52c8fd22d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.76ex; height:6.176ex;" alt="{\displaystyle P^{\mu }=mU^{\mu }=m\gamma (v)(c,v_{x},v_{y},v_{z})=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)=\left({\frac {E}{c}},\mathbf {p} \right).}"> where m is the <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a>. The <a href="/wiki/Four-acceleration" title="Four-acceleration">four-acceleration</a> is the proper time derivative of 4-velocity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4feb9012dfadf24bec25410ef605f0b361f246" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.828ex; height:5.509ex;" alt="{\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}.}"> The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix. The <a href="/wiki/Four-gradient" title="Four-gradient">four-gradient</a> of a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> φ transforms covariantly rather than contravariantly: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mo>+</mo> <mi>β</mi> <mi>γ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>β</mi> <mi>γ</mi> </mtd> <mtd> <mi>γ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2053393b441fc099c6af8f7a058b6e3a83b8e6f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:82.874ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"> which is the transpose of: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\partial _{\mu '}\phi )=\Lambda _{\mu '}{}^{\nu }(\partial _{\nu }\phi )\qquad \partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\partial _{\mu '}\phi )=\Lambda _{\mu '}{}^{\nu }(\partial _{\nu }\phi )\qquad \partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030b3a67ce8bd8597b37a9e18e18199f3ac8bfa8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.842ex; height:5.509ex;" alt="{\displaystyle (\partial _{\mu '}\phi )=\Lambda _{\mu '}{}^{\nu }(\partial _{\nu }\phi )\qquad \partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}.}"> only in Cartesian coordinates. It is the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> that transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates. More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mu '}=\Lambda _{\mu '}{}^{\nu }T_{\nu },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu '}=\Lambda _{\mu '}{}^{\nu }T_{\nu },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1b74089c57487f6dea84a7e4dad8126a9e0786" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.79ex; height:3.009ex;" alt="{\displaystyle T_{\mu '}=\Lambda _{\mu '}{}^{\nu }T_{\nu },}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{\mu '}{}^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{\mu '}{}^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecee6651221846b7ae1c6bb3919356921b4bd66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.472ex; height:3.009ex;" alt="{\displaystyle \Lambda _{\mu '}{}^{\nu }}"> is the reciprocal matrix of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{\mu '}{}_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ^{\mu '}{}_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e3a041aa2487f306dd7bf63846c9ba35a96274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.472ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{\mu '}{}_{\nu }}">⁠. The postulates of special relativity constrain the exact form the Lorentz transformation matrices take. More generally, most physical quantities are best described as (components of) <a href="/wiki/Tensor" title="Tensor">tensors</a>. So to transform from one frame to another, we use the well-known <a href="/wiki/Tensor" title="Tensor">tensor transformation law</a><a href="#cite_note-134">[96]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>θ</mi> <mo>′</mo> </msup> <msup> <mi>ι</mi> <mo>′</mo> </msup> <mo>⋯</mo> <msup> <mi>κ</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>α</mi> <mo>′</mo> </msup> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>⋯</mo> <msup> <mi>ζ</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>α</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>β</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>⋯</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ζ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ι</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> </msup> <mo>⋯</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>κ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>υ</mi> <mo>⋯</mo> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> <mo>⋯</mo> <mi>ρ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103e5eda717d3116b80cde53b524f454365e35aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:51.207ex; height:4.176ex;" alt="{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{\chi '}{}^{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{\chi '}{}^{\psi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02dafdfe9e69b769a54351e53a69c99a2fbd4b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.708ex; height:3.343ex;" alt="{\displaystyle \Lambda _{\chi '}{}^{\psi }}"> is the reciprocal matrix of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{\chi '}{}_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ^{\chi '}{}_{\psi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec14b4b970b90b50f71fe2ca6d3c62f807ed432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.708ex; height:3.509ex;" alt="{\displaystyle \Lambda ^{\chi '}{}_{\psi }}">⁠. All tensors transform by this rule. An example of a four-dimensional second order <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric tensor</a> is the <a href="/wiki/Relativistic_angular_momentum" title="Relativistic angular momentum">relativistic angular momentum</a>, which has six components: three are the classical <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric tensor</a>. The <a href="/wiki/Electromagnetic_field_tensor" class="mw-redirect" title="Electromagnetic field tensor">electromagnetic field tensor</a> is another second order antisymmetric <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a>, with six components: three for the <a href="/wiki/Electric_field" title="Electric field">electric field</a> and another three for the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. There is also the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a> for the electromagnetic field, namely the <a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">electromagnetic stress–energy tensor</a>. <div class="mw-heading mw-heading4"><h4 id="Metric">Metric</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=57" title="Edit section: Metric">edit</a>]</div> The <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> allows one to define the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> η has components (valid with suitably chosen coordinates), which can be arranged in a 4 × 4 matrix: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c66b50fbc4006c9a863be23036b5a23362bde3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.371ex; height:12.509ex;" alt="{\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"> which is equal to its reciprocal, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ^{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ^{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab7443e471121aebf804d0a26ead015592149fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.4ex; height:3.176ex;" alt="{\displaystyle \eta ^{\alpha \beta }}">⁠, in those frames. Throughout we use the signs as above, different authors use different conventions – see <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> alternative signs. The <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> is the most general group of transformations that preserves the Minkowski metric: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> <msup> <mi>ν</mi> <mo>′</mo> </msup> </mrow> </msub> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ν</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342627c7f247ccdc409d6fea260e8145cf1082bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.903ex; height:3.509ex;" alt="{\displaystyle \eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }}"> and this is the physical symmetry underlying special relativity. The metric can be used for <a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">raising and lowering indices</a> on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\alpha }S_{\alpha }=T^{\alpha }\eta _{\alpha \beta }S^{\beta }=T_{\alpha }\eta ^{\alpha \beta }S_{\beta }={\text{invariant scalar}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>invariant scalar</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\alpha }S_{\alpha }=T^{\alpha }\eta _{\alpha \beta }S^{\beta }=T_{\alpha }\eta ^{\alpha \beta }S_{\beta }={\text{invariant scalar}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7c2bf5037ce9e9eba6da255445563f8b09c5ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.391ex; height:3.343ex;" alt="{\displaystyle T^{\alpha }S_{\alpha }=T^{\alpha }\eta _{\alpha \beta }S^{\beta }=T_{\alpha }\eta ^{\alpha \beta }S_{\beta }={\text{invariant scalar}}}"> Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {T} |={\sqrt {T^{\alpha }T_{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {T} |={\sqrt {T^{\alpha }T_{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff32788f487372e490531f994d5067e12a1d367" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.221ex; height:3.343ex;" alt="{\displaystyle |\mathbf {T} |={\sqrt {T^{\alpha }T_{\alpha }}}}"> One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>invariant scalars</mtext> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9dab0cab7e66ed7360d64d7fde2921ed6b280f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.361ex; height:3.343ex;" alt="{\displaystyle T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},}"> similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one does not need to perform Lorentz transformations to determine the invariants. <div class="mw-heading mw-heading4"><h4 id="Relativistic_kinematics_and_invariance">Relativistic kinematics and invariance</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=58" title="Edit section: Relativistic kinematics and invariance">edit</a>]</div> The coordinate differentials transform also contravariantly: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cd97f4f9a5eb85fb7a0d6b3c013f46e9803677" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.854ex; height:3.176ex;" alt="{\displaystyle dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }}"> so the squared length of the differential of the position four-vector dXμ constructed using <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(c\,dt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(c\,dt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e664d8e2708e80fce1f9dfb4a52b13620328724" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.002ex; height:3.343ex;" alt="{\displaystyle d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(c\,dt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}}"> is an invariant. Notice that when the <a href="/wiki/Line_element" title="Line element">line element</a> dX2 is negative that √−dX2 is the differential of <a href="/wiki/Proper_time" title="Proper time">proper time</a>, while when dX2 is positive, √dX2 is differential of the <a href="/wiki/Proper_distance" class="mw-redirect" title="Proper distance">proper distance</a>. The 4-velocity Uμ has an invariant form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>μ</mi> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852710bcd9a908a4aa681614ac68331f9ec1eb9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.47ex; height:3.343ex;" alt="{\displaystyle \mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,}"> which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7c7991bbf5dd698b78868909f9fd25bcbc7a9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.232ex; height:3.009ex;" alt="{\displaystyle 2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.}"> So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal. <div class="mw-heading mw-heading4"><h4 id="Relativistic_dynamics_and_invariance">Relativistic dynamics and invariance</h4>[<a href="/w/index.php?title=Special_relativity&action=edit&section=59" title="Edit section: Relativistic dynamics and invariance">edit</a>]</div> The invariant magnitude of the <a href="/wiki/Four-momentum" title="Four-momentum">momentum 4-vector</a> generates the <a href="/wiki/Energy%E2%80%93momentum_relation" title="Energy–momentum relation">energy–momentum relation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{2}=\eta ^{\mu \nu }P_{\mu }P_{\nu }=-\left({\frac {E}{c}}\right)^{2}+p^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{2}=\eta ^{\mu \nu }P_{\mu }P_{\nu }=-\left({\frac {E}{c}}\right)^{2}+p^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d779f1a47d075607bcb35b628d5c641b52a1e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.264ex; height:6.509ex;" alt="{\displaystyle \mathbf {P} ^{2}=\eta ^{\mu \nu }P_{\mu }P_{\nu }=-\left({\frac {E}{c}}\right)^{2}+p^{2}.}"> We can work out what this invariant is by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{2}=-\left({\frac {E_{\text{rest}}}{c}}\right)^{2}=-(mc)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rest</mtext> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{2}=-\left({\frac {E_{\text{rest}}}{c}}\right)^{2}=-(mc)^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96bbc195fbcaf4e5a23160150ad5665d9322ddc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.173ex; height:6.509ex;" alt="{\displaystyle \mathbf {P} ^{2}=-\left({\frac {E_{\text{rest}}}{c}}\right)^{2}=-(mc)^{2}.}"> We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero. The rest energy is related to the mass according to the celebrated equation discussed above: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{rest}}=mc^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rest</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{rest}}=mc^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd311853870c1943724d1ed66221826827f2fc84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.456ex; height:3.009ex;" alt="{\displaystyle E_{\text{rest}}=mc^{2}.}"> The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames. To use <a href="/wiki/Newton%27s_third_law_of_motion" class="mw-redirect" title="Newton's third law of motion">Newton's third law of motion</a>, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D that contains the components of the 3D force vector among its components. If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the <a href="/wiki/Four-force" title="Four-force">four-force</a>. It is the rate of change of the above energy momentum <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> with respect to proper time. The covariant version of the four-force is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95f7fa73e90c06124463ed81d0885c746ace73e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.33ex; height:5.509ex;" alt="{\displaystyle F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }}"> In the rest frame of the object, the time component of the four-force is zero unless the "<a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a>" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four-force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, dp/dt while the four-force is defined by the rate of change of momentum with respect to proper time, that is, dp/dτ. In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=60" title="Edit section: See also">edit</a>]</div> <dl><dt>People</dt> <dd></dd></dl> <ul><li><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Max von Laue</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Max Born</a></li> <li><a href="/wiki/Mileva_Mari%C4%87" title="Mileva Marić">Mileva Marić</a></li></ul> <dl><dt>Relativity</dt> <dd></dd></dl> <ul><li><a href="/wiki/History_of_special_relativity" title="History of special relativity">History of special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li> <li><a href="/wiki/Bondi_k-calculus" title="Bondi k-calculus">Bondi k-calculus</a></li> <li><a href="/wiki/Einstein_synchronisation" title="Einstein synchronisation">Einstein synchronisation</a></li> <li><a href="/wiki/Rietdijk%E2%80%93Putnam_argument" title="Rietdijk–Putnam argument">Rietdijk–Putnam argument</a></li> <li><a href="/wiki/Special_relativity_(alternative_formulations)" class="mw-redirect" title="Special relativity (alternative formulations)">Special relativity (alternative formulations)</a></li> <li><a href="/wiki/Relativity_priority_dispute" title="Relativity priority dispute">Relativity priority dispute</a></li></ul> <dl><dt>Physics</dt> <dd></dd></dl> <ul><li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a></li> <li><a href="/wiki/Physical_cosmology" title="Physical cosmology">physical cosmology</a></li> <li><a href="/wiki/Relativistic_Euler_equations" title="Relativistic Euler equations">Relativistic Euler equations</a></li> <li><a href="/wiki/Lorentz_ether_theory" title="Lorentz ether theory">Lorentz ether theory</a></li> <li><a href="/wiki/Moving_magnet_and_conductor_problem" title="Moving magnet and conductor problem">Moving magnet and conductor problem</a></li> <li><a href="/wiki/Shape_waves" title="Shape waves">Shape waves</a></li> <li><a href="/wiki/Relativistic_heat_conduction" title="Relativistic heat conduction">Relativistic heat conduction</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">Relativistic disk</a></li> <li><a href="/wiki/Born_rigidity" title="Born rigidity">Born rigidity</a></li> <li><a href="/wiki/Born_coordinates" title="Born coordinates">Born coordinates</a></li></ul> <dl><dt>Mathematics</dt> <dd></dd></dl> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a></li> <li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Relativity in the APS formalism</a></li></ul> <dl><dt>Philosophy</dt> <dd></dd></dl> <ul><li><a href="/wiki/Actualism" title="Actualism">actualism</a></li> <li><a href="/wiki/Conventionalism" title="Conventionalism">conventionalism</a></li></ul> <dl><dt>Paradoxes</dt> <dd></dd></dl> <ul><li><a href="/wiki/Ehrenfest_paradox" title="Ehrenfest paradox">Ehrenfest paradox</a></li> <li><a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a></li> <li><a href="/wiki/Mocanu%27s_velocity_composition_paradox" class="mw-redirect" title="Mocanu's velocity composition paradox">Velocity composition paradox</a></li> <li><a href="/wiki/Lighthouse_paradox" title="Lighthouse paradox">Lighthouse paradox</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=61" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol class="references"> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes "The word 'special' is meant to intimate that the principle is restricted to the case ...". See p. 111 of The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, Dover reprint of 1923 translation by Methuen and Company.] </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Wald, General Relativity, p. 60: "... the special theory of relativity asserts that spacetime is the manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"> with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..." </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> In a spacetime setting, the length of a moving rigid object is the spatial distance between the ends of the object measured at the same time. In the rest frame of the object the simultaneity is not required. </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> The results of the Michelson–Morley experiment led <a href="/wiki/George_Francis_FitzGerald" title="George Francis FitzGerald">George Francis FitzGerald</a> and <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a> independently to propose the phenomenon of <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a>. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.<a href="#cite_note-Miller1998-40">[27]</a>: 62–68    Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation<a href="#cite_note-LorentzPolarization-41">[p 11]</a> and induction of double refraction,<a href="#cite_note-LorentzElectromagnetic-42">[p 12]</a> and the induction of torques on charged condensers moving at an angle with respect to the aether.<a href="#cite_note-LorentzElectromagnetic-42">[p 12]</a> Lorentz was perplexed by experiments such as the <a href="/wiki/Trouton%E2%80%93Noble_experiment" title="Trouton–Noble experiment">Trouton–Noble experiment</a> and the <a href="/wiki/Experiments_of_Rayleigh_and_Brace" title="Experiments of Rayleigh and Brace">experiments of Rayleigh and Brace</a>, which failed to validate his theoretical expectations.<a href="#cite_note-Miller1998-40">[27]</a> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation t′ = t − vx/c2.<a href="#cite_note-Lorentz1895-44">[p 13]</a> Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.<a href="#cite_note-Bernstein2006-45">[28]</a>: 51, 80    Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.<a href="#cite_note-Darrigol2005-46">[29]</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> This concept is counterintuitive at least for the fact that, in contrast to usual concepts of <a href="/wiki/Distance" title="Distance">distance</a>, it may assume negative values (is not <a href="/wiki/Positive-definite_bilinear_form" class="mw-redirect" title="Positive-definite bilinear form">positive definite</a> for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of <a href="/wiki/Minkowski_space#History" title="Minkowski space">imaginary time</a>. It is immediate that the negative of Δs2 is also an invariant, generated by a variant of the <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> of spacetime. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> The invariance of Δs2 under standard Lorentz transformation in analogous to the invariance of squared distances Δr2 under rotations in Euclidean space. Although space and time have an equal footing in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> differs from <a href="/wiki/Four-dimensional_Euclidean_space" class="mw-redirect" title="Four-dimensional Euclidean space">four-dimensional Euclidean space</a>. </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> The refractive index dependence of the presumed partial aether-drag was eventually confirmed by <a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Pieter Zeeman</a> in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to <a href="/wiki/Amsterdam" title="Amsterdam">Amsterdam</a>'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).<a href="#cite_note-zee1-75">[p 17]</a><a href="#cite_note-zee2-76">[p 18]</a> </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein's Cosmos (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length." </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955. </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> Rapidity arises naturally as a coordinates on the pure <a href="/wiki/Representation_theory_of_the_Lorentz_group#Conventions_and_Lie_algebra_bases" title="Representation theory of the Lorentz group">boost generators</a> inside the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo 2π) on the pure <a href="/wiki/Representation_theory_of_the_Lorentz_group#Conventions_and_Lie_algebra_bases" title="Representation theory of the Lorentz group">rotation generators</a> in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance. </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Primary_sources">Primary sources</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=62" title="Edit section: Primary sources">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol class="references"> <li id="cite_note-electro-1">^ <a href="#cite_ref-electro_1-0">a</a> <a href="#cite_ref-electro_1-1">b</a> <a href="#cite_ref-electro_1-2">c</a> <a href="#cite_ref-electro_1-3">d</a> <a href="#cite_ref-electro_1-4">e</a> <a href="#cite_ref-electro_1-5">f</a> <a href="#cite_ref-electro_1-6">g</a> <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> (1905) "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20050220050316/http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf">Zur Elektrodynamik bewegter Körper</a>", Annalen der Physik 17: 891; English translation <a rel="nofollow" class="external text" href="http://www.fourmilab.ch/etexts/einstein/specrel/www/">On the Electrodynamics of Moving Bodies</a> by <a href="/wiki/George_Barker_Jeffery" title="George Barker Jeffery">George Barker Jeffery</a> and Wilfrid Perrett (1923); Another English translation <a href="https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies" class="extiw" title="s:On the Electrodynamics of Moving Bodies">On the Electrodynamics of Moving Bodies</a> by <a href="/wiki/Megh_Nad_Saha" class="mw-redirect" title="Megh Nad Saha">Megh Nad Saha</a> (1920). </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> "Science and Common Sense", P. W. Bridgman, The Scientific Monthly, Vol. 79, No. 1 (Jul. 1954), pp. 32–39. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> The Electromagnetic Mass and Momentum of a Spinning Electron, G. Breit, Proceedings of the National Academy of Sciences, Vol. 12, p. 451, 1926 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Kinematics of an electron with an axis. Phil. Mag. 3:1-22. L. H. Thomas.] </li> <li id="cite_note-autogenerated1-19">^ <a href="#cite_ref-autogenerated1_19-0">a</a> <a href="#cite_ref-autogenerated1_19-1">b</a> Einstein, Autobiographical Notes, 1949. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Einstein, "Fundamental Ideas and Methods of the Theory of Relativity", 1920 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics", 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the Special and General Theory, 1916; The Principal Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949. </li> <li id="cite_note-Friedman-30"><a href="#cite_ref-Friedman_30-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFYaakov_Friedman2004" class="citation book cs1">Yaakov Friedman (2004). Physical Applications of Homogeneous Balls. Progress in Mathematical Physics. Vol. 40. pp. 1–21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3339-4" title="Special:BookSources/978-0-8176-3339-4"><bdi>978-0-8176-3339-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Applications+of+Homogeneous+Balls&rft.series=Progress+in+Mathematical+Physics&rft.pages=1-21&rft.date=2004&rft.isbn=978-0-8176-3339-4&rft.au=Yaakov+Friedman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Das, A. (1993) The Special Theory of Relativity, A Mathematical Exposition, Springer, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94042-1" title="Special:BookSources/0-387-94042-1">0-387-94042-1</a>. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-31760-6" title="Special:BookSources/0-582-31760-6">0-582-31760-6</a>. </li> <li id="cite_note-LorentzPolarization-41"><a href="#cite_ref-LorentzPolarization_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLorentz1902" class="citation journal cs1">Lorentz, H.A. (1902). <a rel="nofollow" class="external text" href="http://www.dwc.knaw.nl/DL/publications/PU00014324.pdf">"The rotation of the plane of polarization in moving media"</a> (PDF). Huygens Institute – Royal Netherlands Academy of Arts and Sciences (KNAW). 4: 669–678. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1901KNAB....4..669L">1901KNAB....4..669L</a>. 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Rev. 43 (6): 491–494. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1933PhRv...43..491A">1933PhRv...43..491A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.43.491">10.1103/PhysRev.43.491</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.&rft.atitle=The+Positive+Electron&rft.volume=43&rft.issue=6&rft.pages=491-494&rft.date=1933&rft_id=info%3Adoi%2F10.1103%2FPhysRev.43.491&rft_id=info%3Abibcode%2F1933PhRv...43..491A&rft.au=C.D.+Anderson&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRev.43.491&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=63" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol class="references"> <li id="cite_note-Griffiths-2013-2">^ <a href="#cite_ref-Griffiths-2013_2-0">a</a> <a href="#cite_ref-Griffiths-2013_2-1">b</a> <a href="#cite_ref-Griffiths-2013_2-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths2013" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2013). "Electrodynamics and Relativity". <a href="/wiki/Introduction_to_Electrodynamics" title="Introduction to Electrodynamics">Introduction to Electrodynamics</a> (4th ed.). Pearson. Chapter 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-85656-2" title="Special:BookSources/978-0-321-85656-2"><bdi>978-0-321-85656-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Electrodynamics+and+Relativity&rft.btitle=Introduction+to+Electrodynamics&rft.pages=Chapter+12&rft.edition=4th&rft.pub=Pearson&rft.date=2013&rft.isbn=978-0-321-85656-2&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Jackson-1999-3">^ <a href="#cite_ref-Jackson-1999_3-0">a</a> <a href="#cite_ref-Jackson-1999_3-1">b</a> <a href="#cite_ref-Jackson-1999_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1999" class="citation book cs1"><a href="/wiki/John_David_Jackson_(physicist)" title="John David Jackson (physicist)">Jackson, John D.</a> (1999). "Special Theory of Relativity". <a href="/wiki/Classical_Electrodynamics_(book)" title="Classical Electrodynamics (book)">Classical Electrodynamics</a> (3rd ed.). John Wiley & Sons. Chapter 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-30932-X" title="Special:BookSources/0-471-30932-X"><bdi>0-471-30932-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Special+Theory+of+Relativity&rft.btitle=Classical+Electrodynamics&rft.pages=Chapter+11&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1999&rft.isbn=0-471-30932-X&rft.aulast=Jackson&rft.aufirst=John+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein1980" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a> (1980). "Chapter 7: Special Relativity in Classical Mechanics". <a href="/wiki/Classical_Mechanics_(Goldstein_book)" class="mw-redirect" title="Classical Mechanics (Goldstein book)">Classical Mechanics</a> (2nd ed.). Addison-Wesley Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02918-9" title="Special:BookSources/0-201-02918-9"><bdi>0-201-02918-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+7%3A+Special+Relativity+in+Classical+Mechanics&rft.btitle=Classical+Mechanics&rft.edition=2nd&rft.pub=Addison-Wesley+Publishing+Company&rft.date=1980&rft.isbn=0-201-02918-9&rft.aulast=Goldstein&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Lanczos-1970-5">^ <a href="#cite_ref-Lanczos-1970_5-0">a</a> <a href="#cite_ref-Lanczos-1970_5-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanczos1970" class="citation book cs1"><a href="/wiki/Cornelius_Lanczos" title="Cornelius Lanczos">Lanczos, Cornelius</a> (1970). "Chapter IX: Relativistic Mechanics". The Variational Principles of Mechanics (4th ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65067-8" title="Special:BookSources/978-0-486-65067-8"><bdi>978-0-486-65067-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+IX%3A+Relativistic+Mechanics&rft.btitle=The+Variational+Principles+of+Mechanics&rft.edition=4th&rft.pub=Dover+Publications&rft.date=1970&rft.isbn=978-0-486-65067-8&rft.aulast=Lanczos&rft.aufirst=Cornelius&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTom_RobertsSiegmar_Schleif2007" class="citation web cs1">Tom Roberts & Siegmar Schleif (October 2007). <a rel="nofollow" class="external text" href="http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html">"What is the experimental basis of Special Relativity?"</a>. Usenet Physics FAQ. Retrieved 2008-09-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Usenet+Physics+FAQ&rft.atitle=What+is+the+experimental+basis+of+Special+Relativity%3F&rft.date=2007-10&rft.au=Tom+Roberts&rft.au=Siegmar+Schleif&rft_id=http%3A%2F%2Fwww.edu-observatory.org%2Fphysics-faq%2FRelativity%2FSR%2Fexperiments.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-relativity-7"><a href="#cite_ref-relativity_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert_Einstein2001" class="citation book cs1">Albert Einstein (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=idb7wJiB6SsC&pg=PA50">Relativity: The Special and the General Theory</a> (Reprint of 1920 translation by Robert W. Lawson ed.). Routledge. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-25384-0" title="Special:BookSources/978-0-415-25384-0"><bdi>978-0-415-25384-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+The+Special+and+the+General+Theory&rft.pages=48&rft.edition=Reprint+of+1920+translation+by+Robert+W.+Lawson&rft.pub=Routledge&rft.date=2001&rft.isbn=978-0-415-25384-0&rft.au=Albert+Einstein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Didb7wJiB6SsC%26pg%3DPA50&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Feynman-8"><a href="#cite_ref-Feynman_8-0">^</a> <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_15.html#Ch15-S9">The Feynman Lectures on Physics Vol. I Ch. 15-9: Equivalence of mass and energy</a> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Sean Carroll, Lecture Notes on General Relativity, ch. 1, "Special relativity and flat spacetime", <a rel="nofollow" class="external free" href="http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html">http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoks2006" class="citation book cs1">Koks, Don (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ObMb7l9-9loC">Explorations in Mathematical Physics: The Concepts Behind an Elegant Language</a> (illustrated ed.). Springer Science & Business Media. p. 234. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-32793-8" title="Special:BookSources/978-0-387-32793-8"><bdi>978-0-387-32793-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Explorations+in+Mathematical+Physics%3A+The+Concepts+Behind+an+Elegant+Language&rft.pages=234&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006&rft.isbn=978-0-387-32793-8&rft.aulast=Koks&rft.aufirst=Don&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DObMb7l9-9loC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ObMb7l9-9loC&pg=PA234">Extract of page 234</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteane2012" class="citation book cs1">Steane, Andrew M. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=75rCErZkh7EC">Relativity Made Relatively Easy</a> (illustrated ed.). OUP Oxford. p. 226. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-966286-9" title="Special:BookSources/978-0-19-966286-9"><bdi>978-0-19-966286-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity+Made+Relatively+Easy&rft.pages=226&rft.edition=illustrated&rft.pub=OUP+Oxford&rft.date=2012&rft.isbn=978-0-19-966286-9&rft.aulast=Steane&rft.aufirst=Andrew+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D75rCErZkh7EC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=75rCErZkh7EC&pg=PA226">Extract of page 226</a> </li> <li id="cite_note-Taylor_1992-17">^ <a href="#cite_ref-Taylor_1992_17-0">a</a> <a href="#cite_ref-Taylor_1992_17-1">b</a> <a href="#cite_ref-Taylor_1992_17-2">c</a> <a href="#cite_ref-Taylor_1992_17-3">d</a> <a href="#cite_ref-Taylor_1992_17-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorWheeler1992" class="citation book cs1">Taylor, Edwin F.; Wheeler, John Archibald (1992). <a rel="nofollow" class="external text" href="https://archive.org/details/spacetime_physics/">Spacetime Physics</a> (2nd ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-2327-1" title="Special:BookSources/0-7167-2327-1"><bdi>0-7167-2327-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+Physics&rft.edition=2nd&rft.pub=W.+H.+Freeman&rft.date=1992&rft.isbn=0-7167-2327-1&rft.aulast=Taylor&rft.aufirst=Edwin+F.&rft.au=Wheeler%2C+John+Archibald&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspacetime_physics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Rindler0-18">^ <a href="#cite_ref-Rindler0_18-0">a</a> <a href="#cite_ref-Rindler0_18-1">b</a> <a href="#cite_ref-Rindler0_18-2">c</a> <a href="#cite_ref-Rindler0_18-3">d</a> <a href="#cite_ref-Rindler0_18-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRindler1977" class="citation book cs1">Rindler, Wolfgang (1977). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0J_dwCmQThgC&pg=PT148">Essential Relativity: Special, General, and Cosmological</a> (illustrated ed.). Springer Science & Business Media. p. §1,11 p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-07970-5" title="Special:BookSources/978-3-540-07970-5"><bdi>978-3-540-07970-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Relativity%3A+Special%2C+General%2C+and+Cosmological&rft.pages=%C2%A71%2C11+p.+7&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=1977&rft.isbn=978-3-540-07970-5&rft.aulast=Rindler&rft.aufirst=Wolfgang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0J_dwCmQThgC%26pg%3DPT148&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physicsworld.com/james-clerk-maxwell-a-force-for-physics/">"James Clerk Maxwell: a force for physics"</a>. Physics World. 2006-12-01. Retrieved 2024-03-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=James+Clerk+Maxwell%3A+a+force+for+physics&rft.date=2006-12-01&rft_id=https%3A%2F%2Fphysicsworld.com%2Fjames-clerk-maxwell-a-force-for-physics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.aps.org/publications/apsnews/200711/physicshistory.cfm">"November 1887: Michelson and Morley report their failure to detect the luminiferous ether"</a>. www.aps.org. Retrieved 2024-03-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.aps.org&rft.atitle=November+1887%3A+Michelson+and+Morley+report+their+failure+to+detect+the+luminiferous+ether&rft_id=http%3A%2F%2Fwww.aps.org%2Fpublications%2Fapsnews%2F200711%2Fphysicshistory.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="/wiki/Michael_Polanyi" title="Michael Polanyi">Michael Polanyi</a> (1974) Personal Knowledge: Towards a Post-Critical Philosophy, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-67288-3" title="Special:BookSources/0-226-67288-3">0-226-67288-3</a>, footnote page 10–11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morley experiment had no role in the foundation of the theory." and "... the theory of relativity was not founded to explain its outcome at all". <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=0Rtu8kCpvz4C&lpg=PP1&pg=PT19">[1]</a> </li> <li id="cite_note-mM1905-23">^ <a href="#cite_ref-mM1905_23-0">a</a> <a href="#cite_ref-mM1905_23-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJeroen_van_Dongen2009" class="citation journal cs1">Jeroen van Dongen (2009). "On the role of the Michelson–Morley experiment: Einstein in Chicago". Archive for History of Exact Sciences. 63 (6): 655–663. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.1545">0908.1545</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009arXiv0908.1545V">2009arXiv0908.1545V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00407-009-0050-5">10.1007/s00407-009-0050-5</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119220040">119220040</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=On+the+role+of+the+Michelson%E2%80%93Morley+experiment%3A+Einstein+in+Chicago&rft.volume=63&rft.issue=6&rft.pages=655-663&rft.date=2009&rft_id=info%3Aarxiv%2F0908.1545&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119220040%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00407-009-0050-5&rft_id=info%3Abibcode%2F2009arXiv0908.1545V&rft.au=Jeroen+van+Dongen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990 </li> <li id="cite_note-Einstein-26"><a href="#cite_ref-Einstein_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein,_A.,_Lorentz,_H._A.,_Minkowski,_H.,_&_Weyl,_H.1952" class="citation book cs1">Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. (1952). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yECokhzsJYIC&pg=PA111">The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity</a>. Courier Dover Publications. p. 111. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60081-9" title="Special:BookSources/978-0-486-60081-9"><bdi>978-0-486-60081-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principle+of+Relativity%3A+a+collection+of+original+memoirs+on+the+special+and+general+theory+of+relativity&rft.pages=111&rft.pub=Courier+Dover+Publications&rft.date=1952&rft.isbn=978-0-486-60081-9&rft.au=Einstein%2C+A.%2C+Lorentz%2C+H.+A.%2C+Minkowski%2C+H.%2C+%26+Weyl%2C+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyECokhzsJYIC%26pg%3DPA111&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) </li> <li id="cite_note-Collier-28"><a href="#cite_ref-Collier_28-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollier2017" class="citation book cs1">Collier, Peter (2017). A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity (3rd ed.). Incomprehensible Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780957389465" title="Special:BookSources/9780957389465"><bdi>9780957389465</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Most+Incomprehensible+Thing%3A+Notes+Towards+a+Very+Gentle+Introduction+to+the+Mathematics+of+Relativity&rft.edition=3rd&rft.pub=Incomprehensible+Books&rft.date=2017&rft.isbn=9780957389465&rft.aulast=Collier&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift", Einstein's generation. The origins of the relativity revolution, Chicago: University of Chicago Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-77057-5" title="Special:BookSources/0-226-77057-5">0-226-77057-5</a> </li> <li id="cite_note-Morin2007-31">^ <a href="#cite_ref-Morin2007_31-0">a</a> <a href="#cite_ref-Morin2007_31-1">b</a> <a href="#cite_ref-Morin2007_31-2">c</a> <a href="#cite_ref-Morin2007_31-3">d</a> <a href="#cite_ref-Morin2007_31-4">e</a> <a href="#cite_ref-Morin2007_31-5">f</a> <a href="#cite_ref-Morin2007_31-6">g</a> <a href="#cite_ref-Morin2007_31-7">h</a> <a href="#cite_ref-Morin2007_31-8">i</a> <a href="#cite_ref-Morin2007_31-9">j</a> <a href="#cite_ref-Morin2007_31-10">k</a> <a href="#cite_ref-Morin2007_31-11">l</a> <a href="#cite_ref-Morin2007_31-12">m</a> <a href="#cite_ref-Morin2007_31-13">n</a> <a href="#cite_ref-Morin2007_31-14">o</a> <a href="#cite_ref-Morin2007_31-15">p</a> David Morin (2007) Introduction to Classical Mechanics, Cambridge University Press, Cambridge, chapter 11, Appendix I, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-139-46837-5" title="Special:BookSources/1-139-46837-5">1-139-46837-5</a>. </li> <li id="cite_note-Miller2009-34"><a href="#cite_ref-Miller2009_34-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller2010" class="citation journal cs1">Miller, D. J. (2010). "A constructive approach to the special theory of relativity". American Journal of Physics. 78 (6): 633–638. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.0902">0907.0902</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010AmJPh..78..633M">2010AmJPh..78..633M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3298908">10.1119/1.3298908</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:20444859">20444859</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=A+constructive+approach+to+the+special+theory+of+relativity&rft.volume=78&rft.issue=6&rft.pages=633-638&rft.date=2010&rft_id=info%3Aarxiv%2F0907.0902&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A20444859%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.3298908&rft_id=info%3Abibcode%2F2010AmJPh..78..633M&rft.aulast=Miller&rft.aufirst=D.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Callahan-35"><a href="#cite_ref-Callahan_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCallahan2011" class="citation book cs1">Callahan, James J. (2011). The Geometry of Spacetime: An Introduction to Special and General Relativity. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781441931429" title="Special:BookSources/9781441931429"><bdi>9781441931429</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Spacetime%3A+An+Introduction+to+Special+and+General+Relativity&rft.place=New+York&rft.pub=Springer&rft.date=2011&rft.isbn=9781441931429&rft.aulast=Callahan&rft.aufirst=James+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> P. G. Bergmann (1976) Introduction to the Theory of Relativity, Dover edition, Chapter IV, page 36 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-63282-2" title="Special:BookSources/0-486-63282-2">0-486-63282-2</a>. </li> <li id="cite_note-Mermin1968-37"><a href="#cite_ref-Mermin1968_37-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMermin1968" class="citation book cs1">Mermin, N. David (1968). <a rel="nofollow" class="external text" href="https://archive.org/details/spacetimeinspeci0000merm">Space and Time in Special Relativity</a>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0881334203" title="Special:BookSources/978-0881334203"><bdi>978-0881334203</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space+and+Time+in+Special+Relativity&rft.pub=McGraw-Hill&rft.date=1968&rft.isbn=978-0881334203&rft.aulast=Mermin&rft.aufirst=N.+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspacetimeinspeci0000merm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_Resnick1968" class="citation book cs1">Robert Resnick (1968). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fsIRAQAAIAAJ">Introduction to special relativity</a>. Wiley. pp. 62–63. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471717249" title="Special:BookSources/9780471717249"><bdi>9780471717249</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+special+relativity&rft.pages=62-63&rft.pub=Wiley&rft.date=1968&rft.isbn=9780471717249&rft.au=Robert+Resnick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfsIRAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Miller1998-40">^ <a href="#cite_ref-Miller1998_40-0">a</a> <a href="#cite_ref-Miller1998_40-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller1998" class="citation book cs1">Miller, Arthur I. (1998). Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911). Mew York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94870-6" title="Special:BookSources/978-0-387-94870-6"><bdi>978-0-387-94870-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Albert+Einstein%27s+Special+Theory+of+Relativity%3A+Emergence+%281905%29+and+Early+Interpretation+%281905%E2%80%931911%29&rft.place=Mew+York&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-94870-6&rft.aulast=Miller&rft.aufirst=Arthur+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Bernstein2006-45"><a href="#cite_ref-Bernstein2006_45-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein2006" class="citation book cs1">Bernstein, Jeremy (2006). Secrets of the Old One: Einstein, 1905. Copernicus Books (imprint of Springer Science + Business Media). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387-26005-1" title="Special:BookSources/978-0387-26005-1"><bdi>978-0387-26005-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Secrets+of+the+Old+One%3A+Einstein%2C+1905&rft.pub=Copernicus+Books+%28imprint+of+Springer+Science+%2B+Business+Media%29&rft.date=2006&rft.isbn=978-0387-26005-1&rft.aulast=Bernstein&rft.aufirst=Jeremy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Darrigol2005-46"><a href="#cite_ref-Darrigol2005_46-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDarrigol2005" class="citation journal cs1">Darrigol, Olivier (2005). <a rel="nofollow" class="external text" href="http://www.bourbaphy.fr/darrigol2.pdf">"The Genesis of the Theory of Relativity"</a> (PDF). Séminaire Poincaré. 1: 1–22. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006eins.book....1D">2006eins.book....1D</a>. Retrieved 15 November 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=S%C3%A9minaire+Poincar%C3%A9&rft.atitle=The+Genesis+of+the+Theory+of+Relativity&rft.volume=1&rft.pages=1-22&rft.date=2005&rft_id=info%3Abibcode%2F2006eins.book....1D&rft.aulast=Darrigol&rft.aufirst=Olivier&rft_id=http%3A%2F%2Fwww.bourbaphy.fr%2Fdarrigol2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Rindler1977-50">^ <a href="#cite_ref-Rindler1977_50-0">a</a> <a href="#cite_ref-Rindler1977_50-1">b</a> <a href="#cite_ref-Rindler1977_50-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRindler1977" class="citation book cs1">Rindler, Wolfgang (1977). Essential Relativity (2nd ed.). New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-10090-6" title="Special:BookSources/978-0-387-10090-6"><bdi>978-0-387-10090-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Relativity&rft.place=New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-10090-6&rft.aulast=Rindler&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Taylor1966-51">^ <a href="#cite_ref-Taylor1966_51-0">a</a> <a href="#cite_ref-Taylor1966_51-1">b</a> <a href="#cite_ref-Taylor1966_51-2">c</a> <a href="#cite_ref-Taylor1966_51-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorWheeler1966" class="citation book cs1">Taylor, Edwin F.; Wheeler, John Archibald (1966). <a rel="nofollow" class="external text" href="https://archive.org/details/spacetimephysics0000tayl">Spacetime Physics</a> (1st ed.). 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Physics FAQ. Department of Mathematics, University of California, Riverside. Retrieved 31 October 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+FAQ&rft.atitle=Is+Faster-Than-Light+Travel+or+Communication+Possible%3F&rft.aulast=Gibbs&rft.aufirst=Philip&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FRelativity%2FSpeedOfLight%2FFTL.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGinsburg1989" class="citation book cs1">Ginsburg, David (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Lh0tjaBNzg0C">Applications of Electrodynamics in Theoretical Physics and Astrophysics</a> (illustrated ed.). CRC Press. p. 206. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989aetp.book.....G">1989aetp.book.....G</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-88124-719-4" title="Special:BookSources/978-2-88124-719-4"><bdi>978-2-88124-719-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Electrodynamics+in+Theoretical+Physics+and+Astrophysics&rft.pages=206&rft.edition=illustrated&rft.pub=CRC+Press&rft.date=1989&rft_id=info%3Abibcode%2F1989aetp.book.....G&rft.isbn=978-2-88124-719-4&rft.aulast=Ginsburg&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLh0tjaBNzg0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Lh0tjaBNzg0C&pg=PA206">Extract of page 206</a> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWesley_C._Salmon2006" class="citation book cs1">Wesley C. Salmon (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FHqOXCd06e8C">Four Decades of Scientific Explanation</a>. University of Pittsburgh. p. 107. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8229-5926-7" title="Special:BookSources/978-0-8229-5926-7"><bdi>978-0-8229-5926-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Four+Decades+of+Scientific+Explanation&rft.pages=107&rft.pub=University+of+Pittsburgh&rft.date=2006&rft.isbn=978-0-8229-5926-7&rft.au=Wesley+C.+Salmon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFHqOXCd06e8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988">, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FHqOXCd06e8C&pg=PA107">Section 3.7 page 107</a> </li> <li id="cite_note-Lauginie2004-74"><a href="#cite_ref-Lauginie2004_74-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLauginie2004" class="citation journal cs1">Lauginie, P. 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(1972). "Review of Experiments on Nucleon-Nucleon Bremsstrahlung". In Austin, S.M.; Crawley, G.M. (eds.). The Two-Body Force in Nuclei. Boston, MA.: Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Review+of+Experiments+on+Nucleon-Nucleon+Bremsstrahlung&rft.btitle=The+Two-Body+Force+in+Nuclei&rft.place=Boston%2C+MA.&rft.pub=Springer&rft.date=1972&rft.aulast=Halbert&rft.aufirst=M.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasWeirHassGiordano2008" class="citation book cs1">Thomas, George B.; Weir, Maurice D.; Hass, Joel; Giordano, Frank R. (2008). Thomas' Calculus: Early Transcendentals (Eleventh ed.). Boston: Pearson Education, Inc. p. 533. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-49575-4" title="Special:BookSources/978-0-321-49575-4"><bdi>978-0-321-49575-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thomas%27+Calculus%3A+Early+Transcendentals&rft.place=Boston&rft.pages=533&rft.edition=Eleventh&rft.pub=Pearson+Education%2C+Inc.&rft.date=2008&rft.isbn=978-0-321-49575-4&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Weir%2C+Maurice+D.&rft.au=Hass%2C+Joel&rft.au=Giordano%2C+Frank+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Post_1962-115">^ <a href="#cite_ref-Post_1962_115-0">a</a> <a href="#cite_ref-Post_1962_115-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE._J._Post1962" class="citation book cs1">E. J. Post (1962). Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65427-0" title="Special:BookSources/978-0-486-65427-0"><bdi>978-0-486-65427-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formal+Structure+of+Electromagnetics%3A+General+Covariance+and+Electromagnetics&rft.pub=Dover+Publications+Inc.&rft.date=1962&rft.isbn=978-0-486-65427-0&rft.au=E.+J.+Post&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Schutz1985-116">^ <a href="#cite_ref-Schutz1985_116-0">a</a> <a href="#cite_ref-Schutz1985_116-1">b</a> <a href="#cite_ref-Schutz1985_116-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchutz1985" class="citation book cs1">Schutz, Bernard F. (1985). A first course in general relativity. Cambridge, UK: Cambridge University Press. p. 26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521277035" title="Special:BookSources/0521277035"><bdi>0521277035</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+general+relativity&rft.place=Cambridge%2C+UK&rft.pages=26&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=0521277035&rft.aulast=Schutz&rft.aufirst=Bernard+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-PhysicsFAQ-117">^ <a href="#cite_ref-PhysicsFAQ_117-0">a</a> <a href="#cite_ref-PhysicsFAQ_117-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbs" class="citation web cs1">Gibbs, Philip. <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html">"Can Special Relativity Handle Acceleration?"</a>. The Physics and Relativity FAQ. math.ucr.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170607102331/http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html">Archived</a> from the original on 7 June 2017. Retrieved 28 May 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Physics+and+Relativity+FAQ&rft.atitle=Can+Special+Relativity+Handle+Acceleration%3F&rft.aulast=Gibbs&rft.aufirst=Philip&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FRelativity%2FSR%2Facceleration.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Franklin-119"><a href="#cite_ref-Franklin_119-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklin,_Jerrold2010" class="citation journal cs1">Franklin, Jerrold (2010). "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity". European Journal of Physics. 31 (2): 291–298. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0906.1919">0906.1919</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010EJPh...31..291F">2010EJPh...31..291F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F31%2F2%2F006">10.1088/0143-0807/31/2/006</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18059490">18059490</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=Lorentz+contraction%2C+Bell%27s+spaceships%2C+and+rigid+body+motion+in+special+relativity&rft.volume=31&rft.issue=2&rft.pages=291-298&rft.date=2010&rft_id=info%3Aarxiv%2F0906.1919&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18059490%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F31%2F2%2F006&rft_id=info%3Abibcode%2F2010EJPh...31..291F&rft.au=Franklin%2C+Jerrold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-Bais-120">^ <a href="#cite_ref-Bais_120-0">a</a> <a href="#cite_ref-Bais_120-1">b</a> <a href="#cite_ref-Bais_120-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBais2007" class="citation book cs1">Bais, Sander (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/veryspecialrelat0000bais">Very Special Relativity: An Illustrated Guide</a>. Cambridge, Massachusetts: Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-02611-7" title="Special:BookSources/978-0-674-02611-7"><bdi>978-0-674-02611-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Very+Special+Relativity%3A+An+Illustrated+Guide&rft.place=Cambridge%2C+Massachusetts&rft.pub=Harvard+University+Press&rft.date=2007&rft.isbn=978-0-674-02611-7&rft.aulast=Bais&rft.aufirst=Sander&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fveryspecialrelat0000bais&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._ResnickR._Eisberg1985" class="citation book cs1">R. Resnick; R. Eisberg (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb/page/114">Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles</a> (2nd ed.). John Wiley & Sons. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb/page/114">114–116</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-87373-0" title="Special:BookSources/978-0-471-87373-0"><bdi>978-0-471-87373-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics+of+Atoms%2C+Molecules%2C+Solids%2C+Nuclei+and+Particles&rft.pages=114-116&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=978-0-471-87373-0&rft.au=R.+Resnick&rft.au=R.+Eisberg&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsof00eisb%2Fpage%2F114&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFØyvind_GrønSigbjørn_Hervik2007" class="citation book cs1">Øyvind Grøn & Sigbjørn Hervik (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IyJhCHAryuUC">Einstein's general theory of relativity: with modern applications in cosmology</a>. Springer. p. 195. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-69199-2" title="Special:BookSources/978-0-387-69199-2"><bdi>978-0-387-69199-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+general+theory+of+relativity%3A+with+modern+applications+in+cosmology&rft.pages=195&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-69199-2&rft.au=%C3%98yvind+Gr%C3%B8n&rft.au=Sigbj%C3%B8rn+Hervik&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIyJhCHAryuUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IyJhCHAryuUC&pg=PA195">Extract of page 195 (with units where c = 1)</a> </li> <li id="cite_note-125"><a href="#cite_ref-125">^</a> The number of works is vast, see as example: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSidney_ColemanSheldon_L._Glashow1997" class="citation journal cs1">Sidney Coleman; Sheldon L. Glashow (1997). "Cosmic Ray and Neutrino Tests of Special Relativity". Physics Letters B. 405 (3–4): 249–252. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/9703240">hep-ph/9703240</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997PhLB..405..249C">1997PhLB..405..249C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-2693%2897%2900638-2">10.1016/S0370-2693(97)00638-2</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17286330">17286330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Cosmic+Ray+and+Neutrino+Tests+of+Special+Relativity&rft.volume=405&rft.issue=3%E2%80%934&rft.pages=249-252&rft.date=1997&rft_id=info%3Aarxiv%2Fhep-ph%2F9703240&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17286330%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS0370-2693%2897%2900638-2&rft_id=info%3Abibcode%2F1997PhLB..405..249C&rft.au=Sidney+Coleman&rft.au=Sheldon+L.+Glashow&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> An overview can be found on <a rel="nofollow" class="external text" href="http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html">this page</a> </li> <li id="cite_note-Roberts_2007-126"><a href="#cite_ref-Roberts_2007_126-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertsSchleif" class="citation web cs1">Roberts, Tom; Schleif, Siegmar. <a rel="nofollow" class="external text" href="https://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Experiments_not_consistent_with_SR">"Experiments that Apparently are NOT Consistent with SR/GR"</a>. What is the experimental basis of Special Relativity?. University of California at Riverside. Retrieved 10 July 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=What+is+the+experimental+basis+of+Special+Relativity%3F&rft.atitle=Experiments+that+Apparently+are+NOT+Consistent+with+SR%2FGR&rft.aulast=Roberts&rft.aufirst=Tom&rft.au=Schleif%2C+Siegmar&rft_id=https%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FRelativity%2FSR%2Fexperiments.html%23Experiments_not_consistent_with_SR&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-127"><a href="#cite_ref-127">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_D._Norton2004" class="citation journal cs1">John D. Norton, John D. (2004). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/archive/00001743/">"Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905"</a>. Archive for History of Exact Sciences. 59 (1): 45–105. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004AHES...59...45N">2004AHES...59...45N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00407-004-0085-6">10.1007/s00407-004-0085-6</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17459755">17459755</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Einstein%27s+Investigations+of+Galilean+Covariant+Electrodynamics+prior+to+1905&rft.volume=59&rft.issue=1&rft.pages=45-105&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17459755%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00407-004-0085-6&rft_id=info%3Abibcode%2F2004AHES...59...45N&rft.aulast=John+D.+Norton&rft.aufirst=John+D.&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2Farchive%2F00001743%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-128"><a href="#cite_ref-128">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ.A._WheelerC._MisnerK.S._Thorne1973" class="citation book cs1">J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0344-0" title="Special:BookSources/978-0-7167-0344-0"><bdi>978-0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=58&rft.pub=W.H.+Freeman+%26+Co&rft.date=1973&rft.isbn=978-0-7167-0344-0&rft.au=J.A.+Wheeler&rft.au=C.+Misner&rft.au=K.S.+Thorne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-129"><a href="#cite_ref-129">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ.R._ForshawA.G._Smith2009" class="citation book cs1">J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. p. 247. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-01460-8" title="Special:BookSources/978-0-470-01460-8"><bdi>978-0-470-01460-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+and+Relativity&rft.pages=247&rft.pub=Wiley&rft.date=2009&rft.isbn=978-0-470-01460-8&rft.au=J.R.+Forshaw&rft.au=A.G.+Smith&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-130"><a href="#cite_ref-130">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._Penrose2007" class="citation book cs1">R. Penrose (2007). <a href="/wiki/The_Road_to_Reality" title="The Road to Reality">The Road to Reality</a>. Vintage books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-679-77631-4" title="Special:BookSources/978-0-679-77631-4"><bdi>978-0-679-77631-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality&rft.pub=Vintage+books&rft.date=2007&rft.isbn=978-0-679-77631-4&rft.au=R.+Penrose&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> <li id="cite_note-131"><a href="#cite_ref-131">^</a> Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-032071-3" title="Special:BookSources/0-07-032071-3">0-07-032071-3</a> </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> <a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Charles W. Misner</a>, <a href="/wiki/Kip_S._Thorne" class="mw-redirect" title="Kip S. Thorne">Kip S. Thorne</a> & <a href="/wiki/John_A._Wheeler" class="mw-redirect" title="John A. Wheeler">John A. Wheeler</a>, Gravitation, pg 51, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0">0-7167-0344-0</a> </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> <a href="/wiki/George_Sterman" title="George Sterman">George Sterman</a>, An Introduction to Quantum Field Theory, pg 4, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-31132-2" title="Special:BookSources/0-521-31132-2">0-521-31132-2</a> </li> <li id="cite_note-134"><a href="#cite_ref-134">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSean_M._Carroll2004" class="citation book cs1">Sean M. Carroll (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1SKFQgAACAAJ">Spacetime and Geometry: An Introduction to General Relativity</a>. Addison Wesley. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-8732-2" title="Special:BookSources/978-0-8053-8732-2"><bdi>978-0-8053-8732-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+and+Geometry%3A+An+Introduction+to+General+Relativity&rft.pages=22&rft.pub=Addison+Wesley&rft.date=2004&rft.isbn=978-0-8053-8732-2&rft.au=Sean+M.+Carroll&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1SKFQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=64" title="Edit section: Further reading">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Texts_by_Einstein_and_text_about_history_of_special_relativity">Texts by Einstein and text about history of special relativity</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=65" title="Edit section: Texts by Einstein and text about history of special relativity">edit</a>]</div> <ul><li>Einstein, Albert (1920). <a href="https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory" class="extiw" title="s:Relativity: The Special and General Theory">Relativity: The Special and General Theory</a>.</li> <li>Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56731-136-9" title="Special:BookSources/1-56731-136-9">1-56731-136-9</a></li> <li>Logunov, Anatoly A. (2005). <a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0408077">Henri Poincaré and the Relativity Theory</a> (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=66" title="Edit section: Textbooks">edit</a>]</div> <ul><li><a href="/wiki/Charles_Misner" class="mw-redirect" title="Charles Misner">Charles Misner</a>, <a href="/wiki/Kip_Thorne" title="Kip Thorne">Kip Thorne</a>, and <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> (1971) Gravitation. W. H. Freeman & Co. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0334-3" title="Special:BookSources/0-7167-0334-3">0-7167-0334-3</a></li> <li>Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.</li> <li><a href="/wiki/Wolfgang_Rindler" title="Wolfgang Rindler">Wolfgang Rindler</a> (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853952-0" title="Special:BookSources/978-0-19-853952-0">978-0-19-853952-0</a>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853952-5" title="Special:BookSources/0-19-853952-5">0-19-853952-5</a></li> <li>Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-927583-1" title="Special:BookSources/0-19-927583-1">0-19-927583-1</a>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-927583-0" title="Special:BookSources/978-0-19-927583-0">978-0-19-927583-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQadir1989" class="citation book cs1"><a href="/wiki/Asghar_Qadir" title="Asghar Qadir">Qadir, Asghar</a> (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X5YofYrqFoAC">Relativity: An Introduction to the Special Theory</a>. Singapore: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific Publications</a>. p. 128. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989rist.book.....Q">1989rist.book.....Q</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9971-5-0612-4" title="Special:BookSources/978-9971-5-0612-4"><bdi>978-9971-5-0612-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+An+Introduction+to+the+Special+Theory&rft.place=Singapore&rft.pages=128&rft.pub=World+Scientific+Publications&rft.date=1989&rft_id=info%3Abibcode%2F1989rist.book.....Q&rft.isbn=978-9971-5-0612-4&rft.aulast=Qadir&rft.aufirst=Asghar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX5YofYrqFoAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrench1968" class="citation book cs1">French, A. P. (1968). Special Relativity (M.I.T. Introductory Physics) (1st ed.). W. W. Norton & Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0393097931" title="Special:BookSources/978-0393097931"><bdi>978-0393097931</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+Relativity+%28M.I.T.+Introductory+Physics%29&rft.edition=1st&rft.pub=W.+W.+Norton+%26+Company&rft.date=1968&rft.isbn=978-0393097931&rft.aulast=French&rft.aufirst=A.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li>Silberstein, Ludwik (1914). The Theory of Relativity.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawrence_Sklar1977" class="citation book cs1">Lawrence Sklar (1977). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cPLXqV3QwuMC&pg=PA206">Space, Time and Spacetime</a>. University of California Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-520-03174-6" title="Special:BookSources/978-0-520-03174-6"><bdi>978-0-520-03174-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space%2C+Time+and+Spacetime&rft.pub=University+of+California+Press&rft.date=1977&rft.isbn=978-0-520-03174-6&rft.au=Lawrence+Sklar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcPLXqV3QwuMC%26pg%3DPA206&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawrence_Sklar1992" class="citation book cs1">Lawrence Sklar (1992). Philosophy of Physics. Westview Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8133-0625-4" title="Special:BookSources/978-0-8133-0625-4"><bdi>978-0-8133-0625-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Physics&rft.pub=Westview+Press&rft.date=1992&rft.isbn=978-0-8133-0625-4&rft.au=Lawrence+Sklar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSergey_Stepanov2018" class="citation book cs1">Sergey Stepanov (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Rc6PtAEACAAJ">Relativistic World</a>. De Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783110515879" title="Special:BookSources/9783110515879"><bdi>9783110515879</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+World&rft.pub=De+Gruyter&rft.date=2018&rft.isbn=9783110515879&rft.au=Sergey+Stepanov&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRc6PtAEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li>Taylor, Edwin, and <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> (1992). Spacetime Physics (2nd ed.). W. H. Freeman & Co. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-2327-1" title="Special:BookSources/0-7167-2327-1">0-7167-2327-1</a>.</li> <li>Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-4345-0" title="Special:BookSources/0-7167-4345-0">0-7167-4345-0</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Journal_articles">Journal articles</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=67" title="Edit section: Journal articles">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlvager,_T.Farley,_F._J._M.Kjellman,_J.Wallin,_L.1964" class="citation journal cs1">Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters. 12 (3): 260–262. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhL....12..260A">1964PhL....12..260A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0031-9163%2864%2991095-9">10.1016/0031-9163(64)91095-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters&rft.atitle=Test+of+the+Second+Postulate+of+Special+Relativity+in+the+GeV+region&rft.volume=12&rft.issue=3&rft.pages=260-262&rft.date=1964&rft_id=info%3Adoi%2F10.1016%2F0031-9163%2864%2991095-9&rft_id=info%3Abibcode%2F1964PhL....12..260A&rft.au=Alvager%2C+T.&rft.au=Farley%2C+F.+J.+M.&rft.au=Kjellman%2C+J.&rft.au=Wallin%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDarrigol2004" class="citation journal cs1">Darrigol, Olivier (2004). "The Mystery of the Poincaré–Einstein Connection". Isis. 95 (4): 614–26. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F430652">10.1086/430652</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16011297">16011297</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:26997100">26997100</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=The+Mystery+of+the+Poincar%C3%A9%E2%80%93Einstein+Connection&rft.volume=95&rft.issue=4&rft.pages=614-26&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A26997100%23id-name%3DS2CID&rft_id=info%3Apmid%2F16011297&rft_id=info%3Adoi%2F10.1086%2F430652&rft.aulast=Darrigol&rft.aufirst=Olivier&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfPetit1997" class="citation journal cs1">Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System". Physical Review A. 56 (6): 4405–09. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997PhRvA..56.4405W">1997PhRvA..56.4405W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.56.4405">10.1103/PhysRevA.56.4405</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Satellite+test+of+Special+Relativity+using+the+Global+Positioning+System&rft.volume=56&rft.issue=6&rft.pages=4405-09&rft.date=1997&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.56.4405&rft_id=info%3Abibcode%2F1997PhRvA..56.4405W&rft.aulast=Wolf&rft.aufirst=Peter&rft.au=Petit%2C+Gerard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://scholarpedia.org/article/Special_relativity">Special Relativity</a> <a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRindler2011" class="citation journal cs1"><a href="/wiki/Wolfgang_Rindler" title="Wolfgang Rindler">Rindler, Wolfgang</a> (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8520">"Special relativity: Kinematics"</a>. <a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a>. 6 (2): 8520. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011SchpJ...6.8520R">2011SchpJ...6.8520R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8520">10.4249/scholarpedia.8520</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Special+relativity%3A+Kinematics&rft.volume=6&rft.issue=2&rft.pages=8520&rft.date=2011&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.8520&rft_id=info%3Abibcode%2F2011SchpJ...6.8520R&rft.aulast=Rindler&rft.aufirst=Wolfgang&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.8520&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpecial+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Special_relativity&action=edit&section=68" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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10px;"><a href="https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory" class="extiw" title="wikisource:Relativity: The Special and General Theory">Relativity: The Special and General Theory</a></div></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, 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src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <a href="https://en.wikiversity.org/wiki/Special_Relativity" class="extiw" title="v:Special Relativity">Special Relativity</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/special_relativity" class="extiw" title="wiktionary:special relativity">special relativity</a> in Wiktionary, the free dictionary.</div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Original_works">Original works</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=69" title="Edit section: Original works">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf">Zur Elektrodynamik bewegter Körper</a> Einstein's original work in German, <a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a>, <a href="/wiki/Bern" title="Bern">Bern</a> 1905</li> <li><a rel="nofollow" class="external text" href="http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf">On the Electrodynamics of Moving Bodies</a> English Translation as published in the 1923 book The Principle of Relativity.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Special_relativity_for_a_general_audience_(no_mathematical_knowledge_required)">Special relativity for a general audience (no mathematical knowledge required)</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=70" title="Edit section: Special relativity for a general audience (no mathematical knowledge required)">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.phys.unsw.edu.au/einsteinlight">Einstein Light</a> An <a rel="nofollow" class="external text" href="http://www.sciam.com/article.cfm?chanID=sa004&articleID=0005CFF9-524F-1340-924F83414B7F0000">award</a>-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.</li> <li><a rel="nofollow" class="external text" href="http://www.einstein-online.info/en/elementary/index.html">Einstein Online</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100201234156/http://www.einstein-online.info/en/elementary/index.html">Archived</a> 2010-02-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.</li> <li>Audio: Cain/Gay (2006) – <a rel="nofollow" class="external text" href="http://www.astronomycast.com/astronomy/einsteins-theory-of-special-relativity/">Astronomy Cast</a>. Einstein's Theory of Special Relativity</li></ul> <div class="mw-heading mw-heading3"><h3 id="Special_relativity_explained_(using_simple_or_more_advanced_mathematics)">Special relativity explained (using simple or more advanced mathematics)</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=71" title="Edit section: Special relativity explained (using simple or more advanced mathematics)">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040206110957/http://www.geocities.com/autotheist/Bondi/intro.htm">Bondi K-Calculus</a> – A simple introduction to the special theory of relativity.</li> <li><a rel="nofollow" class="external text" href="http://gregegan.customer.netspace.net.au/FOUNDATIONS/01/found01.html">Greg Egan's Foundations</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130425091908/http://gregegan.customer.netspace.net.au/FOUNDATIONS/01/found01.html">Archived</a> 2013-04-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><a rel="nofollow" class="external text" href="http://cosmo.nyu.edu/hogg/sr/">The Hogg Notes on Special Relativity</a> A good introduction to special relativity at the undergraduate level, using calculus.</li> <li><a rel="nofollow" class="external text" href="http://www.relativitycalculator.com/E=mc2.shtml">Relativity Calculator: Special Relativity</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130321052504/http://www.relativitycalculator.com/E=mc2.shtml">Archived</a> 2013-03-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – An algebraic and integral calculus derivation for E = mc2.</li> <li><a rel="nofollow" class="external text" href="http://www.mathpages.com/rr/rrtoc.htm">MathPages – Reflections on Relativity</a> A complete online book on relativity with an extensive bibliography.</li> <li><a rel="nofollow" class="external text" href="http://lightandmatter.com/sr/">Special Relativity</a> An introduction to special relativity at the undergraduate level.</li> <li class="mw-empty-elt"></li> <li><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style> <a rel="nofollow" class="external text" href="https://gutenberg.org/ebooks/5001">Relativity: the Special and General Theory</a> at <a href="/wiki/Project_Gutenberg" title="Project Gutenberg">Project Gutenberg</a>, by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a rel="nofollow" class="external text" href="http://www.phys.vt.edu/~takeuchi/relativity/notes">Special Relativity Lecture Notes</a> is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.</li> <li><a rel="nofollow" class="external text" href="http://www.rafimoor.com/english/SRE.htm">Understanding Special Relativity</a> The theory of special relativity in an easily understandable way.</li> <li><a rel="nofollow" class="external text" href="http://digitalcommons.unl.edu/physicskatz/49/">An Introduction to the Special Theory of Relativity</a> (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).</li> <li><a rel="nofollow" class="external text" href="http://www.physics.mq.edu.au/~jcresser/Phys378/LectureNotes/VectorsTensorsSR.pdf">Lecture Notes on Special Relativity</a> by J D Cresser Department of Physics Macquarie University.</li> <li><a rel="nofollow" class="external text" href="http://specialrelativity.net/">SpecialRelativity.net</a> – An overview with visualizations and minimal mathematics.</li> <li><a rel="nofollow" class="external text" href="https://www.mdpi.com/2624-8174/4/2/28">Relativity 4-ever?</a> The problem of superluminal motion is discussed in an entertaining manner.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Visualization">Visualization</h3>[<a href="/w/index.php?title=Special_relativity&action=edit&section=72" title="Edit section: Visualization">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.hakenberg.de/diffgeo/special_relativity.htm">Raytracing Special Relativity</a> Software visualizing several scenarios under the influence of special relativity.</li> <li><a rel="nofollow" class="external text" href="http://www.anu.edu.au/Physics/Savage/RTR/">Real Time Relativity</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130508021027/http://www.anu.edu.au/Physics/Savage/RTR/">Archived</a> 2013-05-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> The Australian National University. Relativistic visual effects experienced through an interactive program.</li> <li><a rel="nofollow" class="external text" href="http://www.spacetimetravel.org/">Spacetime travel</a> A variety of visualizations of relativistic effects, from relativistic motion to black holes.</li> <li><a rel="nofollow" class="external text" href="http://www.anu.edu.au/Physics/Savage/TEE/">Through Einstein's Eyes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130514175026/http://www.anu.edu.au/Physics/Savage/TEE/">Archived</a> 2013-05-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> The Australian National University. Relativistic visual effects explained with movies and images.</li> <li><a rel="nofollow" class="external text" href="http://www.adamauton.com/warp/">Warp Special Relativity Simulator</a> A computer program to show the effects of traveling close to the speed of light.</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=C2VMO7pcWhg">Animation clip</a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> visualizing the Lorentz transformation.</li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/~jdp/Relativity/SpecialRelativity.html">Original interactive FLASH Animations</a> from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.</li> <li><a rel="nofollow" class="external text" href="http://lightspeed.sourceforge.net/">lightspeed</a> An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.</li> <li><a rel="nofollow" class="external text" href="http://specialrelativity.net/animations/starfield/starfield.html?beta=0.8&color=on&circles=on&avgstellardensity=0.11&starpopulation=yalebsc&limitingMag=5&projection=stereographic&anim=on&runningTime=8">Animation</a> showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a class="mw-selflink selflink">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Molecular_physics" title="Molecular physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Albert_Einstein" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Albert_Einstein" title="Template:Albert Einstein"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Albert_Einstein" title="Template talk:Albert Einstein"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Albert_Einstein" title="Special:EditPage/Template:Albert Einstein"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Albert_Einstein" style="font-size:114%;margin:0 4em"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a> <ul><li><a class="mw-selflink selflink">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li></ul></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc2)</a></li> <li><a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a></li> <li><a href="/wiki/Photoelectric_effect" title="Photoelectric effect">Photoelectric effect</a></li> <li><a href="/wiki/Einstein_coefficients" title="Einstein coefficients">Einstein coefficients</a></li> <li><a href="/wiki/Einstein_solid" title="Einstein solid">Einstein solid</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Einstein_radius" title="Einstein radius">Einstein radius</a></li> <li><a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a></li> <li><a href="/wiki/Cosmological_constant" title="Cosmological constant">Cosmological constant</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_correlations" title="Bose–Einstein correlations">Bose–Einstein correlations</a></li> <li><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a></li> <li><a href="/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations" title="Einstein–Infeld–Hoffmann equations">Einstein–Infeld–Hoffmann equations</a></li> <li><a href="/wiki/Einstein%E2%80%93de_Haas_effect" title="Einstein–de Haas effect">Einstein–de Haas effect</a></li> <li><a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a></li> <li><a href="/wiki/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein debates</a></li> <li><a href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</a></li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Thought experiments</a></li> <li><a href="/wiki/Einstein%27s_unsuccessful_investigations" title="Einstein's unsuccessful investigations">Unsuccessful investigations</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></li> <li><a href="/wiki/Tea_leaf_paradox" title="Tea leaf paradox">Tea leaf paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_scientific_publications_by_Albert_Einstein" title="List of scientific publications by Albert Einstein">Works</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Annus_mirabilis_papers" title="Annus mirabilis papers">Annus mirabilis papers</a> (1905)</li> <li>"<a href="/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen" title="Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen">Investigations on the Theory of Brownian Movement</a>" (1905)</li> <li><a href="/wiki/Relativity:_The_Special_and_the_General_Theory" title="Relativity: The Special and the General Theory">Relativity: The Special and the General Theory</a> (1916)</li> <li><a href="/wiki/The_Meaning_of_Relativity" title="The Meaning of Relativity">The Meaning of Relativity</a> (1922)</li> <li><a href="/wiki/The_World_as_I_See_It_(book)" title="The World as I See It (book)">The World as I See It</a> (1934)</li> <li><a href="/wiki/The_Evolution_of_Physics" title="The Evolution of Physics">The Evolution of Physics</a> (1938)</li> <li>"<a href="/wiki/Why_Socialism%3F" title="Why Socialism?">Why Socialism?</a>" (1949)</li> <li><a href="/wiki/Russell%E2%80%93Einstein_Manifesto" title="Russell–Einstein Manifesto">Russell–Einstein Manifesto</a> (1955)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Albert_Einstein_in_popular_culture" title="Albert Einstein in popular culture">In popular culture</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Die_Grundlagen_der_Einsteinschen_Relativit%C3%A4ts-Theorie" title="Die Grundlagen der Einsteinschen Relativitäts-Theorie">Die Grundlagen der Einsteinschen Relativitäts-Theorie</a> (1922 documentary)</li> <li><a href="/wiki/The_Einstein_Theory_of_Relativity" title="The Einstein Theory of Relativity">The Einstein Theory of Relativity</a> (1923 documentary)</li> <li><a href="/wiki/Relics:_Einstein%27s_Brain" title="Relics: Einstein's Brain">Relics: Einstein's Brain</a> (1994 documentary)</li> <li><a href="/wiki/Insignificance_(film)" title="Insignificance (film)">Insignificance</a> (1985 film)</li> <li><a href="/wiki/Young_Einstein" title="Young Einstein">Young Einstein</a> (1988 film)</li> <li><a href="/wiki/Picasso_at_the_Lapin_Agile" title="Picasso at the Lapin Agile">Picasso at the Lapin Agile</a> (1993 play)</li> <li><a href="/wiki/I.Q._(film)" title="I.Q. (film)">I.Q.</a> (1994 film)</li> <li><a href="/wiki/Einstein%27s_Gift" title="Einstein's Gift">Einstein's Gift</a> (2003 play)</li> <li><a href="/wiki/Einstein_and_Eddington" title="Einstein and Eddington">Einstein and Eddington</a> (2008 TV film)</li> <li><a href="/wiki/Genius_(American_TV_series)" title="Genius (American TV series)">Genius</a> (2017 series)</li> <li><a href="/wiki/Oppenheimer_(film)" title="Oppenheimer (film)">Oppenheimer</a> (2023 film)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Prizes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_Einstein_Award" title="Albert Einstein Award">Albert Einstein Award</a></li> <li><a href="/wiki/Albert_Einstein_Medal" title="Albert Einstein Medal">Albert Einstein Medal</a></li> <li><a href="/wiki/Kalinga_Prize" title="Kalinga Prize">Kalinga Prize</a></li> <li><a href="/wiki/Albert_Einstein_Peace_Prize" title="Albert Einstein Peace Prize">Albert Einstein Peace Prize</a></li> <li><a href="/wiki/Albert_Einstein_World_Award_of_Science" title="Albert Einstein World Award of Science">Albert Einstein World Award of Science</a></li> <li><a href="/wiki/Einstein_Prize_for_Laser_Science" title="Einstein Prize for Laser Science">Einstein Prize for Laser Science</a></li> <li><a href="/wiki/Einstein_Prize_(APS)" title="Einstein Prize (APS)">Einstein Prize (APS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books about Einstein</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_Einstein:_Creator_and_Rebel" title="Albert Einstein: Creator and Rebel">Albert Einstein: Creator and Rebel</a></li> <li><a href="/wiki/Einstein_and_Religion" title="Einstein and Religion">Einstein and Religion</a></li> <li><a href="/wiki/Einstein_for_Beginners" title="Einstein for Beginners">Einstein for Beginners</a></li> <li><a href="/wiki/Einstein:_His_Life_and_Universe" title="Einstein: His Life and Universe">Einstein: His Life and Universe</a></li> <li><a href="/wiki/Einstein%27s_Cosmos" title="Einstein's Cosmos">Einstein's Cosmos</a></li> <li><a href="/wiki/I_Am_Albert_Einstein" title="I Am Albert Einstein">I Am Albert Einstein</a></li> <li><a href="/wiki/Introducing_Relativity" title="Introducing Relativity">Introducing Relativity</a></li> <li><a href="/wiki/Subtle_is_the_Lord" title="Subtle is the Lord">Subtle is the Lord</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Einstein_family" title="Einstein family">Family</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mileva_Mari%C4%87" title="Mileva Marić">Mileva Marić</a> (first wife)</li> <li><a href="/wiki/Elsa_Einstein" title="Elsa Einstein">Elsa Einstein</a> (second wife; cousin)</li> <li><a href="/wiki/Lieserl_Einstein" class="mw-redirect" title="Lieserl Einstein">Lieserl Einstein</a> (daughter)</li> <li><a href="/wiki/Hans_Albert_Einstein" title="Hans Albert Einstein">Hans Albert Einstein</a> (son)</li> <li><a href="/wiki/Pauline_Koch" class="mw-redirect" title="Pauline Koch">Pauline Koch</a> (mother)</li> <li><a href="/wiki/Hermann_Einstein" class="mw-redirect" title="Hermann Einstein">Hermann Einstein</a> (father)</li> <li><a href="/wiki/Maja_Einstein" title="Maja Einstein">Maja Einstein</a> (sister)</li> <li><a href="/wiki/Einstein_family#Eduard_"Tete"_Einstein_(Albert's_second_son)" title="Einstein family">Eduard Einstein</a> (son)</li> <li><a href="/wiki/Murder_of_the_family_of_Robert_Einstein" title="Murder of the family of Robert Einstein">Robert Einstein</a> (cousin)</li> <li><a href="/wiki/Bernhard_Caesar_Einstein" title="Bernhard Caesar Einstein">Bernhard Caesar Einstein</a> (grandson)</li> <li><a href="/wiki/Evelyn_Einstein" title="Evelyn Einstein">Evelyn Einstein</a> (granddaughter)</li> <li><a href="/wiki/Thomas_Martin_Einstein" class="mw-redirect" title="Thomas Martin Einstein">Thomas Martin Einstein</a> (great-grandson)</li> <li><a href="/wiki/Siegbert_Einstein" title="Siegbert Einstein">Siegbert Einstein</a> (distant cousin)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_awards_and_honors_received_by_Albert_Einstein" title="List of awards and honors received by Albert Einstein">Awards and honors</a></li> <li><a href="/wiki/Brain_of_Albert_Einstein" title="Brain of Albert Einstein">Brain</a></li> <li><a href="/wiki/Albert_Einstein_House" title="Albert Einstein House">House</a></li> <li><a href="/wiki/Albert_Einstein_Memorial" title="Albert Einstein Memorial">Memorial</a></li> <li><a href="/wiki/Political_views_of_Albert_Einstein" title="Political views of Albert Einstein">Political views</a></li> <li><a href="/wiki/Religious_and_philosophical_views_of_Albert_Einstein" title="Religious and philosophical views of Albert Einstein">Religious views</a></li> <li><a href="/wiki/List_of_things_named_after_Albert_Einstein" title="List of things named after Albert Einstein">Things named after</a></li> <li><a href="/wiki/Einstein%E2%80%93Oppenheimer_relationship" title="Einstein–Oppenheimer relationship">Einstein–Oppenheimer relationship</a></li> <li><a href="/wiki/Albert_Einstein_Archives" title="Albert Einstein Archives">Albert Einstein Archives</a></li> <li><a href="/wiki/Einstein%27s_Blackboard" title="Einstein's Blackboard">Einstein's Blackboard</a></li> <li><a href="/wiki/Einstein_Papers_Project" title="Einstein Papers Project">Einstein Papers Project</a></li> <li><a href="/wiki/Einstein_refrigerator" title="Einstein refrigerator">Einstein refrigerator</a></li> <li><a href="/wiki/Einsteinhaus" title="Einsteinhaus">Einsteinhaus</a></li> <li><a href="/wiki/Einsteinium" title="Einsteinium">Einsteinium</a></li> <li><a href="/wiki/Max_Talmey" title="Max Talmey">Max Talmey</a></li> <li><a href="/wiki/Emergency_Committee_of_Atomic_Scientists" title="Emergency Committee of Atomic Scientists">Emergency Committee of Atomic Scientists</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Albert_Einstein" title="Category:Albert Einstein">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Relativity" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a class="mw-selflink selflink">Special relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a class="mw-selflink selflink">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc2)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity">others</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Special_relativity" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Special_relativity" title="Template:Special relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Special_relativity" class="mw-redirect" title="Template talk:Special relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Special_relativity" title="Special:EditPage/Template:Special relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Special_relativity" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Special relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overviews</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/History_of_special_relativity" title="History of special relativity">History</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a></li> <li><a href="/wiki/Aether_theories" title="Aether theories">Aether theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kinematics" title="Kinematics">Relative motion</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Consequences</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">Relativistic disk</a></li> <li><a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a></li> <li><a href="/wiki/Ehrenfest_paradox" title="Ehrenfest paradox">Ehrenfest paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">Dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Invariant_mass" title="Invariant mass">Proper mass</a></li> <li><a href="/wiki/Four-momentum" title="Four-momentum">4-momentum</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tests_of_special_relativity" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tests_of_special_relativity" title="Template:Tests of special relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tests_of_special_relativity" title="Template talk:Tests of special relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tests_of_special_relativity" title="Special:EditPage/Template:Tests of special relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tests_of_special_relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Tests_of_special_relativity" title="Tests of special relativity">Tests of special relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Speed/isotropy</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a></li> <li><a href="/wiki/Kennedy%E2%80%93Thorndike_experiment" title="Kennedy–Thorndike experiment">Kennedy–Thorndike experiment</a></li> <li><a href="/wiki/Ives%E2%80%93Stilwell_experiment#Mössbauer_rotor_experiments" title="Ives–Stilwell experiment">Moessbauer rotor experiments</a></li> <li><a href="/wiki/Michelson%E2%80%93Morley_experiment#Recent_experiments" title="Michelson–Morley experiment">Resonator experiments</a></li> <li><a href="/wiki/De_Sitter_double_star_experiment" title="De Sitter double star experiment">de Sitter double star experiment</a></li> <li><a href="/wiki/Hammar_experiment" title="Hammar experiment">Hammar experiment</a></li> <li><a href="/wiki/Measurements_of_neutrino_speed" title="Measurements of neutrino speed">Measurements of neutrino speed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lorentz invariance</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Modern_searches_for_Lorentz_violation" title="Modern searches for Lorentz violation">Modern searches for Lorentz violation</a></li> <li><a href="/wiki/Hughes%E2%80%93Drever_experiment" title="Hughes–Drever experiment">Hughes–Drever experiment</a></li> <li><a href="/wiki/Trouton%E2%80%93Noble_experiment" title="Trouton–Noble experiment">Trouton–Noble experiment</a></li> <li><a href="/wiki/Experiments_of_Rayleigh_and_Brace" title="Experiments of Rayleigh and Brace">Experiments of Rayleigh and Brace</a></li> <li><a href="/wiki/Trouton%E2%80%93Rankine_experiment" title="Trouton–Rankine experiment">Trouton–Rankine experiment</a></li> <li><a href="/wiki/Antimatter_tests_of_Lorentz_violation" title="Antimatter tests of Lorentz violation">Antimatter tests of Lorentz violation</a></li> <li><a href="/wiki/Lorentz-violating_neutrino_oscillations" title="Lorentz-violating neutrino oscillations">Lorentz-violating neutrino oscillations</a></li> <li><a href="/wiki/Lorentz-violating_electrodynamics" title="Lorentz-violating electrodynamics">Lorentz-violating electrodynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a> <a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ives%E2%80%93Stilwell_experiment" title="Ives–Stilwell experiment">Ives–Stilwell experiment</a></li> <li><a href="/wiki/Ives%E2%80%93Stilwell_experiment#Mössbauer_rotor_experiments" title="Ives–Stilwell experiment">Moessbauer rotor experiments</a></li> <li><a href="/wiki/Experimental_testing_of_time_dilation" title="Experimental testing of time dilation">Experimental testing of time dilation</a></li> <li><a href="/wiki/Hafele%E2%80%93Keating_experiment" title="Hafele–Keating experiment">Hafele–Keating experiment</a></li> <li><a href="/wiki/Length_contraction#Experimental_verifications" title="Length contraction">Length contraction confirmations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relativistic energy</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tests_of_relativistic_energy_and_momentum" title="Tests of relativistic energy and momentum">Tests of relativistic energy and momentum</a></li> <li><a href="/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments" title="Kaufmann–Bucherer–Neumann experiments">Kaufmann–Bucherer–Neumann experiments</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fizeau/Sagnac</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a></li> <li><a href="/wiki/Sagnac_effect" title="Sagnac effect">Sagnac experiment</a></li> <li><a href="/wiki/Michelson%E2%80%93Gale%E2%80%93Pearson_experiment" title="Michelson–Gale–Pearson experiment">Michelson–Gale–Pearson experiment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Alternatives</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Luminiferous_aether#Negative_aether-drift_experiments" title="Luminiferous aether">Refutations of aether theory</a></li> <li><a href="/wiki/Emission_theory_(relativity)#Refutations_of_emission_theory" title="Emission theory (relativity)">Refutations of emission theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-way_speed_of_light" title="One-way speed of light">One-way speed of light</a></li> <li><a href="/wiki/Test_theories_of_special_relativity" title="Test theories of special relativity">Test theories of special relativity</a></li> <li><a href="/wiki/Standard-Model_Extension" title="Standard-Model Extension">Standard-Model Extension</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tensors" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tensors" title="Template:Tensors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tensors" title="Template talk:Tensors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tensors" style="font-size:114%;margin:0 4em"><a href="/wiki/Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 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