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vector-toc-toggle"> Toggle From classical mechanics to general relativity subsection </button> <ul id="toc-From_classical_mechanics_to_general_relativity-sublist" class="vector-toc-list"> <li id="toc-Geometry_of_Newtonian_gravity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_of_Newtonian_gravity"> <div class="vector-toc-text"> 2.1 Geometry of Newtonian gravity </div> </a> <ul id="toc-Geometry_of_Newtonian_gravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_generalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_generalization"> <div class="vector-toc-text"> 2.2 Relativistic generalization </div> </a> <ul id="toc-Relativistic_generalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einstein's_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein's_equations"> <div class="vector-toc-text"> 2.3 Einstein's equations </div> </a> <ul id="toc-Einstein's_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_force_in_general_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_force_in_general_relativity"> <div class="vector-toc-text"> 2.4 Total force in general relativity </div> </a> <ul id="toc-Total_force_in_general_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternatives_to_general_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternatives_to_general_relativity"> <div class="vector-toc-text"> 2.5 Alternatives to general relativity </div> </a> <ul id="toc-Alternatives_to_general_relativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition_and_basic_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_and_basic_applications"> <div class="vector-toc-text"> 3 Definition and basic applications </div> </a> <button aria-controls="toc-Definition_and_basic_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition and basic applications subsection </button> <ul id="toc-Definition_and_basic_applications-sublist" class="vector-toc-list"> <li id="toc-Definition_and_basic_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_and_basic_properties"> <div class="vector-toc-text"> 3.1 Definition and basic properties </div> </a> <ul id="toc-Definition_and_basic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model-building" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model-building"> <div class="vector-toc-text"> 3.2 Model-building </div> </a> <ul id="toc-Model-building-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Consequences_of_Einstein's_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Consequences_of_Einstein's_theory"> <div class="vector-toc-text"> 4 Consequences of Einstein's theory </div> </a> <button aria-controls="toc-Consequences_of_Einstein's_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Consequences of Einstein's theory subsection </button> <ul id="toc-Consequences_of_Einstein's_theory-sublist" class="vector-toc-list"> <li id="toc-Gravitational_time_dilation_and_frequency_shift" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gravitational_time_dilation_and_frequency_shift"> <div class="vector-toc-text"> 4.1 Gravitational time dilation and frequency shift </div> </a> <ul id="toc-Gravitational_time_dilation_and_frequency_shift-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Light_deflection_and_gravitational_time_delay" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Light_deflection_and_gravitational_time_delay"> <div class="vector-toc-text"> 4.2 Light deflection and gravitational time delay </div> </a> <ul id="toc-Light_deflection_and_gravitational_time_delay-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gravitational_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gravitational_waves"> <div class="vector-toc-text"> 4.3 Gravitational waves </div> </a> <ul id="toc-Gravitational_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_effects_and_the_relativity_of_direction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_effects_and_the_relativity_of_direction"> <div class="vector-toc-text"> 4.4 Orbital effects and the relativity of direction </div> </a> <ul id="toc-Orbital_effects_and_the_relativity_of_direction-sublist" class="vector-toc-list"> <li id="toc-Precession_of_apsides" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Precession_of_apsides"> <div class="vector-toc-text"> 4.4.1 Precession of apsides </div> </a> <ul id="toc-Precession_of_apsides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_decay" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orbital_decay"> <div class="vector-toc-text"> 4.4.2 Orbital decay </div> </a> <ul id="toc-Orbital_decay-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geodetic_precession_and_frame-dragging" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geodetic_precession_and_frame-dragging"> <div class="vector-toc-text"> 4.4.3 Geodetic precession and frame-dragging </div> </a> <ul id="toc-Geodetic_precession_and_frame-dragging-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Astrophysical_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Astrophysical_applications"> <div class="vector-toc-text"> 5 Astrophysical applications </div> </a> <button aria-controls="toc-Astrophysical_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Astrophysical applications subsection </button> <ul id="toc-Astrophysical_applications-sublist" class="vector-toc-list"> <li id="toc-Gravitational_lensing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gravitational_lensing"> <div class="vector-toc-text"> 5.1 Gravitational lensing </div> </a> <ul id="toc-Gravitational_lensing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gravitational-wave_astronomy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gravitational-wave_astronomy"> <div class="vector-toc-text"> 5.2 Gravitational-wave astronomy </div> </a> <ul id="toc-Gravitational-wave_astronomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Black_holes_and_other_compact_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Black_holes_and_other_compact_objects"> <div class="vector-toc-text"> 5.3 Black holes and other compact objects </div> </a> <ul id="toc-Black_holes_and_other_compact_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosmology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosmology"> <div class="vector-toc-text"> 5.4 Cosmology </div> </a> <ul id="toc-Cosmology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exotic_solutions:_time_travel,_warp_drives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exotic_solutions:_time_travel,_warp_drives"> <div class="vector-toc-text"> 5.5 Exotic solutions: time travel, warp drives </div> </a> <ul id="toc-Exotic_solutions:_time_travel,_warp_drives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Advanced_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Advanced_concepts"> <div class="vector-toc-text"> 6 Advanced concepts </div> </a> <button aria-controls="toc-Advanced_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Advanced concepts subsection </button> <ul id="toc-Advanced_concepts-sublist" class="vector-toc-list"> <li id="toc-Asymptotic_symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_symmetries"> <div class="vector-toc-text"> 6.1 Asymptotic symmetries </div> </a> <ul id="toc-Asymptotic_symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Causal_structure_and_global_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Causal_structure_and_global_geometry"> <div class="vector-toc-text"> 6.2 Causal structure and global geometry </div> </a> <ul id="toc-Causal_structure_and_global_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Horizons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Horizons"> <div class="vector-toc-text"> 6.3 Horizons </div> </a> <ul id="toc-Horizons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singularities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singularities"> <div class="vector-toc-text"> 6.4 Singularities </div> </a> <ul id="toc-Singularities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evolution_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evolution_equations"> <div class="vector-toc-text"> 6.5 Evolution equations </div> </a> <ul id="toc-Evolution_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Global_and_quasi-local_quantities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Global_and_quasi-local_quantities"> <div class="vector-toc-text"> 6.6 Global and quasi-local quantities </div> </a> <ul id="toc-Global_and_quasi-local_quantities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_with_quantum_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_with_quantum_theory"> <div class="vector-toc-text"> 7 Relationship with quantum theory </div> </a> <button aria-controls="toc-Relationship_with_quantum_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relationship with quantum theory subsection </button> <ul id="toc-Relationship_with_quantum_theory-sublist" class="vector-toc-list"> <li id="toc-Quantum_field_theory_in_curved_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_field_theory_in_curved_spacetime"> <div class="vector-toc-text"> 7.1 Quantum field theory in curved spacetime </div> </a> <ul id="toc-Quantum_field_theory_in_curved_spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_gravity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_gravity"> <div class="vector-toc-text"> 7.2 Quantum gravity </div> </a> <ul id="toc-Quantum_gravity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Current_status" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Current_status"> <div class="vector-toc-text"> 8 Current status </div> </a> <ul id="toc-Current_status-sublist" class="vector-toc-list"> </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"> 9 See also </div> </a> <ul id="toc-See_also-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"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 11 Bibliography </div> </a> <ul id="toc-Bibliography-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"> 12 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-Popular_books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Popular_books"> <div class="vector-toc-text"> 12.1 Popular books </div> </a> <ul id="toc-Popular_books-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beginning_undergraduate_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beginning_undergraduate_textbooks"> <div class="vector-toc-text"> 12.2 Beginning undergraduate textbooks </div> </a> <ul id="toc-Beginning_undergraduate_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_undergraduate_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Advanced_undergraduate_textbooks"> <div class="vector-toc-text"> 12.3 Advanced undergraduate textbooks </div> </a> <ul id="toc-Advanced_undergraduate_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graduate_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graduate_textbooks"> <div class="vector-toc-text"> 12.4 Graduate textbooks </div> </a> <ul id="toc-Graduate_textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specialists'_books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specialists'_books"> <div class="vector-toc-text"> 12.5 Specialists' books </div> </a> <ul id="toc-Specialists'_books-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"> 12.6 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"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </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">General 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 111 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-111" aria-hidden="true" > 111 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/Algemene_relatiwiteit" title="Algemene relatiwiteit – Afrikaans" lang="af" hreflang="af" data-title="Algemene 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/Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie – Alemannic" lang="gsw" hreflang="gsw" data-title="Allgemeine 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%8A%A0%E1%8C%A0%E1%89%83%E1%88%8B%E1%8B%AD_%E1%8A%A0%E1%8A%95%E1%8C%BB%E1%88%AB%E1%8B%8A%E1%8A%90%E1%89%B5" title="አጠቃላይ አንጻራዊነት – Amharic" lang="am" hreflang="am" data-title="አጠቃላይ አንጻራዊነት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><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%B9%D8%A7%D9%85%D8%A9" title="النسبية العامة – Arabic" lang="ar" hreflang="ar" data-title="النسبية العامة" 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_cheneral" title="Relatividat cheneral – Aragonese" lang="an" hreflang="an" data-title="Relatividat cheneral" 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%B8%E0%A6%BE%E0%A6%A7%E0%A6%BE%E0%A7%B0%E0%A6%A3_%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" title="সাধাৰণ আপেক্ষিকতাবাদ – Assamese" lang="as" hreflang="as" data-title="সাধাৰণ আপেক্ষিকতাবাদ" 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/Relativid%C3%A1_xeneral" title="Relatividá xeneral – Asturian" lang="ast" hreflang="ast" data-title="Relatividá xeneral" 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_pav%E1%BA%BD" title="Mba'ekuaarã joguerahaviárava pavẽ – Guarani" lang="gn" hreflang="gn" data-title="Mba'ekuaarã joguerahaviárava pavẽ" 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/%C3%9Cmumi_nisbilik_n%C9%99z%C9%99riyy%C9%99si" title="Ümumi nisbilik nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Ümumi 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%B9%D8%A7%D9%85_%D9%86%DB%8C%D8%B3%D8%A8%DB%8C%D8%AA" title="عام نیسبیت – South Azerbaijani" lang="azb" hreflang="azb" data-title="عام 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_umum" title="Rélativitas umum – Balinese" lang="ban" hreflang="ban" data-title="Rélativitas umum" 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%B8%E0%A6%BE%E0%A6%A7%E0%A6%BE%E0%A6%B0%E0%A6%A3_%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="সাধারণ আপেক্ষিকতা – Bangla" lang="bn" hreflang="bn" data-title="সাধারণ আপেক্ষিকতা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/It-poa%E2%81%BF_siong-t%C3%B9i-s%C3%A8ng_l%C3%AD-l%C5%ABn" title="It-poaⁿ siong-tùi-sèng lí-lūn – Minnan" lang="nan" hreflang="nan" data-title="It-poaⁿ siong-tùi-sèng lí-lūn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ba.wikipedia.org/wiki/%D0%94%D3%A9%D0%B9%D3%A9%D0%BC_%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="Дөйөм сағыштырмалыҡ теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Дөйөм 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%90%D0%B3%D1%83%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="Агульная тэорыя адноснасці – Belarusian" lang="be" hreflang="be" data-title="Агульная 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%90%D0%B3%D1%83%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="Агульная тэорыя адноснасьці – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Агульная 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%B8%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%A8_%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="सधारन सापेक्षता – Bhojpuri" lang="bh" hreflang="bh" data-title="सधारन 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%9E%D0%B1%D1%89%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="Обща теория на относителността – Bulgarian" lang="bg" hreflang="bg" data-title="Обща 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/Oigmoane_Relativitetstheorie" title="Oigmoane Relativitetstheorie – Bavarian" lang="bar" hreflang="bar" data-title="Oigmoane 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/Op%C4%87a_teorija_relativnosti" title="Opća teorija relativnosti – Bosnian" lang="bs" hreflang="bs" data-title="Opća teorija relativnosti" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Relativelezh_hollek" title="Relativelezh hollek – Breton" lang="br" hreflang="br" data-title="Relativelezh hollek" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</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%BD_%D1%8E%D1%80%D1%8D%D0%BD%D1%85%D1%8B_%D0%BE%D0%BD%D0%BE%D0%BB" title="Харисангын юрэнхы онол – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Харисангын юрэнхы 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_general" title="Relativitat general – Catalan" lang="ca" hreflang="ca" data-title="Relativitat general" 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_%D0%BF%C4%95%D1%82%C4%95%D0%BC%C4%95%D1%88%D0%BB%D0%B5_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Танлаштарулăхăн пĕтĕмĕшле теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Танлаштарулăхăн пĕтĕмĕшле 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/Obecn%C3%A1_teorie_relativity" title="Obecná teorie relativity – Czech" lang="cs" hreflang="cs" data-title="Obecná 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_cyffredinol" title="Perthnasedd cyffredinol – Welsh" lang="cy" hreflang="cy" data-title="Perthnasedd cyffredinol" 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/Generel_relativitetsteori" title="Generel relativitetsteori – Danish" lang="da" hreflang="da" data-title="Generel relativitetsteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://de.wikipedia.org/wiki/Allgemeine_Relativit%C3%A4tstheorie" title="Allgemeine Relativitätstheorie – German" lang="de" hreflang="de" data-title="Allgemeine 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/%C3%9Cldrelatiivsusteooria" title="Üldrelatiivsusteooria – Estonian" lang="et" hreflang="et" data-title="Üldrelatiivsusteooria" 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%93%CE%B5%CE%BD%CE%B9%CE%BA%CE%AE_%CE%B8%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%84%CE%B7%CF%82_%CF%83%CF%87%CE%B5%CF%84%CE%B9%CE%BA%CF%8C%CF%84%CE%B7%CF%84%CE%B1%CF%82" title="Γενική θεωρία της σχετικότητας – Greek" lang="el" hreflang="el" data-title="Γενική θεωρία της σχετικότητας" 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/Relatividad_general" title="Relatividad general – Spanish" lang="es" hreflang="es" data-title="Relatividad general" 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/%C4%9Cenerala_teorio_de_relativeco" title="Ĝenerala teorio de relativeco – Esperanto" lang="eo" hreflang="eo" data-title="Ĝenerala 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_orokorra" title="Erlatibitate orokorra – Basque" lang="eu" hreflang="eu" data-title="Erlatibitate orokorra" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%DB%8C%D8%AA_%D8%B9%D8%A7%D9%85" title="نسبیت عام – Persian" lang="fa" hreflang="fa" data-title="نسبیت عام" 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/General_relativity" title="General relativity – Fiji Hindi" lang="hif" hreflang="hif" data-title="General 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_g%C3%A9n%C3%A9rale" title="Relativité générale – French" lang="fr" hreflang="fr" data-title="Relativité générale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teoiric_ghinear%C3%A1lta_na_coibhneasachta" title="Teoiric ghinearálta na coibhneasachta – Irish" lang="ga" hreflang="ga" data-title="Teoiric ghinearálta na coibhneasachta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Relatividade_xeral" title="Relatividade xeral – Galician" lang="gl" hreflang="gl" data-title="Relatividade xeral" 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/%EC%9D%BC%EB%B0%98_%EC%83%81%EB%8C%80%EC%84%B1%EC%9D%B4%EB%A1%A0" title="일반 상대성이론 – Korean" lang="ko" hreflang="ko" data-title="일반 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%A8%D5%B6%D5%A4%D5%B0%D5%A1%D5%B6%D5%B8%D6%82%D6%80_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հարաբերականության ընդհանուր տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Հարաբերականության ընդհանուր տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%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="सामान्य आपेक्षिकता – Hindi" lang="hi" hreflang="hi" data-title="सामान्य आपेक्षिकता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Op%C4%87a_teorija_relativnosti" title="Opća teorija relativnosti – Croatian" lang="hr" hreflang="hr" data-title="Opća 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/Generala_relativeso" title="Generala relativeso – Ido" lang="io" hreflang="io" data-title="Generala relativeso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Sapasap_a_relatibidad" title="Sapasap a relatibidad – Iloko" lang="ilo" hreflang="ilo" data-title="Sapasap a relatibidad" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Relativitas_umum" title="Relativitas umum – Indonesian" lang="id" hreflang="id" data-title="Relativitas umum" 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_general" title="Relativitate general – Interlingua" lang="ia" hreflang="ia" data-title="Relativitate general" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Relativit%C3%A1_general" title="Relativitá general – Interlingue" lang="ie" hreflang="ie" data-title="Relativitá general" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Almenna_afst%C3%A6%C3%B0iskenningin" title="Almenna afstæðiskenningin – Icelandic" lang="is" hreflang="is" data-title="Almenna 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_generale" title="Relatività generale – Italian" lang="it" hreflang="it" data-title="Relatività generale" 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%9B%D7%9C%D7%9C%D7%99%D7%AA" title="תורת היחסות הכללית – Hebrew" lang="he" hreflang="he" data-title="תורת היחסות הכללית" 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%96%E1%83%9D%E1%83%92%E1%83%90%E1%83%93%E1%83%98_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ფარდობითობის ზოგადი თეორია – Georgian" lang="ka" hreflang="ka" data-title="ფარდობითობის ზოგადი 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%96%D0%B0%D0%BB%D0%BF%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="Жалпы салыстырмалылық теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Жалпы салыстырмалылық теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A1%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%D1%83%D0%BD_%D0%B6%D0%B0%D0%BB%D0%BF%D1%8B_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Салыштырмалуулуктун жалпы теориясы – Kyrgyz" lang="ky" hreflang="ky" data-title="Салыштырмалуулуктун жалпы теориясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Relativitas_generalis" title="Relativitas generalis – Latin" lang="la" hreflang="la" data-title="Relativitas generalis" 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/Visp%C4%81r%C4%ABg%C4%81_relativit%C4%81tes_teorija" title="Vispārīgā relativitātes teorija – Latvian" lang="lv" hreflang="lv" data-title="Vispārīgā 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/Bendroji_reliatyvumo_teorija" title="Bendroji reliatyvumo teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Bendroji 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/%C3%81ltal%C3%A1nos_relativit%C3%A1selm%C3%A9let" title="Általános relativitáselmélet – Hungarian" lang="hu" hreflang="hu" data-title="Általános relativitáselmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D0%BF%D1%88%D1%82%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="Општа теорија за релативноста – Macedonian" lang="mk" hreflang="mk" data-title="Општа теорија за релативноста" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Relativit%C3%A9_general" title="Relativité general – Malagasy" lang="mg" hreflang="mg" data-title="Relativité general" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%BE%E0%B4%AE%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%AF_%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="സാമാന്യ ആപേക്ഷികതാസിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="സാമാന്യ ആപേക്ഷികതാസിദ്ധാന്തം" 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_%C4%A1enerali" title="Relattività ġenerali – Maltese" lang="mt" hreflang="mt" data-title="Relattività ġenerali" 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%B8%E0%A4%BE%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%A3_%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="साधारण सापेक्षता – Marathi" lang="mr" hreflang="mr" data-title="साधारण 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%B9%D8%A7%D9%85%D9%87" title="نسبيه عامه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="نسبيه عامه" 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_am" title="Kerelatifan am – Malay" lang="ms" hreflang="ms" data-title="Kerelatifan am" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9A%E1%80%B1%E1%80%98%E1%80%AF%E1%80%9A%E1%80%BB%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="ယေဘုယျနှိုင်းရသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="ယေဘုယျနှိုင်းရသီအိုရီ" 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/Algemene_relativiteitstheorie" title="Algemene relativiteitstheorie – Dutch" lang="nl" hreflang="nl" data-title="Algemene relativiteitstheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%AF_%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="सामान्य सापेक्षता – Newari" lang="new" hreflang="new" data-title="सामान्य सापेक्षता" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%80%E8%88%AC%E7%9B%B8%E5%AF%BE%E6%80%A7%E7%90%86%E8%AB%96" title="一般相対性理論 – Japanese" lang="ja" hreflang="ja" data-title="一般相対性理論" 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_generelle_relativitetsteorien" title="Den generelle relativitetsteorien – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Den generelle 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_generelle_relativitetsteorien" title="Den generelle relativitetsteorien – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Den generelle 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_generala" title="Relativitat generala – Occitan" lang="oc" hreflang="oc" data-title="Relativitat generala" 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%B8%E0%AC%BE%E0%AC%A7%E0%AC%BE%E0%AC%B0%E0%AC%A3_%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="ସାଧାରଣ ଆପେକ୍ଷିକ ତତ୍ତ୍ୱ – Odia" lang="or" hreflang="or" data-title="ସାଧାରଣ ଆପେକ୍ଷିକ ତତ୍ତ୍ୱ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Sadhata_waliigalaa" title="Sadhata waliigalaa – Oromo" lang="om" hreflang="om" data-title="Sadhata waliigalaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Umumiy_nisbiylik_nazariyasi" title="Umumiy nisbiylik nazariyasi – Uzbek" lang="uz" hreflang="uz" data-title="Umumiy 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%9C%E0%A8%A8%E0%A8%B0%E0%A8%B2_%E0%A8%B0%E0%A8%BF%E0%A8%B2%E0%A9%87%E0%A8%9F%E0%A9%80%E0%A8%B5%E0%A8%BF%E0%A8%9F%E0%A9%80" title="ਜਨਰਲ ਰਿਲੇਟੀਵਿਟੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਜਨਰਲ ਰਿਲੇਟੀਵਿਟੀ" 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%AC%D9%86%D8%B1%D9%84_%D8%B1%DB%8C%D9%84%DB%8C%D9%B9%D9%88%DB%8C%D9%B9%DB%8C" title="جنرل ریلیٹویٹی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="جنرل ریلیٹویٹی" 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/%D8%B9%D8%A7%D9%85_%D9%86%D8%B3%D8%A8%D9%8A%D8%AA" title="عام نسبيت – Pashto" lang="ps" hreflang="ps" data-title="عام نسبيت" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Allgemene_Relativit%C3%A4tstheorie" title="Allgemene Relativitätstheorie – Low German" lang="nds" hreflang="nds" data-title="Allgemene 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/Og%C3%B3lna_teoria_wzgl%C4%99dno%C5%9Bci" title="Ogólna teoria względności – Polish" lang="pl" hreflang="pl" data-title="Ogólna 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_geral" title="Relatividade geral – Portuguese" lang="pt" hreflang="pt" data-title="Relatividade geral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ro.wikipedia.org/wiki/Teoria_relativit%C4%83%C8%9Bii_generale" title="Teoria relativității generale – Romanian" lang="ro" hreflang="ro" data-title="Teoria relativității generale" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%89%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="Общая теория относительности – Russian" lang="ru" hreflang="ru" data-title="Общая 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/General_relativity" title="General relativity – Scots" lang="sco" hreflang="sco" data-title="General 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/Relativiteti_i_p%C3%ABrgjithsh%C3%ABm" title="Relativiteti i përgjithshëm – Albanian" lang="sq" hreflang="sq" data-title="Relativiteti i përgjithshëm" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B7%8F%E0%B6%B8%E0%B7%8F%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA_%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="සාමාන්‍ය සාපේක්ෂතාවාදය – Sinhala" lang="si" hreflang="si" data-title="සාමාන්‍ය 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/General_relativity" title="General relativity – Simple English" lang="en-simple" hreflang="en-simple" data-title="General 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/V%C5%A1eobecn%C3%A1_te%C3%B3ria_relativity" title="Všeobecná teória relativity – Slovak" lang="sk" hreflang="sk" data-title="Všeobecná 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/Splo%C5%A1na_teorija_relativnosti" title="Splošna teorija relativnosti – Slovenian" lang="sl" hreflang="sl" data-title="Splošna 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_%DA%AF%D8%B4%D8%AA%DB%8C" title="ڕێژەییی گشتی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕێژەییی گشتی" 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/%D0%9E%D0%BF%D1%88%D1%82%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%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%B8" title="Општа теорија релативности – Serbian" lang="sr" hreflang="sr" data-title="Општа теорија релативности" 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/Op%C5%A1ta_teorija_relativnosti" title="Opšta teorija relativnosti – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Opšta 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yleinen_suhteellisuusteoria" title="Yleinen suhteellisuusteoria – Finnish" lang="fi" hreflang="fi" data-title="Yleinen suhteellisuusteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://sv.wikipedia.org/wiki/Allm%C3%A4nna_relativitetsteorin" title="Allmänna relativitetsteorin – Swedish" lang="sv" hreflang="sv" data-title="Allmänna 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_pangkalahatang_relatibidad" title="Teorya ng pangkalahatang relatibidad – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng pangkalahatang 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%AA%E0%AF%8A%E0%AE%A4%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="பொதுச் சார்புக் கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="பொதுச் சார்புக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D1%83%D0%BC%D0%B8_%D1%87%D0%B0%D0%B3%D1%8B%D1%88%D1%82%D1%8B%D1%80%D0%BC%D0%B0%D0%BB%D1%8B%D0%BB%D1%8B%D0%BA_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D0%B5" title="Гомуми чагыштырмалылык теориясе – Tatar" lang="tt" hreflang="tt" data-title="Гомуми чагыштырмалылык теориясе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%BE%E0%B0%A7%E0%B0%BE%E0%B0%B0%E0%B0%A3_%E0%B0%B8%E0%B0%BE%E0%B0%AA%E0%B1%87%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B0%A4" title="సాధారణ సాపేక్షత – Telugu" lang="te" hreflang="te" data-title="సాధారణ సాపేక్షత" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</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%97%E0%B8%B1%E0%B9%88%E0%B8%A7%E0%B9%84%E0%B8%9B" title="ทฤษฎีสัมพัทธภาพทั่วไป – Thai" lang="th" hreflang="th" data-title="ทฤษฎีสัมพัทธภาพทั่วไป" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B7%D0%B0%D1%80%D0%B8%D1%8F%D0%B8_%D0%BD%D0%B8%D1%81%D0%B1%D0%B8%D1%8F%D1%82%D0%B8_%D0%BE%D0%BC" title="Назарияи нисбияти ом – Tajik" lang="tg" hreflang="tg" data-title="Назарияи нисбияти ом" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Genel_g%C3%B6relilik" title="Genel görelilik – Turkish" lang="tr" hreflang="tr" data-title="Genel 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%97%D0%B0%D0%B3%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="Загальна теорія відносності – Ukrainian" lang="uk" hreflang="uk" data-title="Загальна 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%B9%D9%85%D9%88%D9%85%DB%8C_%D8%A7%D8%B6%D8%A7%D9%81%DB%8C%D8%AA" title="عمومی اضافیت – Urdu" lang="ur" hreflang="ur" data-title="عمومی 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Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theory of gravitation as curved spacetime</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">For the graduate textbook by Robert Wald, see <a href="/wiki/General_Relativity_(book)" title="General Relativity (book)">General Relativity (book)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a more accessible and less technical introduction to this topic, see <a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction to general relativity</a>.</div> <figure typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/BBH_gravitational_lensing_of_gw150914.webm/266px--BBH_gravitational_lensing_of_gw150914.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="266" height="150" data-durationhint="35" data-mwtitle="BBH_gravitational_lensing_of_gw150914.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:BBH_gravitational_lensing_of_gw150914.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/a/a4/BBH_gravitational_lensing_of_gw150914.webm" type="video/webm; codecs="vp8"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/BBH_gravitational_lensing_of_gw150914.webm/BBH_gravitational_lensing_of_gw150914.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video><figcaption>Slow motion computer simulation of the black hole <a href="/wiki/Binary_system" title="Binary system">binary system</a> GW150914 as seen by a nearby observer, during 0.33 s of its final <a href="/wiki/Binary_black_hole#Inspiral" title="Binary black hole">inspiral</a>, <a href="/wiki/Binary_black_hole#Merger" title="Binary black hole">merge</a>, and <a href="/wiki/Binary_black_hole#Ringdown" title="Binary black hole">ringdown</a>. The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extreme <a href="/wiki/Gravitational_lens" title="Gravitational lens">gravitational lensing</a>, as <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> itself is distorted and dragged around by the rotating <a href="/wiki/Black_hole" title="Black hole">black holes</a>.<a href="#cite_note-SXSproject-1">[1]</a></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">General relativity</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Spacetime_lattice_analogy.svg" class="mw-file-description" title="Spacetime curvature schematic"><img alt="Spacetime curvature schematic" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/220px-Spacetime_lattice_analogy.svg.png" decoding="async" width="220" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/330px-Spacetime_lattice_analogy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/440px-Spacetime_lattice_analogy.svg.png 2x" data-file-width="1260" data-file-height="469" /></a><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</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="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</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/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><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/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</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/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</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="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</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/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</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/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</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/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</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/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</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/List_of_contributors_to_general_relativity" title="List of contributors to general relativity">others</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a> <a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></li> <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:General_relativity" title="Category:General relativity">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the <a href="/wiki/Differential_geometry" title="Differential geometry">geometric</a> <a href="/wiki/Scientific_theory" title="Scientific theory">theory</a> of <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> published by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> in 1915 and is the current description of gravitation in <a href="/wiki/Modern_physics" title="Modern physics">modern physics</a>. General <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</a> generalizes <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and refines <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>, providing a unified description of gravity as a geometric property of <a href="/wiki/Space" title="Space">space</a> and <a href="/wiki/Time_in_physics" title="Time in physics">time</a>, or <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">four-dimensional</a> <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. In particular, the <a href="/wiki/Curvature" title="Curvature">curvature</a> of spacetime is directly related to the <a href="/wiki/Energy" title="Energy">energy</a> and <a href="/wiki/Momentum" title="Momentum">momentum</a> of whatever present <a href="/wiki/Matter" title="Matter">matter</a> and <a href="/wiki/Radiation" title="Radiation">radiation</a>. The relation is specified by the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>, a system of second-order <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a>, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> in <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>. These predictions concern the passage of time, the <a href="/wiki/Geometry" title="Geometry">geometry</a> of space, the motion of bodies in <a href="/wiki/Free_fall" title="Free fall">free fall</a>, and the propagation of light, and include <a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">gravitational time dilation</a>, <a href="/wiki/Gravitational_lens" title="Gravitational lens">gravitational lensing</a>, the <a href="/wiki/Gravitational_redshift" title="Gravitational redshift">gravitational redshift</a> of light, the <a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro time delay</a> and <a href="/wiki/Gravitational_singularity" title="Gravitational singularity">singularities</a>/<a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">black holes</a>. So far, all <a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">tests of general relativity</a> have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>, thus leading to the discovery of the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a> and <a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">cosmic microwave background</a> radiation. Despite the introduction of a number of <a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">alternative theories</a>, general relativity continues to be the simplest theory consistent with <a href="/wiki/Experimental_data" title="Experimental data">experimental data</a>. Reconciliation of general relativity with the laws of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum physics</a> remains a problem, however, as there is a lack of a self-consistent theory of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>. It is not yet known how gravity can be <a href="/wiki/Theory_of_everything" title="Theory of everything">unified</a> with the three non-gravitational forces: <a href="/wiki/Strong_interaction" title="Strong interaction">strong</a>, <a href="/wiki/Weak_interaction" title="Weak interaction">weak</a> and <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetic</a>. Einstein's theory has <a href="/wiki/Astrophysical" class="mw-redirect" title="Astrophysical">astrophysical</a> implications, including the prediction of <a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">black holes</a>—regions of space in which space and time are distorted in such a way that nothing, not even <a href="/wiki/Photon" title="Photon">light</a>, can escape from them. Black holes are the end-state for <a href="/wiki/Massive_star" class="mw-redirect" title="Massive star">massive stars</a>. <a href="/wiki/Microquasar" title="Microquasar">Microquasars</a> and <a href="/wiki/Active_galactic_nucleus" title="Active galactic nucleus">active galactic nuclei</a> are believed to be <a href="/wiki/Stellar_black_hole" title="Stellar black hole">stellar black holes</a> and <a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">supermassive black holes</a>. It also predicts <a href="/wiki/Gravitational_lensing" class="mw-redirect" title="Gravitational lensing">gravitational lensing</a>, where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>, which have been <a href="/wiki/List_of_gravitational_wave_observations" title="List of gravitational wave observations">observed directly</a> by the physics collaboration <a href="/wiki/LIGO" title="LIGO">LIGO</a> and other observatories. In addition, general relativity has provided the base of <a href="/wiki/Physical_cosmology" title="Physical cosmology">cosmological</a> models of an <a href="/wiki/Metric_expansion_of_space" class="mw-redirect" title="Metric expansion of space">expanding universe</a>. Widely acknowledged as a theory of extraordinary <a href="/wiki/Mathematical_beauty" title="Mathematical beauty">beauty</a>, general relativity has often been described as the most beautiful of all existing physical theories.<a href="#cite_note-:0-2">[2]</a> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=1" title="Edit section: History">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/History_of_general_relativity" title="History of general relativity">History of general relativity</a> and <a href="/wiki/Classical_theories_of_gravitation" class="mw-redirect" title="Classical theories of gravitation">Classical theories of gravitation</a></div> <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>'s 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that <a href="/wiki/Gravitational_waves" class="mw-redirect" title="Gravitational waves">gravitational waves</a> propagate at the speed of light.<a href="#cite_note-3">[3]</a> Soon afterwards, Einstein started thinking about how to incorporate <a href="/wiki/Gravity" title="Gravity">gravity</a> into his relativistic framework. In 1907, beginning with a simple <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a> involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the <a href="/wiki/Prussian_Academy_of_Science" class="mw-redirect" title="Prussian Academy of Science">Prussian Academy of Science</a> in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity.<a href="#cite_note-4">[4]</a> These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present.<a href="#cite_note-5">[5]</a> A version of <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, called <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.<a href="#cite_note-6">[6]</a> This idea was pointed out by mathematician <a href="/wiki/Marcel_Grossmann" title="Marcel Grossmann">Marcel Grossmann</a> and published by Grossmann and Einstein in 1913.<a href="#cite_note-7">[7]</a> The Einstein field equations are <a href="/wiki/Nonlinear_differential_equation" class="mw-redirect" title="Nonlinear differential equation">nonlinear</a> and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> found the first non-trivial exact solution to the Einstein field equations, the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a>. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to <a href="/wiki/Electrical_charge" class="mw-redirect" title="Electrical charge">electrically charged</a> objects were taken, eventually resulting in the <a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström solution</a>, which is now associated with <a href="/wiki/Charged_black_hole" title="Charged black hole">electrically charged black holes</a>.<a href="#cite_note-8">[8]</a> In 1917, Einstein applied his theory to the <a href="/wiki/Universe" title="Universe">universe</a> as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a>—to match that observational presumption.<a href="#cite_note-9">[9]</a> By 1929, however, the work of <a href="/wiki/Edwin_Hubble" title="Edwin Hubble">Hubble</a> and others had shown that the universe is expanding. This is readily described by the expanding cosmological solutions found by <a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a> in 1922, which do not require a cosmological constant. <a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a> used these solutions to formulate the earliest version of the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a> models, in which the universe has evolved from an extremely hot and dense earlier state.<a href="#cite_note-10">[10]</a> Einstein later declared the cosmological constant the biggest blunder of his life.<a href="#cite_note-11">[11]</a> During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to <a href="/wiki/Newtonian_gravity" class="mw-redirect" title="Newtonian gravity">Newtonian gravity</a>, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the <a href="/wiki/Perihelion_precession_of_Mercury" class="mw-redirect" title="Perihelion precession of Mercury">anomalous perihelion advance</a> of the planet <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a> without any arbitrary parameters ("<a href="https://en.wiktionary.org/wiki/fudge_factor" class="extiw" title="wikt:fudge factor">fudge factors</a>"),<a href="#cite_note-12">[12]</a> and in 1919 an expedition led by <a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a> confirmed general relativity's prediction for the deflection of starlight by the Sun during the total <a href="/wiki/Solar_eclipse_of_29_May_1919" class="mw-redirect" title="Solar eclipse of 29 May 1919">solar eclipse of 29 May 1919</a>,<a href="#cite_note-13">[13]</a> instantly making Einstein famous.<a href="#cite_note-14">[14]</a> Yet the theory remained outside the mainstream of <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> and astrophysics until developments between approximately 1960 and 1975, now known as the <a href="/wiki/History_of_general_relativity#Golden_age" title="History of general relativity">golden age of general relativity</a>.<a href="#cite_note-15">[15]</a> Physicists began to understand the concept of a black hole, and to identify <a href="/wiki/Quasar" title="Quasar">quasars</a> as one of these objects' astrophysical manifestations.<a href="#cite_note-16">[16]</a> Ever more precise solar system tests confirmed the theory's predictive power,<a href="#cite_note-17">[17]</a> and relativistic cosmology also became amenable to direct observational tests.<a href="#cite_note-18">[18]</a> General relativity has acquired a reputation as a theory of extraordinary beauty.<a href="#cite_note-:0-2">[2]</a><a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> <a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Subrahmanyan Chandrasekhar</a> has noted that at multiple levels, general relativity exhibits what <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a> has termed a "strangeness in the proportion" (i.e. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory.<a href="#cite_note-21">[21]</a> Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.<a href="#cite_note-22">[22]</a> In the preface to <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>, Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."<a href="#cite_note-23">[23]</a> <div class="mw-heading mw-heading2"><h2 id="From_classical_mechanics_to_general_relativity">From classical mechanics to general relativity</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=2" title="Edit section: From classical mechanics to general relativity">edit</a>]</div> General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> derivation of general relativity.<a href="#cite_note-24">[24]</a><a href="#cite_note-25">[25]</a> <div class="mw-heading mw-heading3"><h3 id="Geometry_of_Newtonian_gravity">Geometry of Newtonian gravity</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=3" title="Edit section: Geometry of Newtonian gravity">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Elevator_gravity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Elevator_gravity.svg/220px-Elevator_gravity.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Elevator_gravity.svg/330px-Elevator_gravity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Elevator_gravity.svg/440px-Elevator_gravity.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity on the surface of the Earth).</figcaption></figure> At the base of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> is the notion that a <a href="/wiki/Physical_body" class="mw-redirect" title="Physical body">body</a>'s motion can be described as a combination of free (or <a href="/wiki/Inertia" title="Inertia">inertial</a>) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">law of motion</a>, which states that the net <a href="/wiki/Force" title="Force">force</a> acting on a body is equal to that body's (inertial) <a href="/wiki/Mass" title="Mass">mass</a> multiplied by its <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>.<a href="#cite_note-26">[26]</a> The preferred inertial motions are related to the geometry of space and time: in the standard <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frames</a> of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are <a href="/wiki/Geodesic" title="Geodesic">geodesics</a>, straight <a href="/wiki/World_lines" class="mw-redirect" title="World lines">world lines</a> in <a href="/wiki/Curved_spacetime" title="Curved spacetime">curved spacetime</a>.<a href="#cite_note-27">[27]</a> Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> or <a href="/wiki/Friction" title="Friction">friction</a>), can be used to define the geometry of space, as well as a time <a href="/wiki/Coordinate" class="mw-redirect" title="Coordinate">coordinate</a>. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of <a href="/wiki/Lor%C3%A1nd_E%C3%B6tv%C3%B6s" title="Loránd Eötvös">Eötvös</a> and its successors (see <a href="/wiki/E%C3%B6tv%C3%B6s_experiment" title="Eötvös experiment">Eötvös experiment</a>), there is a universality of free fall (also known as the weak <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>, or the universal equality of inertial and passive-gravitational mass): the trajectory of a <a href="/wiki/Test_body" class="mw-redirect" title="Test body">test body</a> in free fall depends only on its position and initial speed, but not on any of its material properties.<a href="#cite_note-28">[28]</a> A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.<a href="#cite_note-29">[29]</a> Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connection</a> which depends on the <a href="/wiki/Gradient" title="Gradient">gradient</a> of the <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a>. Space, in this construction, still has the ordinary <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not <a href="/wiki/Integrable_systems" class="mw-redirect" title="Integrable systems">integrable</a>. From this, one can deduce that spacetime is curved. The resulting <a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a> is a geometric formulation of Newtonian gravity using only <a href="/wiki/Covariance_and_contravariance_of_vectors#Informal_usage" title="Covariance and contravariance of vectors">covariant</a> concepts, i.e. a description which is valid in any desired coordinate system.<a href="#cite_note-30">[30]</a> In this geometric description, <a href="/wiki/Tidal_effect" class="mw-redirect" title="Tidal effect">tidal effects</a>—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.<a href="#cite_note-31">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Relativistic_generalization">Relativistic generalization</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=4" title="Edit section: Relativistic generalization">edit</a>]</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Light_cone.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Light_cone.svg/170px-Light_cone.svg.png" decoding="async" width="170" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Light_cone.svg/255px-Light_cone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Light_cone.svg/340px-Light_cone.svg.png 2x" data-file-width="333" data-file-height="515" /></a><figcaption><a href="/wiki/Light_cone" title="Light cone">Light cone</a></figcaption></figure> As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a <a href="/wiki/Limiting_case_(philosophy_of_science)" title="Limiting case (philosophy of science)">limiting case</a> of (special) relativistic mechanics.<a href="#cite_note-32">[32]</a> In the language of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>: where gravity can be neglected, physics is <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariant</a> as in special relativity rather than <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilei invariant</a> as in classical mechanics. (The defining symmetry of special relativity is the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>, which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, and with high-energy phenomena.<a href="#cite_note-33">[33]</a> With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each <a href="/wiki/Event_(relativity)" title="Event (relativity)">event</a> A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are <a href="/wiki/Frame_of_reference" title="Frame of reference">observer</a>-independent.<a href="#cite_note-34">[34]</a> In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a <a href="/wiki/Conformal_structure" class="mw-redirect" title="Conformal structure">conformal structure</a><a href="#cite_note-35">[35]</a> or conformal geometry. Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frames</a>. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight <a href="/wiki/Time-like" class="mw-redirect" title="Time-like">time-like</a> lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.<a href="#cite_note-36">[36]</a> A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of <a href="/wiki/Preferred_frame" title="Preferred frame">preferred frames</a>. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. <a href="#Gravitational_time_dilation_and_frequency_shift">below</a>). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.<a href="#cite_note-37">[37]</a> The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the <a href="/wiki/Equivalence_Principle#The_Einstein_equivalence_principle" class="mw-redirect" title="Equivalence Principle">Einstein equivalence principle</a>, a crucial guiding principle for generalizing special-relativistic physics to include gravity.<a href="#cite_note-38">[38]</a> The same experimental data shows that time as measured by clocks in a gravitational field—<a href="/wiki/Proper_time" title="Proper time">proper time</a>, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a> that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or <a href="/wiki/Pseudo-Riemannian" class="mw-redirect" title="Pseudo-Riemannian">pseudo-Riemannian</a> metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a>, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable <a href="/wiki/Local_reference_frame" title="Local reference frame">locally inertial coordinates</a>, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).<a href="#cite_note-39">[39]</a> <div class="mw-heading mw-heading3"><h3 id="Einstein's_equations">Einstein's equations</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=5" title="Edit section: Einstein's equations">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/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> and <a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></div> Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the <a href="/wiki/Energy%E2%80%93momentum_tensor" class="mw-redirect" title="Energy–momentum tensor">energy–momentum tensor</a>, which includes both <a href="/wiki/Energy_density" title="Energy density">energy</a> and momentum <a href="/wiki/Density" title="Density">densities</a> as well as <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>: <a href="/wiki/Pressure" title="Pressure">pressure</a> and shear.<a href="#cite_note-40">[40]</a> Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the <a href="/wiki/Field_equation" title="Field equation">field equation</a> for gravity relates this tensor and the <a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci tensor</a>, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>–momentum corresponds to the statement that the energy–momentum tensor is <a href="/wiki/Divergence" title="Divergence">divergence</a>-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-<a href="/wiki/Manifold" title="Manifold">manifold</a> counterparts, <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivatives</a> studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table">Einstein's field equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> </mstyle> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>κ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5278a80ae076d56ed32ff7cea0b45172014fccca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.349ex; height:5.176ex;" alt="{\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,}"> </div> On the left-hand side is the <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce4a5b59de7eda449c1f08ed7a84ae5de88884a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.921ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }}">, which is symmetric and a specific divergence-free combination of the Ricci tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ee22d1a052bee0115efb8b5ffdaf10b04e42aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.859ex; height:2.843ex;" alt="{\displaystyle R_{\mu \nu }}"> and the metric. In particular, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=g^{\mu \nu }R_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=g^{\mu \nu }R_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f33af69081f0596df0623dbe7fe6ab2f0aa21b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.934ex; height:3.009ex;" alt="{\displaystyle R=g^{\mu \nu }R_{\mu \nu }}"></dd></dl> is the curvature scalar. The Ricci tensor itself is related to the more general <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a> as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>α</mi> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38523c5a829f82fe2996b348aff739baa9cb0c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.799ex; height:3.009ex;" alt="{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.}"></dd></dl> On the right-hand side, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> is a constant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463ab8cef859ece28e33b8460ebd4a6699834dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.452ex; height:2.843ex;" alt="{\displaystyle T_{\mu \nu }}"> is the energy–momentum tensor. All tensors are written in <a href="/wiki/Abstract_index_notation" title="Abstract index notation">abstract index notation</a>.<a href="#cite_note-41">[41]</a> Matching the theory's prediction to observational results for <a href="/wiki/Planet" title="Planet">planetary</a> <a href="/wiki/Orbit" title="Orbit">orbits</a> or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> is found to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \kappa ={\frac {8\pi G}{c^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \kappa ={\frac {8\pi G}{c^{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dda558b0fd8c094978203f4279a3a8b127734f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.329ex; height:4.009ex;" alt="{\textstyle \kappa ={\frac {8\pi G}{c^{4}}}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is the <a href="/wiki/Newtonian_constant_of_gravitation" class="mw-redirect" title="Newtonian constant of gravitation">Newtonian constant of gravitation</a> and <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}"> the speed of light in vacuum.<a href="#cite_note-42">[42]</a> When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mu \nu }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f767d26db25b97adb1a1bb7264bac5df6653f344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.766ex; height:2.843ex;" alt="{\displaystyle R_{\mu \nu }=0.}"></dd></dl> In general relativity, the <a href="/wiki/World_line" title="World line">world line</a> of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The <a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">geodesic equation</a> is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f8e1493ceea6bf01d246b6118cc2c3a928d592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.534ex; height:6.009ex;" alt="{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,}"></dd></dl> where <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/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is a scalar parameter of motion (e.g. the <a href="/wiki/Proper_time" title="Proper time">proper time</a>), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67659b8257da8da050c9c2e3fa6b26fa8ea28216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.902ex; height:3.009ex;" alt="{\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }}"> are <a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a> (sometimes called the <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a> coefficients or <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a> coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the <a href="/wiki/Summation_convention" class="mw-redirect" title="Summation convention">summation convention</a> is used for repeated indices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> and <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 }">. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> which likewise provide formulae for the acceleration of a particle. This equation of motion employs the <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a>, meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a <a href="/wiki/Test_particle" title="Test particle">test particle</a> whose motion is described by the geodesic equation. <div class="mw-heading mw-heading3"><h3 id="Total_force_in_general_relativity">Total force in general relativity</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=6" title="Edit section: Total force in general relativity">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/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem in general relativity</a></div> In general relativity, the effective <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a> of an object of mass m revolving around a massive central body M is given by<a href="#cite_note-43">[43]</a><a href="#cite_note-44">[44]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d170a889110dfde67f6205eba7a128f92eb2503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.199ex; height:6.009ex;" alt="{\displaystyle U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}}"></dd></dl> A conservative total <a href="/wiki/Force" title="Force">force</a> can then be obtained as its <a href="/wiki/Force#Potential_energy" title="Force">negative gradient</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>G</mi> <mi>M</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed629da3fa58eb1ecd154750f469514153f9a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.106ex; height:6.009ex;" alt="{\displaystyle F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}}"></dd></dl> where L is the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. The first term represents the <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">force of Newtonian gravity</a>, which is described by the inverse-square law. The second term represents the <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a> in the circular motion. The third term represents the relativistic effect. <div class="mw-heading mw-heading3"><h3 id="Alternatives_to_general_relativity">Alternatives to general relativity</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=7" title="Edit section: Alternatives to general 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/Alternatives_to_general_relativity" title="Alternatives to general relativity">Alternatives to general relativity</a></div> There are <a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">alternatives to general relativity</a> built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are <a href="/wiki/Whitehead%27s_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead's theory</a>, <a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a>, <a href="/wiki/Teleparallelism" title="Teleparallelism">teleparallelism</a>, <a href="/wiki/F(R)_gravity" title="F(R) gravity">f(R) gravity</a> and <a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a>.<a href="#cite_note-45">[45]</a> <div class="mw-heading mw-heading2"><h2 id="Definition_and_basic_applications">Definition and basic applications</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=8" title="Edit section: Definition and basic applications">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/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a> and <a href="/wiki/Physical_theories_modified_by_general_relativity" title="Physical theories modified by general relativity">Physical theories modified by general relativity</a></div> The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building. <div class="mw-heading mw-heading3"><h3 id="Definition_and_basic_properties">Definition and basic properties</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=9" title="Edit section: Definition and basic properties">edit</a>]</div> General relativity is a <a href="/wiki/Metric_(general_relativity)" class="mw-redirect" title="Metric (general relativity)">metric</a> theory of gravitation. At its core are <a href="/wiki/Einstein%27s_equations" class="mw-redirect" title="Einstein's equations">Einstein's equations</a>, which describe the relation between the geometry of a four-dimensional <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> representing spacetime, and the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">energy–momentum</a> contained in that spacetime.<a href="#cite_note-46">[46]</a> Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as <a href="/wiki/Free-fall" class="mw-redirect" title="Free-fall">free-fall</a>, orbital motion, and <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a> <a href="/wiki/Trajectory" title="Trajectory">trajectories</a>), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.<a href="#cite_note-47">[47]</a> The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a>, spacetime tells matter how to move; matter tells spacetime how to curve.<a href="#cite_note-48">[48]</a> While general relativity replaces the <a href="/wiki/Scalar_field" title="Scalar field">scalar</a> gravitational potential of classical physics by a symmetric <a href="/wiki/Tensor#As_multidimensional_arrays" title="Tensor">rank</a>-two <a href="/wiki/Tensor" title="Tensor">tensor</a>, the latter reduces to the former in certain <a href="/wiki/Correspondence_principle#Other_scientific_theories" title="Correspondence principle">limiting cases</a>. For <a href="/wiki/Weak-field_approximation" class="mw-redirect" title="Weak-field approximation">weak gravitational fields</a> and <a href="/wiki/Slow-motion_approximation" class="mw-redirect" title="Slow-motion approximation">slow speed</a> relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.<a href="#cite_note-49">[49]</a> As it is constructed using tensors, general relativity exhibits <a href="/wiki/General_covariance" title="General covariance">general covariance</a>: its laws—and further laws formulated within the general relativistic framework—take on the same form in all <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate systems</a>.<a href="#cite_note-50">[50]</a> Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is <a href="/wiki/Background_independence" title="Background independence">background independent</a>. It thus satisfies a more stringent <a href="/wiki/General_principle_of_relativity" class="mw-redirect" title="General principle of relativity">general principle of relativity</a>, namely that the <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">laws of physics</a> are the same for all observers.<a href="#cite_note-51">[51]</a> <a href="/wiki/Local_spacetime_structure" class="mw-redirect" title="Local spacetime structure">Locally</a>, as expressed in the equivalence principle, spacetime is <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowskian</a>, and the laws of physics exhibit <a href="/wiki/Local_Lorentz_invariance" class="mw-redirect" title="Local Lorentz invariance">local Lorentz invariance</a>.<a href="#cite_note-52">[52]</a> <div class="mw-heading mw-heading3"><h3 id="Model-building">Model-building</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=10" title="Edit section: Model-building">edit</a>]</div> The core concept of general-relativistic model-building is that of a <a href="/wiki/Solutions_of_the_Einstein_field_equations" title="Solutions of the Einstein field equations">solution of Einstein's equations</a>. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-<a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.<a href="#cite_note-53">[53]</a> Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.<a href="#cite_note-54">[54]</a> Nevertheless, a number of <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solutions</a> are known, although only a few have direct physical applications.<a href="#cite_note-55">[55]</a> The best-known exact solutions, and also those most interesting from a physics point of view, are the <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a>, the <a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_solution" class="mw-redirect" title="Reissner–Nordström solution">Reissner–Nordström solution</a> and the <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metric</a>, each corresponding to a certain type of black hole in an otherwise empty universe,<a href="#cite_note-56">[56]</a> and the <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> and <a href="/wiki/De_Sitter_universe" title="De Sitter universe">de Sitter universes</a>, each describing an expanding cosmos.<a href="#cite_note-57">[57]</a> Exact solutions of great theoretical interest include the <a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel universe</a> (which opens up the intriguing possibility of <a href="/wiki/Time_travel" title="Time travel">time travel</a> in curved spacetimes), the <a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT solution</a> (a model universe that is <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneous</a>, but <a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a>), and <a href="/wiki/Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a> (which has recently come to prominence in the context of what is called the <a href="/wiki/Maldacena_conjecture" class="mw-redirect" title="Maldacena conjecture">Maldacena conjecture</a>).<a href="#cite_note-58">[58]</a> Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a> on a computer, or by considering small perturbations of exact solutions. In the field of <a href="/wiki/Numerical_relativity" title="Numerical relativity">numerical relativity</a>, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.<a href="#cite_note-59">[59]</a> In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as <a href="/wiki/Naked_singularity" title="Naked singularity">naked singularities</a>. Approximate solutions may also be found by <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theories</a> such as <a href="/wiki/Linearized_gravity" title="Linearized gravity">linearized gravity</a><a href="#cite_note-60">[60]</a> and its generalization, the <a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a>, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.<a href="#cite_note-61">[61]</a> An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.<a href="#cite_note-62">[62]</a> <div class="mw-heading mw-heading2"><h2 id="Consequences_of_Einstein's_theory">Consequences of Einstein's theory</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=11" title="Edit section: Consequences of Einstein's theory">edit</a>]</div> General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. <div class="mw-heading mw-heading3"><h3 id="Gravitational_time_dilation_and_frequency_shift">Gravitational time dilation and frequency shift</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=12" title="Edit section: Gravitational time dilation and frequency shift">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/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravitational_red-shifting.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitational_red-shifting.png/220px-Gravitational_red-shifting.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitational_red-shifting.png/330px-Gravitational_red-shifting.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitational_red-shifting.png/440px-Gravitational_red-shifting.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body</figcaption></figure> Assuming that the equivalence principle holds,<a href="#cite_note-63">[63]</a> gravity influences the passage of time. Light sent down into a <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravity well</a> is <a href="/wiki/Blueshift" class="mw-redirect" title="Blueshift">blueshifted</a>, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is <a href="/wiki/Redshift" title="Redshift">redshifted</a>; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.<a href="#cite_note-64">[64]</a> Gravitational redshift has been measured in the laboratory<a href="#cite_note-65">[65]</a> and using astronomical observations.<a href="#cite_note-66">[66]</a> Gravitational time dilation in the Earth's gravitational field has been measured numerous times using <a href="/wiki/Atomic_clocks" class="mw-redirect" title="Atomic clocks">atomic clocks</a>,<a href="#cite_note-67">[67]</a> while ongoing validation is provided as a side effect of the operation of the <a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Positioning System</a> (GPS).<a href="#cite_note-68">[68]</a> Tests in stronger gravitational fields are provided by the observation of <a href="/wiki/Binary_pulsar" title="Binary pulsar">binary pulsars</a>.<a href="#cite_note-69">[69]</a> All results are in agreement with general relativity.<a href="#cite_note-70">[70]</a> However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.<a href="#cite_note-71">[71]</a> <div class="mw-heading mw-heading3"><h3 id="Light_deflection_and_gravitational_time_delay">Light deflection and gravitational time delay</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=13" title="Edit section: Light deflection and gravitational time delay">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/Schwarzschild_geodesics" title="Schwarzschild geodesics">Schwarzschild geodesics</a>, <a href="/wiki/Kepler_problem_in_general_relativity" class="mw-redirect" title="Kepler problem in general relativity">Kepler problem in general relativity</a>, <a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lens</a>, and <a href="/wiki/Shapiro_delay" class="mw-redirect" title="Shapiro delay">Shapiro delay</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Light_deflection.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Light_deflection.png/170px-Light_deflection.png" decoding="async" width="170" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Light_deflection.png/255px-Light_deflection.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c2/Light_deflection.png 2x" data-file-width="300" data-file-height="352" /></a><figcaption>Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)</figcaption></figure> General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the <a href="/wiki/Sun" title="Sun">Sun</a>.<a href="#cite_note-72">[72]</a> This and related predictions follow from the fact that light follows what is called a light-like or <a href="/wiki/Geodesic_(general_relativity)" class="mw-redirect" title="Geodesic (general relativity)">null geodesic</a>—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariance</a> of lightspeed in special relativity.<a href="#cite_note-73">[73]</a> As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),<a href="#cite_note-74">[74]</a> several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,<a href="#cite_note-75">[75]</a> the angle of deflection resulting from such calculations is only half the value given by general relativity.<a href="#cite_note-76">[76]</a> Closely related to light deflection is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.<a href="#cite_note-77">[77]</a> In the <a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized post-Newtonian formalism</a> (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.<a href="#cite_note-78">[78]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Gravitational_waves">Gravitational waves</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=14" title="Edit section: Gravitational waves">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/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravwav.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravwav.gif/220px-Gravwav.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5c/Gravwav.gif 1.5x" data-file-width="250" data-file-height="250" /></a><figcaption>Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave</figcaption></figure> Predicted in 1916<a href="#cite_note-79">[79]</a><a href="#cite_note-80">[80]</a> by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a>. On 11 February 2016, the Advanced LIGO team announced that they had <a href="/wiki/Gravitational_wave_observation" class="mw-redirect" title="Gravitational wave observation">directly detected gravitational waves</a> from a <a href="/wiki/Binary_black_hole" title="Binary black hole">pair</a> of black holes <a href="/wiki/Stellar_collision" title="Stellar collision">merging</a>.<a href="#cite_note-Discovery_2016-81">[81]</a><a href="#cite_note-Abbot-82">[82]</a><a href="#cite_note-NSF-83">[83]</a> The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).<a href="#cite_note-84">[84]</a> Since Einstein's equations are <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a>, arbitrarily strong gravitational waves do not obey <a href="/wiki/Linear_superposition" class="mw-redirect" title="Linear superposition">linear superposition</a>, making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>21</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-21}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6897cc272146f9756290c0ed5875bc68e49b3077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.48ex; height:2.676ex;" alt="{\displaystyle 10^{-21}}"> or less. Data analysis methods routinely make use of the fact that these linearized waves can be <a href="/wiki/Fourier_decomposition" class="mw-redirect" title="Fourier decomposition">Fourier decomposed</a>.<a href="#cite_note-85">[85]</a> Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space<a href="#cite_note-86">[86]</a> or <a href="/wiki/Gowdy_universe" class="mw-redirect" title="Gowdy universe">Gowdy universes</a>, varieties of an expanding cosmos filled with gravitational waves.<a href="#cite_note-87">[87]</a> But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.<a href="#cite_note-88">[88]</a> <div class="mw-heading mw-heading3"><h3 id="Orbital_effects_and_the_relativity_of_direction">Orbital effects and the relativity of direction</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=15" title="Edit section: Orbital effects and the relativity of direction">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/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem in general relativity</a></div> General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (<a href="/wiki/Precession" title="Precession">precession</a>) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction. <div class="mw-heading mw-heading4"><h4 id="Precession_of_apsides">Precession of apsides</h4>[<a href="/w/index.php?title=General_relativity&action=edit&section=16" title="Edit section: Precession of apsides">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relativistic_precession.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/230px-Relativistic_precession.svg.png" decoding="async" width="230" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/345px-Relativistic_precession.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/460px-Relativistic_precession.svg.png 2x" data-file-width="420" data-file-height="390" /></a><figcaption>Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Apsidal_precession" title="Apsidal precession">Apsidal precession</a></div> In general relativity, the <a href="/wiki/Apsis" title="Apsis">apsides</a> of any orbit (the point of the orbiting body's closest approach to the system's <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>) will <a href="/wiki/Apsidal_precession" title="Apsidal precession">precess</a>; the orbit is not an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>, but akin to an ellipse that rotates on its focus, resulting in a <a href="/wiki/Rose_(mathematics)" title="Rose (mathematics)">rose curve</a>-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a <a href="/wiki/Test_particle" title="Test particle">test particle</a>. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by <a href="/wiki/Urbain_Le_Verrier" title="Urbain Le Verrier">Urbain Le Verrier</a> in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.<a href="#cite_note-89">[89]</a> The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)<a href="#cite_note-90">[90]</a> or the much more general <a href="/wiki/Post-Newtonian_formalism" class="mw-redirect" title="Post-Newtonian formalism">post-Newtonian formalism</a>.<a href="#cite_note-91">[91]</a> It is due to the influence of gravity on the geometry of space and to the contribution of <a href="/wiki/Self-energy" title="Self-energy">self-energy</a> to a body's gravity (encoded in the <a href="/wiki/Nonlinearity" class="mw-redirect" title="Nonlinearity">nonlinearity</a> of Einstein's equations).<a href="#cite_note-92">[92]</a> Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),<a href="#cite_note-93">[93]</a> as well as in binary pulsar systems, where it is larger by five <a href="/wiki/Order_of_magnitude" title="Order of magnitude">orders of magnitude</a>.<a href="#cite_note-94">[94]</a> In general relativity the perihelion shift <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 }">, expressed in radians per revolution, is approximately given by<a href="#cite_note-FOOTNOTEDediuMagdalenaMartín-Vide2015[httpsbooksgooglecombooksidXmwiCwAAQBAJpgPA141_141]-95">[95]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>24</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>T</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> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58b27c973a8dcd18661d17dbd71493077b077ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.277ex; height:6.509ex;" alt="{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,}"></dd></dl> where: <ul><li><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}"> is the <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a></li> <li><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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is the <a href="/wiki/Orbital_period" title="Orbital period">orbital period</a></li> <li><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 speed of light in vacuum</li> <li><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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> is the <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Orbital_decay">Orbital decay</h4>[<a href="/w/index.php?title=General_relativity&action=edit&section=17" title="Edit section: Orbital decay">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PSRJ0737%E2%88%923039shift2021.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/PSRJ0737%E2%88%923039shift2021.png/180px-PSRJ0737%E2%88%923039shift2021.png" decoding="async" width="180" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/PSRJ0737%E2%88%923039shift2021.png/270px-PSRJ0737%E2%88%923039shift2021.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f3/PSRJ0737%E2%88%923039shift2021.png 2x" data-file-width="351" data-file-height="500" /></a><figcaption>Orbital decay for PSR J0737−3039: time shift (in <a href="/wiki/Second" title="Second">s</a>), tracked over 16 years (2021).<a href="#cite_note-:1-96">[96]</a></figcaption></figure> According to general relativity, a <a href="/wiki/Binary_system_(astronomy)" class="mw-redirect" title="Binary system (astronomy)">binary system</a> will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> or for ordinary <a href="/wiki/Double_star" title="Double star">double stars</a>, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a>, one of which is a <a href="/wiki/Pulsar" title="Pulsar">pulsar</a>: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.<a href="#cite_note-97">[97]</a> The first observation of a decrease in orbital period due to the emission of gravitational waves was made by <a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a> and <a href="/wiki/Joseph_Hooton_Taylor,_Jr." class="mw-redirect" title="Joseph Hooton Taylor, Jr.">Taylor</a>, using the binary pulsar <a href="/wiki/PSR1913%2B16" class="mw-redirect" title="PSR1913+16">PSR1913+16</a> they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a> in physics.<a href="#cite_note-98">[98]</a> Since then, several other binary pulsars have been found, in particular the double pulsar <a href="/wiki/PSR_J0737%E2%88%923039" title="PSR J0737−3039">PSR J0737−3039</a>, where both stars are pulsars<a href="#cite_note-99">[99]</a> and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.<a href="#cite_note-:1-96">[96]</a> <div class="mw-heading mw-heading4"><h4 id="Geodetic_precession_and_frame-dragging">Geodetic precession and frame-dragging</h4>[<a href="/w/index.php?title=General_relativity&action=edit&section=18" title="Edit section: Geodetic precession and frame-dragging">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/Geodetic_precession" class="mw-redirect" title="Geodetic precession">Geodetic precession</a> and <a href="/wiki/Frame_dragging" class="mw-redirect" title="Frame dragging">Frame dragging</a></div> Several relativistic effects are directly related to the relativity of direction.<a href="#cite_note-100">[100]</a> One is <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic precession</a>: the axis direction of a <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("<a href="/wiki/Parallel_transport" title="Parallel transport">parallel transport</a>").<a href="#cite_note-101">[101]</a> For the Moon–Earth system, this effect has been measured with the help of <a href="/wiki/Lunar_laser_ranging" class="mw-redirect" title="Lunar laser ranging">lunar laser ranging</a>.<a href="#cite_note-102">[102]</a> More recently, it has been measured for test masses aboard the satellite <a href="/wiki/Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> to a precision of better than 0.3%.<a href="#cite_note-103">[103]</a><a href="#cite_note-104">[104]</a> Near a rotating mass, there are gravitomagnetic or <a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for <a href="/wiki/Kerr_solution" class="mw-redirect" title="Kerr solution">rotating black holes</a> where, for any object entering a zone known as the <a href="/wiki/Ergosphere" title="Ergosphere">ergosphere</a>, rotation is inevitable.<a href="#cite_note-105">[105]</a> Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.<a href="#cite_note-106">[106]</a> Somewhat controversial tests have been performed using the <a href="/wiki/LAGEOS" title="LAGEOS">LAGEOS</a> satellites, confirming the relativistic prediction.<a href="#cite_note-107">[107]</a> Also the <a href="/wiki/Mars_Global_Surveyor" title="Mars Global Surveyor">Mars Global Surveyor</a> probe around Mars has been used.<a href="#cite_note-108">[108]</a> <div class="mw-heading mw-heading2"><h2 id="Astrophysical_applications">Astrophysical applications</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=19" title="Edit section: Astrophysical applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Gravitational_lensing">Gravitational lensing</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=20" title="Edit section: Gravitational lensing">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/Gravitational_lensing" class="mw-redirect" title="Gravitational lensing">Gravitational lensing</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Einstein_cross_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Einstein_cross_%28cropped%29.jpg/190px-Einstein_cross_%28cropped%29.jpg" decoding="async" width="190" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Einstein_cross_%28cropped%29.jpg/285px-Einstein_cross_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Einstein_cross_%28cropped%29.jpg/380px-Einstein_cross_%28cropped%29.jpg 2x" data-file-width="1600" data-file-height="1600" /></a><figcaption><a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a>: four images of the same astronomical object, produced by a gravitational lens</figcaption></figure> The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.<a href="#cite_note-109">[109]</a> Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an <a href="/wiki/Einstein_ring" title="Einstein ring">Einstein ring</a>, or partial rings called arcs.<a href="#cite_note-110">[110]</a> The <a href="/wiki/Twin_Quasar" title="Twin Quasar">earliest example</a> was discovered in 1979;<a href="#cite_note-111">[111]</a> since then, more than a hundred gravitational lenses have been observed.<a href="#cite_note-112">[112]</a> Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "<a href="/wiki/Microlensing" class="mw-redirect" title="Microlensing">microlensing</a> events" have been observed.<a href="#cite_note-113">[113]</a> Gravitational lensing has developed into a tool of <a href="/wiki/Observational_astronomy" title="Observational astronomy">observational astronomy</a>. It is used to detect the presence and distribution of <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a>, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the <a href="/wiki/Hubble_constant" class="mw-redirect" title="Hubble constant">Hubble constant</a>. Statistical evaluations of lensing data provide valuable insight into the structural evolution of <a href="/wiki/Galaxy" title="Galaxy">galaxies</a>.<a href="#cite_note-114">[114]</a> <div class="mw-heading mw-heading3"><h3 id="Gravitational-wave_astronomy">Gravitational-wave astronomy</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=21" title="Edit section: Gravitational-wave astronomy">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/Gravitational_wave" title="Gravitational wave">Gravitational wave</a> and <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">Gravitational-wave astronomy</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:LISA.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/LISA.jpg/180px-LISA.jpg" decoding="async" width="180" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/LISA.jpg/270px-LISA.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/LISA.jpg/360px-LISA.jpg 2x" data-file-width="450" data-file-height="579" /></a><figcaption>Artist's impression of the space-borne gravitational wave detector <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">LISA</a></figcaption></figure> Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see <a href="#Orbital_decay">Orbital decay</a>, above). Detection of these waves is a major goal of current relativity-related research.<a href="#cite_note-115">[115]</a> Several land-based <a href="/wiki/Gravitational_wave_detector" class="mw-redirect" title="Gravitational wave detector">gravitational wave detectors</a> are currently in operation, most notably the <a href="/wiki/Interferometric_gravitational_wave_detector" class="mw-redirect" title="Interferometric gravitational wave detector">interferometric detectors</a> <a href="/wiki/GEO_600" class="mw-redirect" title="GEO 600">GEO 600</a>, <a href="/wiki/LIGO" title="LIGO">LIGO</a> (two detectors), <a href="/wiki/TAMA_300" title="TAMA 300">TAMA 300</a> and <a href="/wiki/Virgo_interferometer" title="Virgo interferometer">VIRGO</a>.<a href="#cite_note-116">[116]</a> Various <a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a> are using <a href="/wiki/Millisecond_pulsar" title="Millisecond pulsar">millisecond pulsars</a> to detect gravitational waves in the 10−9 to 10−6 <a href="/wiki/Hertz" title="Hertz">hertz</a> frequency range, which originate from binary supermassive blackholes.<a href="#cite_note-117">[117]</a> A European space-based detector, <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">eLISA / NGO</a>, is currently under development,<a href="#cite_note-118">[118]</a> with a precursor mission (<a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a>) having launched in December 2015.<a href="#cite_note-119">[119]</a> Observations of gravitational waves promise to complement observations in the <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a>.<a href="#cite_note-120">[120]</a> They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of <a href="/wiki/Supernova" title="Supernova">supernova</a> implosions, and about processes in the very early universe, including the signature of certain types of hypothetical <a href="/wiki/Cosmic_string" title="Cosmic string">cosmic string</a>.<a href="#cite_note-121">[121]</a> In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.<a href="#cite_note-Discovery_2016-81">[81]</a><a href="#cite_note-Abbot-82">[82]</a><a href="#cite_note-NSF-83">[83]</a> <div class="mw-heading mw-heading3"><h3 id="Black_holes_and_other_compact_objects">Black holes and other compact objects</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=22" title="Edit section: Black holes and other compact objects">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/Black_hole" title="Black hole">Black hole</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Star_collapse_to_black_hole.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Star_collapse_to_black_hole.png/220px-Star_collapse_to_black_hole.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Star_collapse_to_black_hole.png/330px-Star_collapse_to_black_hole.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Star_collapse_to_black_hole.png/440px-Star_collapse_to_black_hole.png 2x" data-file-width="500" data-file-height="308" /></a><figcaption>Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves</figcaption></figure> Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of <a href="/wiki/Stellar_evolution" title="Stellar evolution">stellar evolution</a>, neutron stars of around 1.4 <a href="/wiki/Solar_mass" title="Solar mass">solar masses</a>, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars.<a href="#cite_note-122">[122]</a> Usually a galaxy has one <a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">supermassive black hole</a> with a few million to a few <a href="/wiki/1000000000_(number)" class="mw-redirect" title="1000000000 (number)">billion</a> solar masses in its center,<a href="#cite_note-123">[123]</a> and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.<a href="#cite_note-124">[124]</a> Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.<a href="#cite_note-125">[125]</a> <a href="/wiki/Accretion_(astrophysics)" title="Accretion (astrophysics)">Accretion</a>, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.<a href="#cite_note-126">[126]</a> In particular, accretion can lead to <a href="/wiki/Relativistic_jet" class="mw-redirect" title="Relativistic jet">relativistic jets</a>, focused beams of highly energetic particles that are being flung into space at almost light speed.<a href="#cite_note-127">[127]</a> General relativity plays a central role in modelling all these phenomena,<a href="#cite_note-128">[128]</a> and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.<a href="#cite_note-129">[129]</a> Black holes are also sought-after targets in the search for gravitational waves (cf. <a href="#Gravitational_waves">Gravitational waves</a>, above). Merging <a href="/wiki/Binary_black_hole" title="Binary black hole">black hole binaries</a> should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "<a href="/wiki/Standard_candle" class="mw-redirect" title="Standard candle">standard candle</a>" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.<a href="#cite_note-130">[130]</a> The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.<a href="#cite_note-131">[131]</a> <div class="mw-heading mw-heading3"><h3 id="Cosmology">Cosmology</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=23" title="Edit section: Cosmology">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/Physical_cosmology" title="Physical cosmology">Physical cosmology</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lensshoe_hubble.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Lensshoe_hubble.jpg/220px-Lensshoe_hubble.jpg" decoding="async" width="220" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Lensshoe_hubble.jpg/330px-Lensshoe_hubble.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Lensshoe_hubble.jpg/440px-Lensshoe_hubble.jpg 2x" data-file-width="3235" data-file-height="2572" /></a><figcaption>This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.</figcaption></figure> The current models of cosmology are based on <a href="/wiki/Einstein%27s_field_equations" class="mw-redirect" title="Einstein's field equations">Einstein's field equations</a>, which include the cosmological constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"> since it has important influence on the large-scale dynamics of the cosmos, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> </mstyle> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <mtext> </mtext> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90cd8fb6edf598f64c45bf2bfce258dce51de6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.385ex; height:5.676ex;" alt="{\displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf4140993a891f5782167dc8a0c236dc7667b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}"> is the spacetime metric.<a href="#cite_note-132">[132]</a> <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">Isotropic</a> and homogeneous solutions of these enhanced equations, the <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 solutions</a>,<a href="#cite_note-133">[133]</a> allow physicists to model a universe that has evolved over the past 14 <a href="/wiki/1000000000_(number)" class="mw-redirect" title="1000000000 (number)">billion</a> years from a hot, early Big Bang phase.<a href="#cite_note-134">[134]</a> Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,<a href="#cite_note-135">[135]</a> further observational data can be used to put the models to the test.<a href="#cite_note-136">[136]</a> Predictions, all successful, include the initial abundance of chemical elements formed in a period of <a href="/wiki/Big_Bang_nucleosynthesis" title="Big Bang nucleosynthesis">primordial nucleosynthesis</a>,<a href="#cite_note-137">[137]</a> the large-scale structure of the universe,<a href="#cite_note-138">[138]</a> and the existence and properties of a "<a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal</a> echo" from the early cosmos, the <a href="/wiki/Cosmic_background_radiation" title="Cosmic background radiation">cosmic background radiation</a>.<a href="#cite_note-139">[139]</a> Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.<a href="#cite_note-140">[140]</a> There is no generally accepted description of this new kind of matter, within the framework of known <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a><a href="#cite_note-141">[141]</a> or otherwise.<a href="#cite_note-142">[142]</a> Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a>, known as <a href="/wiki/Dark_energy" title="Dark energy">dark energy</a>, the nature of which remains unclear.<a href="#cite_note-143">[143]</a> An <a href="/wiki/Cosmic_inflation" title="Cosmic inflation">inflationary phase</a>,<a href="#cite_note-144">[144]</a> an additional phase of strongly accelerated expansion at cosmic times of around 10−33 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.<a href="#cite_note-145">[145]</a> Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.<a href="#cite_note-146">[146]</a> However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.<a href="#cite_note-147">[147]</a> An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang <a href="/wiki/Gravitational_singularity" title="Gravitational singularity">singularity</a>. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed<a href="#cite_note-148">[148]</a> (cf. the section on <a href="#Quantum_gravity">quantum gravity</a>, below). <div class="mw-heading mw-heading3"><h3 id="Exotic_solutions:_time_travel,_warp_drives">Exotic solutions: time travel, warp drives</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=24" title="Edit section: Exotic solutions: time travel, warp drives">edit</a>]</div> <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> showed<a href="#cite_note-149">[149]</a> that solutions to Einstein's equations exist that contain <a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">closed timelike curves</a> (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the <a href="/wiki/Tipler_cylinder" title="Tipler cylinder">Tipler cylinder</a> and <a href="/wiki/Wormhole#Traversable_wormholes" title="Wormhole">traversable wormholes</a>. <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a> introduced <a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">chronology protection conjecture</a>, which is an assumption beyond those of standard general relativity to prevent <a href="/wiki/Time_travel" title="Time travel">time travel</a>. Some <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solutions in general relativity</a> such as <a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre drive</a> present examples of <a href="/wiki/Warp_drive" title="Warp drive">warp drive</a> but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability. <a href="#cite_note-150">[150]</a> <div class="mw-heading mw-heading2"><h2 id="Advanced_concepts">Advanced concepts</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=25" title="Edit section: Advanced concepts">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Asymptotic_symmetries">Asymptotic symmetries</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=26" title="Edit section: Asymptotic symmetries">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/Bondi%E2%80%93Metzner%E2%80%93Sachs_group" title="Bondi–Metzner–Sachs group">Bondi–Metzner–Sachs group</a></div> The spacetime symmetry group for <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> is the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries, if any, might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group. In 1962 <a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Hermann Bondi</a>, M. G. van der Burg, A. W. Metzner<a href="#cite_note-bondi_etal_1962-151">[151]</a> and <a href="/wiki/Rainer_K._Sachs" title="Rainer K. Sachs">Rainer K. Sachs</a><a href="#cite_note-sachs1962-152">[152]</a> addressed this <a href="/wiki/Bondi%E2%80%93Metzner%E2%80%93Sachs_group" title="Bondi–Metzner–Sachs group">asymptotic symmetry</a> problem in order to investigate the flow of energy at infinity due to propagating <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal <a href="/wiki/Soft_graviton_theorem" title="Soft graviton theorem">soft graviton theorem</a> in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.<a href="#cite_note-strominger2017-153">[153]</a> <div class="mw-heading mw-heading3"><h3 id="Causal_structure_and_global_geometry">Causal structure and global geometry</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=27" title="Edit section: Causal structure and global geometry">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/Causal_structure" title="Causal structure">Causal structure</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Penrose.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Penrose.svg/220px-Penrose.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Penrose.svg/330px-Penrose.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Penrose.svg/440px-Penrose.svg.png 2x" data-file-width="651" data-file-height="588" /></a><figcaption>Penrose–Carter diagram of an infinite <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski universe</a></figcaption></figure> In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (<a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">null geodesics</a>) yields key information about the spacetime's causal structure. This structure can be displayed using <a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose–Carter diagrams</a> in which infinitely large regions of space and infinite time intervals are shrunk ("<a href="/wiki/Compactification_(mathematics)" title="Compactification (mathematics)">compactified</a>") so as to fit onto a finite map, while light still travels along diagonals as in standard <a href="/wiki/Spacetime_diagram" title="Spacetime diagram">spacetime diagrams</a>.<a href="#cite_note-154">[154]</a> Aware of the importance of causal structure, <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> and others developed what is known as <a href="/wiki/Spacetime_topology" title="Spacetime topology">global geometry</a>. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the <a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a>, and additional non-specific assumptions about the nature of matter (usually in the form of <a href="/wiki/Energy_conditions" class="mw-redirect" title="Energy conditions">energy conditions</a>) are used to derive general results.<a href="#cite_note-155">[155]</a> <div class="mw-heading mw-heading3"><h3 id="Horizons">Horizons</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=28" title="Edit section: Horizons">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/Horizon_(general_relativity)" title="Horizon (general relativity)">Horizon (general relativity)</a>, <a href="/wiki/No_hair_theorem" class="mw-redirect" title="No hair theorem">No hair theorem</a>, and <a href="/wiki/Black_hole_mechanics" class="mw-redirect" title="Black hole mechanics">Black hole mechanics</a></div> Using global geometry, some spacetimes can be shown to contain boundaries called <a href="/wiki/Event_horizon" title="Event horizon">horizons</a>, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the <a href="/wiki/Hoop_conjecture" title="Hoop conjecture">hoop conjecture</a>, the relevant length scale is the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a><a href="#cite_note-156">[156]</a>), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.<a href="#cite_note-157">[157]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ergosphere_of_a_rotating_black_hole.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Ergosphere_of_a_rotating_black_hole.svg/220px-Ergosphere_of_a_rotating_black_hole.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Ergosphere_of_a_rotating_black_hole.svg/330px-Ergosphere_of_a_rotating_black_hole.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Ergosphere_of_a_rotating_black_hole.svg/440px-Ergosphere_of_a_rotating_black_hole.svg.png 2x" data-file-width="1000" data-file-height="700" /></a><figcaption>The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole</figcaption></figure> Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a <a href="/wiki/Static_spacetime" title="Static spacetime">static</a> black hole) and the axisymmetric <a href="/wiki/Kerr_solution" class="mw-redirect" title="Kerr solution">Kerr solution</a> (used to describe a rotating, <a href="/wiki/Stationary_spacetime" title="Stationary spacetime">stationary</a> black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, <a href="/wiki/Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a>, <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, and location at a specified time. This is stated by the <a href="/wiki/No_hair_theorem" class="mw-redirect" title="No hair theorem">black hole uniqueness theorem</a>: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.<a href="#cite_note-158">[158]</a> Even more remarkably, there is a general set of laws known as <a href="/wiki/Black_hole_mechanics" class="mw-redirect" title="Black hole mechanics">black hole mechanics</a>, which is analogous to the <a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">laws of thermodynamics</a>. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the <a href="/wiki/Entropy" title="Entropy">entropy</a> of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the <a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a>).<a href="#cite_note-159">[159]</a> There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.<a href="#cite_note-160">[160]</a> This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for the black hole area to decrease as long as other processes ensure that entropy increases overall. As thermodynamical objects with nonzero temperature, black holes should emit <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a>. Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in <a href="/wiki/Planck%27s_law" title="Planck's law">Planck's law</a>. This radiation is known as <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a> (cf. the <a href="#Quantum_field_theory_in_curved_spacetime">quantum theory section</a>, below).<a href="#cite_note-161">[161]</a> There are many other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("<a href="/wiki/Particle_horizon" title="Particle horizon">particle horizon</a>"), and some regions of the future cannot be influenced (event horizon).<a href="#cite_note-162">[162]</a> Even in flat Minkowski space, when described by an accelerated observer (<a href="/wiki/Rindler_space" class="mw-redirect" title="Rindler space">Rindler space</a>), there will be horizons associated with a semiclassical radiation known as <a href="/wiki/Unruh_effect" title="Unruh effect">Unruh radiation</a>.<a href="#cite_note-163">[163]</a> <div class="mw-heading mw-heading3"><h3 id="Singularities">Singularities</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=29" title="Edit section: Singularities">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/Spacetime_singularity" class="mw-redirect" title="Spacetime singularity">Spacetime singularity</a></div> Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as <a href="/wiki/Spacetime_singularity" class="mw-redirect" title="Spacetime singularity">spacetime singularities</a>, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the <a href="/wiki/Ricci_scalar" class="mw-redirect" title="Ricci scalar">Ricci scalar</a>, take on infinite values.<a href="#cite_note-164">[164]</a> Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,<a href="#cite_note-165">[165]</a> or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.<a href="#cite_note-166">[166]</a> The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (<a href="/wiki/Big_Crunch" title="Big Crunch">Big Crunch</a>) as well.<a href="#cite_note-167">[167]</a> Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.<a href="#cite_note-168">[168]</a> The famous <a href="/wiki/Singularity_theorems" class="mw-redirect" title="Singularity theorems">singularity theorems</a>, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage<a href="#cite_note-169">[169]</a> and also at the beginning of a wide class of expanding universes.<a href="#cite_note-170">[170]</a> However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the <a href="/wiki/BKL_singularity" title="BKL singularity">BKL conjecture</a>).<a href="#cite_note-171">[171]</a> The <a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">cosmic censorship hypothesis</a> states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.<a href="#cite_note-172">[172]</a> <div class="mw-heading mw-heading3"><h3 id="Evolution_equations">Evolution equations</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=30" title="Edit section: Evolution equations">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/Initial_value_formulation_(general_relativity)" title="Initial value formulation (general relativity)">Initial value formulation (general relativity)</a></div> Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the <a href="/wiki/Time_evolution" title="Time evolution">time evolution</a> of the metric tensor. It must be combined with a <a href="/wiki/Coordinate_condition" class="mw-redirect" title="Coordinate condition">coordinate condition</a>, which is analogous to <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a> in other field theories.<a href="#cite_note-173">[173]</a> To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the <a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a>.<a href="#cite_note-174">[174]</a> These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always <a href="/wiki/Existence_theorem" title="Existence theorem">exist</a>, and are uniquely defined, once suitable initial conditions have been specified.<a href="#cite_note-175">[175]</a> Such formulations of Einstein's field equations are the basis of numerical relativity.<a href="#cite_note-176">[176]</a> <div class="mw-heading mw-heading3"><h3 id="Global_and_quasi-local_quantities">Global and quasi-local quantities</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=31" title="Edit section: Global and quasi-local quantities">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_in_general_relativity" title="Mass in general relativity">Mass in general relativity</a></div> The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.<a href="#cite_note-177">[177]</a> Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (<a href="/wiki/ADM_mass" class="mw-redirect" title="ADM mass">ADM mass</a>)<a href="#cite_note-178">[178]</a> or suitable symmetries (<a href="/wiki/Komar_mass" title="Komar mass">Komar mass</a>).<a href="#cite_note-179">[179]</a> If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the <a href="/wiki/Mass_in_general_relativity#ADM_and_Bondi_masses_in_asymptotically_flat_space-times" title="Mass in general relativity">Bondi mass</a> at null infinity.<a href="#cite_note-180">[180]</a> Just as in <a href="/wiki/Physics_in_the_Classical_Limit" class="mw-redirect" title="Physics in the Classical Limit">classical physics</a>, it can be shown that these masses are positive.<a href="#cite_note-181">[181]</a> Corresponding global definitions exist for momentum and angular momentum.<a href="#cite_note-182">[182]</a> There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about <a href="/wiki/Isolated_system" title="Isolated system">isolated systems</a>, such as a more precise formulation of the hoop conjecture.<a href="#cite_note-183">[183]</a> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_quantum_theory">Relationship with quantum theory</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=32" title="Edit section: Relationship with quantum theory">edit</a>]</div> If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, would be the other.<a href="#cite_note-184">[184]</a> However, how to reconcile quantum theory with general relativity is still an open question. <div class="mw-heading mw-heading3"><h3 id="Quantum_field_theory_in_curved_spacetime">Quantum field theory in curved spacetime</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=33" title="Edit section: Quantum field theory in curved 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/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">Quantum field theory in curved spacetime</a></div> Ordinary <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theories</a>, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.<a href="#cite_note-185">[185]</a> In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.<a href="#cite_note-186">[186]</a> Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a> leading to the possibility that they <a href="/wiki/Black_hole_evaporation" class="mw-redirect" title="Black hole evaporation">evaporate</a> over time.<a href="#cite_note-187">[187]</a> As briefly mentioned <a href="#Horizons">above</a>, this radiation plays an important role for the thermodynamics of black holes.<a href="#cite_note-188">[188]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_gravity">Quantum gravity</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=34" title="Edit section: Quantum gravity">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/Quantum_gravity" title="Quantum gravity">Quantum gravity</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/String_theory" title="String theory">String theory</a>, <a href="/wiki/Canonical_general_relativity" class="mw-redirect" title="Canonical general relativity">Canonical general relativity</a>, <a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a>, <a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a>, and <a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Calabi_yau.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/170px-Calabi_yau.jpg" decoding="async" width="170" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/255px-Calabi_yau.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/340px-Calabi_yau.jpg 2x" data-file-width="1432" data-file-height="1493" /></a><figcaption>Projection of a <a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a>, one of the ways of <a href="/wiki/Compactification_(physics)" title="Compactification (physics)">compactifying</a> the extra dimensions posited by string theory</figcaption></figure> The demand for consistency between a quantum description of matter and a geometric description of spacetime,<a href="#cite_note-189">[189]</a> as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.<a href="#cite_note-190">[190]</a> Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.<a href="#cite_note-Hamber_2009-191">[191]</a><a href="#cite_note-192">[192]</a> Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.<a href="#cite_note-193">[193]</a> Some have argued that at low energies, this approach proves successful, in that it results in an acceptable <a href="/wiki/Effective_field_theory" title="Effective field theory">effective (quantum) field theory</a> of gravity.<a href="#cite_note-194">[194]</a> At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative <a href="/wiki/Non-renormalizable" class="mw-redirect" title="Non-renormalizable">non-renormalizability</a>").<a href="#cite_note-195">[195]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spin_network.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Spin_network.svg/170px-Spin_network.svg.png" decoding="async" width="170" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Spin_network.svg/255px-Spin_network.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Spin_network.svg/340px-Spin_network.svg.png 2x" data-file-width="702" data-file-height="964" /></a><figcaption>Simple <a href="/wiki/Spin_network" title="Spin network">spin network</a> of the type used in loop quantum gravity</figcaption></figure> One attempt to overcome these limitations is <a href="/wiki/String_theory" title="String theory">string theory</a>, a quantum theory not of <a href="/wiki/Point_particle" title="Point particle">point particles</a>, but of minute one-dimensional extended objects.<a href="#cite_note-196">[196]</a> The theory promises to be a <a href="/wiki/Theory_of_everything" title="Theory of everything">unified description</a> of all particles and interactions, including gravity;<a href="#cite_note-197">[197]</a> the price to pay is unusual features such as six <a href="/wiki/Superstring_theory#Extra_dimensions" title="Superstring theory">extra dimensions</a> of space in addition to the usual three.<a href="#cite_note-198">[198]</a> In what is called the <a href="/wiki/Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">second superstring revolution</a>, it was conjectured that both string theory and a unification of general relativity and <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> known as <a href="/wiki/Supergravity" title="Supergravity">supergravity</a><a href="#cite_note-199">[199]</a> form part of a hypothesized eleven-dimensional model known as <a href="/wiki/M-theory" title="M-theory">M-theory</a>, which would constitute a uniquely defined and consistent theory of quantum gravity.<a href="#cite_note-200">[200]</a> Another approach starts with the <a href="/wiki/Canonical_quantization" title="Canonical quantization">canonical quantization</a> procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. <a href="#Evolution_equations">evolution equations</a> above), the result is the <a href="/wiki/Wheeler%E2%80%93deWitt_equation" class="mw-redirect" title="Wheeler–deWitt equation">Wheeler–deWitt equation</a> (an analogue of the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.<a href="#cite_note-201">[201]</a> However, with the introduction of what are now known as <a href="/wiki/Ashtekar_variables" title="Ashtekar variables">Ashtekar variables</a>,<a href="#cite_note-202">[202]</a> this leads to a promising model known as <a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">loop quantum gravity</a>. Space is represented by a web-like structure called a <a href="/wiki/Spin_network" title="Spin network">spin network</a>, evolving over time in discrete steps.<a href="#cite_note-203">[203]</a> Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,<a href="#cite_note-204">[204]</a> there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman <a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path Integral</a> approach and <a href="/wiki/Regge_calculus" title="Regge calculus">Regge calculus</a>,<a href="#cite_note-Hamber_2009-191">[191]</a> <a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">dynamical triangulations</a>,<a href="#cite_note-205">[205]</a> <a href="/wiki/Causal_sets" title="Causal sets">causal sets</a>,<a href="#cite_note-206">[206]</a> twistor models<a href="#cite_note-207">[207]</a> or the path integral based models of <a href="/wiki/Quantum_cosmology" title="Quantum cosmology">quantum cosmology</a>.<a href="#cite_note-208">[208]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:LIGO_measurement_of_gravitational_waves.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/LIGO_measurement_of_gravitational_waves.svg/220px-LIGO_measurement_of_gravitational_waves.svg.png" decoding="async" width="220" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/LIGO_measurement_of_gravitational_waves.svg/330px-LIGO_measurement_of_gravitational_waves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/LIGO_measurement_of_gravitational_waves.svg/440px-LIGO_measurement_of_gravitational_waves.svg.png 2x" data-file-width="499" data-file-height="422" /></a><figcaption>Observation of gravitational waves from binary black hole merger GW150914</figcaption></figure> All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.<a href="#cite_note-209">[209]</a> <div class="mw-heading mw-heading2"><h2 id="Current_status">Current status</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=35" title="Edit section: Current status">edit</a>]</div> General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete.<a href="#cite_note-210">[210]</a> The problem of quantum gravity and the question of the reality of spacetime singularities remain open.<a href="#cite_note-211">[211]</a> Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.<a href="#cite_note-212">[212]</a> Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,<a href="#cite_note-213">[213]</a> while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes).<a href="#cite_note-214">[214]</a> In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015.<a href="#cite_note-NSF-83">[83]</a><a href="#cite_note-215">[215]</a><a href="#cite_note-216">[216]</a> A century after its introduction, general relativity remains a highly active area of research.<a href="#cite_note-217">[217]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=36" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre drive</a> – Hypothetical FTL transportation by warping space (warp drive)</li> <li><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Alternatives to general relativity</a> – Proposed theories of gravity</li> <li><a href="/wiki/Contributors_to_general_relativity" class="mw-redirect" title="Contributors to general relativity">Contributors to general relativity</a></li> <li><a href="/wiki/Derivations_of_the_Lorentz_transformations" title="Derivations of the Lorentz transformations">Derivations of the Lorentz transformations</a></li> <li><a href="/wiki/Ehrenfest_paradox" title="Ehrenfest paradox">Ehrenfest paradox</a> – Paradox in special relativity</li> <li><a href="/wiki/Einstein%E2%80%93Hilbert_action" title="Einstein–Hilbert action">Einstein–Hilbert action</a> – Concept in general relativity</li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a> – Albert Einstein's hypothetical situations to argue scientific points</li> <li><a href="/wiki/General_relativity_priority_dispute" title="General relativity priority dispute">General relativity priority dispute</a> – Debate about credit for general relativity</li> <li><a href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Introduction to the mathematics of general relativity</a></li> <li><a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström's theory of gravitation</a> – Predecessor to the theory of relativity</li> <li><a href="/w/index.php?title=Problems_with_Einstein%27s_general_theory_of_relativity&action=edit&redlink=1" class="new" title="Problems with Einstein's general theory of relativity (page does not exist)">Problems with Einstein's general theory of relativity</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a> – Tensor index notation for tensor-based calculations</li> <li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline of gravitational physics and relativity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=37" title="Edit section: References">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: 30em;"> <ol class="references"> <li id="cite_note-SXSproject-1"><a href="#cite_ref-SXSproject_1-0">^</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.black-holes.org/gw150914">"GW150914: LIGO Detects Gravitational Waves"</a>. Black-holes.org. Retrieved 18 April 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Black-holes.org&rft.atitle=GW150914%3A+LIGO+Detects+Gravitational+Waves&rft_id=http%3A%2F%2Fwww.black-holes.org%2Fgw150914&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-:0-2">^ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> <a href="#CITEREFLandauLifshitz1975">Landau & Lifshitz 1975</a>, p. 228 "...the general theory of relativity...was established by Einstein, and represents probably the most beautiful of all existing physical theories." </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFPoincaré1905">Poincaré 1905</a> </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="CITEREFO'ConnorRobertson1996" class="citation web cs1">O'Connor, J.J.; Robertson, E.F. (May 1996). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/HistTopics/General_relativity.html">"General relativity]"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=General+relativity%5D&rft.date=1996-05&rft.aulast=O%27Connor&rft.aufirst=J.J.&rft.au=Robertson%2C+E.F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FHistTopics%2FGeneral_relativity.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150204231934/http://www-history.mcs.st-and.ac.uk/Indexes/Math_Physics.html">History Topics: Mathematical Physics Index</a>, Scotland: School of Mathematics and Statistics, <a href="/wiki/University_of_St._Andrews" class="mw-redirect" title="University of St. Andrews">University of St. Andrews</a>, archived from <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Indexes/Math_Physics.html">the original</a> on 4 February 2015, retrieved 4 February 2015</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+Topics%3A+Mathematical+Physics+Index&rft.place=Scotland&rft.pub=School+of+Mathematics+and+Statistics%2C+University+of+St.+Andrews&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FIndexes%2FMath_Physics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFPais1982">Pais 1982</a>, ch. 9 to 15, <a href="#CITEREFJanssen2005">Janssen 2005</a>; an up-to-date collection of current research, including reprints of many of the original articles, is <a href="#CITEREFRenn2007">Renn 2007</a>; an accessible overview can be found in <a href="#CITEREFRenn2005">Renn 2005</a>, pp. 110ff. Einstein's original papers are found in <a rel="nofollow" class="external text" href="http://einsteinpapers.press.princeton.edu/">Digital Einstein</a>, volumes 4 and 6. An early key article is <a href="#CITEREFEinstein1907">Einstein 1907</a>, cf. <a href="#CITEREFPais1982">Pais 1982</a>, ch. 9. The publication featuring the field equations is <a href="#CITEREFEinstein1915">Einstein 1915</a>, cf. <a href="#CITEREFPais1982">Pais 1982</a>, ch. 11–15 </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Moshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261. <a rel="nofollow" class="external text" href="http://www.pitt.edu/~jdnorton/teaching/GR&Grav_2007/pdf/Einstein_Entwurf_1913.pdf">English translate</a> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFSchwarzschild1916a">Schwarzschild 1916a</a>, <a href="#CITEREFSchwarzschild1916b">Schwarzschild 1916b</a> and <a href="#CITEREFReissner1916">Reissner 1916</a> (later complemented in <a href="#CITEREFNordström1918">Nordström 1918</a>) </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFEinstein1917">Einstein 1917</a>, cf. <a href="#CITEREFPais1982">Pais 1982</a>, ch. 15e </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Hubble's original article is <a href="#CITEREFHubble1929">Hubble 1929</a>; an accessible overview is given in <a href="#CITEREFSingh2004">Singh 2004</a>, ch. 2–4 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> As reported in <a href="#CITEREFGamow1970">Gamow 1970</a>. Einstein's condemnation would prove to be premature, cf. the section <a href="#Cosmology">Cosmology</a>, below </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFPais1982">Pais 1982</a>, pp. 253–254 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFKennefick2005">Kennefick 2005</a>, <a href="#CITEREFKennefick2007">Kennefick 2007</a> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFPais1982">Pais 1982</a>, ch. 16 </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFThorne2003">Thorne 2003</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yLy4b61rfPwC&pg=PA74">74</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFIsrael1987">Israel 1987</a>, ch. 7.8–7.10, <a href="#CITEREFThorne1994">Thorne 1994</a>, ch. 3–9 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Sections <a href="#Orbital_effects_and_the_relativity_of_direction">Orbital effects and the relativity of direction</a>, <a href="#Gravitational_time_dilation_and_frequency_shift">Gravitational time dilation and frequency shift</a> and <a href="#Light_deflection_and_gravitational_time_delay">Light deflection and gravitational time delay</a>, and references therein </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Section <a href="#Cosmology">Cosmology</a> and references therein; the historical development is in <a href="#CITEREFOverbye1999">Overbye 1999</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, p. 3 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFRovelli2015">Rovelli 2015</a>, pp. 1–6 "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more." </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFChandrasekhar1984">Chandrasekhar 1984</a>, p. 6 </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFEngler2002">Engler 2002</a> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert_Einstein2011" class="citation book cs1">Albert Einstein (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yhN9CgAAQBAJ">Relativity – The Special and General Theory</a>. Read Books Ltd. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4474-9358-7" title="Special:BookSources/978-1-4474-9358-7"><bdi>978-1-4474-9358-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity+%E2%80%93+The+Special+and+General+Theory&rft.pages=4&rft.pub=Read+Books+Ltd&rft.date=2011&rft.isbn=978-1-4474-9358-7&rft.au=Albert+Einstein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyhN9CgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yhN9CgAAQBAJ&pg=PT4">Extract of page 4</a> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> The following exposition re-traces that of <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 1 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAl-Khalili2021" class="citation web cs1">Al-Khalili, Jim (26 March 2021). <a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b08kgv7f">"Gravity and Me: The force that shapes our lives"</a>. www.bbc.co.uk. Retrieved 9 April 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.bbc.co.uk&rft.atitle=Gravity+and+Me%3A+The+force+that+shapes+our+lives&rft.date=2021-03-26&rft.aulast=Al-Khalili&rft.aufirst=Jim&rft_id=https%3A%2F%2Fwww.bbc.co.uk%2Fprogrammes%2Fb08kgv7f&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFArnold1989">Arnold 1989</a>, ch. 1 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, pp. 5f </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFWill1993">Will 1993</a>, sec. 2.4, <a href="#CITEREFWill2006">Will 2006</a>, sec. 2 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFWheeler1990">Wheeler 1990</a>, ch. 2 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 1.2, <a href="#CITEREFHavas1964">Havas 1964</a>, <a href="#CITEREFKünzle1972">Künzle 1972</a>. The simple thought experiment in question was first described in <a href="#CITEREFHeckmannSchücking1959">Heckmann & Schücking 1959</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, pp. 10f </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Good introductions are, in order of increasing presupposed knowledge of mathematics, <a href="#CITEREFGiulini2005">Giulini 2005</a>, <a href="#CITEREFMermin2005">Mermin 2005</a>, and <a href="#CITEREFRindler1991">Rindler 1991</a>; for accounts of precision experiments, cf. part IV of <a href="#CITEREFEhlersLämmerzahl2006">Ehlers & Lämmerzahl 2006</a> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> An in-depth comparison between the two symmetry groups can be found in <a href="#CITEREFGiulini2006">Giulini 2006</a> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a href="#CITEREFRindler1991">Rindler 1991</a>, sec. 22, <a href="#CITEREFSynge1972">Synge 1972</a>, ch. 1 and 2 </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 2.3 </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 1.4, <a href="#CITEREFSchutz1985">Schutz 1985</a>, sec. 5.1 </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, pp. 17ff; a derivation can be found in <a href="#CITEREFMermin2005">Mermin 2005</a>, ch. 12. For the experimental evidence, cf. the section <a href="#Gravitational_time_dilation_and_frequency_shift">Gravitational time dilation and frequency shift</a>, below </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, sec. 1.13; for an elementary account, see <a href="#CITEREFWheeler1990">Wheeler 1990</a>, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. <a href="#CITEREFNorton1985">Norton 1985</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 1.4 for the experimental evidence, see once more section <a href="#Gravitational_time_dilation_and_frequency_shift">Gravitational time dilation and frequency shift</a>. Choosing a different connection with non-zero <a href="/wiki/Torsion_tensor" title="Torsion tensor">torsion</a> leads to a modified theory known as <a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, p. 16, <a href="#CITEREFKenyon1990">Kenyon 1990</a>, sec. 7.2, <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, sec. 2.8 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <a href="#CITEREFEhlers1973">Ehlers 1973</a>, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in <a href="#CITEREFWeinberg1972">Weinberg 1972</a>. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. <a href="#CITEREFLovelock1972">Lovelock 1972</a>. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as <a href="/wiki/Bianchi_identities" class="mw-redirect" title="Bianchi identities">Bianchi identities</a>, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. <a href="#CITEREFSchutz1985">Schutz 1985</a>, sec. 8.3 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="#CITEREFKenyon1990">Kenyon 1990</a>, sec. 7.4 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg,_Steven1972" class="citation book cs1">Weinberg, Steven (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92567-5" title="Special:BookSources/978-0-471-92567-5"><bdi>978-0-471-92567-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Cosmology%3A+Principles+and+Applications+of+the+General+Theory+of+Relativity&rft.pub=John+Wiley&rft.date=1972&rft.isbn=978-0-471-92567-5&rft.au=Weinberg%2C+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng,_Ta-Pei2005" class="citation book cs1">Cheng, Ta-Pei (2005). Relativity, Gravitation and Cosmology: a Basic Introduction. Oxford and New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852957-6" title="Special:BookSources/978-0-19-852957-6"><bdi>978-0-19-852957-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%2C+Gravitation+and+Cosmology%3A+a+Basic+Introduction&rft.pub=Oxford+and+New+York%3A+Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-852957-6&rft.au=Cheng%2C+Ta-Pei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a href="#CITEREFBransDicke1961">Brans & Dicke 1961</a>, <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, sec. 3 in ch. 7, <a href="#CITEREFGoenner2004">Goenner 2004</a>, sec. 7.2, and <a href="#CITEREFTrautman2006">Trautman 2006</a>, respectively </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, ch. 4, <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, ch. 7 or, in fact, any other textbook on general relativity </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> At least approximately, cf. <a href="#CITEREFPoisson2004a">Poisson 2004a</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <a href="#CITEREFWheeler1990">Wheeler 1990</a>, p. xi </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, sec. 4.4 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, sec. 4.1 </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see <a href="#CITEREFGiulini2007">Giulini 2007</a> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> section 5 in ch. 12 of <a href="#CITEREFWeinberg1972">Weinberg 1972</a> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Introductory chapters of <a href="#CITEREFStephaniKramerMacCallumHoenselaers2003">Stephani et al. 2003</a> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> A review showing Einstein's equation in the broader context of other PDEs with physical significance is <a href="#CITEREFGeroch1996">Geroch 1996</a> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> For background information and a list of solutions, cf. <a href="#CITEREFStephaniKramerMacCallumHoenselaers2003">Stephani et al. 2003</a>; a more recent review can be found in <a href="#CITEREFMacCallum2006">MacCallum 2006</a> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFChandrasekhar1983">Chandrasekhar 1983</a>, ch. 3,5,6 </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFNarlikar1993">Narlikar 1993</a>, ch. 4, sec. 3.3 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Brief descriptions of these and further interesting solutions can be found in <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, ch. 5 </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <a href="#CITEREFLehner2002">Lehner 2002</a> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> For instance <a href="#CITEREFWald1984">Wald 1984</a>, sec. 4.4 </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <a href="#CITEREFWill1993">Will 1993</a>, sec. 4.1 and 4.2 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <a href="#CITEREFWill2006">Will 2006</a>, sec. 3.2, <a href="#CITEREFWill1993">Will 1993</a>, ch. 4 </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, pp. 24–26 vs. pp. 236–237 and <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. <a href="#CITEREFEinstein1907">Einstein 1907</a> and the description in <a href="#CITEREFPais1982">Pais 1982</a>, pp. 196–198 </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, pp. 24–26; <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, § 38.5 </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <a href="/wiki/Pound%E2%80%93Rebka_experiment" title="Pound–Rebka experiment">Pound–Rebka experiment</a>, see <a href="#CITEREFPoundRebka1959">Pound & Rebka 1959</a>, <a href="#CITEREFPoundRebka1960">Pound & Rebka 1960</a>; <a href="#CITEREFPoundSnider1964">Pound & Snider 1964</a>; a list of further experiments is given in <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, table 4.1 on p. 186 </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFGreensteinOkeShipman1971">Greenstein, Oke & Shipman 1971</a>; the most recent and most accurate Sirius B measurements are published in <a href="#CITEREFBarstow,_Bond_et_al.2005">Barstow, Bond et al. 2005</a>. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> Starting with the <a href="/wiki/Hafele%E2%80%93Keating_experiment" title="Hafele–Keating experiment">Hafele–Keating experiment</a>, <a href="#CITEREFHafeleKeating1972a">Hafele & Keating 1972a</a> and <a href="#CITEREFHafeleKeating1972b">Hafele & Keating 1972b</a>, and culminating in the <a href="/wiki/Gravity_Probe_A" title="Gravity Probe A">Gravity Probe A</a> experiment; an overview of experiments can be found in <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, table 4.1 on p. 186 </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see <a href="#CITEREFAshby2002">Ashby 2002</a> and <a href="#CITEREFAshby2003">Ashby 2003</a> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a href="#CITEREFStairs2003">Stairs 2003</a> and <a href="#CITEREFKramer2004">Kramer 2004</a> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, sec. 4.2 </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, pp. 164–172 </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> Cf. <a href="#CITEREFKennefick2005">Kennefick 2005</a> for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, ch. 4.3. For the most precise direct modern observations using quasars, cf. <a href="#CITEREFShapiroDavisLebachGregory2004">Shapiro et al. 2004</a> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> using a <a href="/wiki/WKB_approximation" title="WKB approximation">WKB approximation</a>, cf. <a href="#CITEREFEhlers1973">Ehlers 1973</a>, sec. 5 </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <a href="#CITEREFBlanchet2006">Blanchet 2006</a>, sec. 1.3 </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, sec. 1.16; for the historical examples, <a href="#CITEREFIsrael1987">Israel 1987</a>, pp. 202–204; in fact, Einstein published one such derivation as <a href="#CITEREFEinstein1907">Einstein 1907</a>. Such calculations tacitly assume that the geometry of space is <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a>, cf. <a href="#CITEREFEhlersRindler1997">Ehlers & Rindler 1997</a> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. <a href="#CITEREFRindler2001">Rindler 2001</a>, sec. 11.11 </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> For the Sun's gravitational field using radar signals reflected from planets such as <a href="/wiki/Venus" title="Venus">Venus</a> and Mercury, cf. <a href="#CITEREFShapiro1964">Shapiro 1964</a>, <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, ch. 8, sec. 7; for signals actively sent back by space probes (<a href="/wiki/Transponder" title="Transponder">transponder</a> measurements), cf. <a href="#CITEREFBertottiIessTortora2003">Bertotti, Iess & Tortora 2003</a>; for an overview, see <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, table 4.4 on p. 200; for more recent measurements using signals received from a <a href="/wiki/Pulsar" title="Pulsar">pulsar</a> that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. <a href="#CITEREFStairs2003">Stairs 2003</a>, sec. 4.4 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <a href="#CITEREFWill1993">Will 1993</a>, sec. 7.1 and 7.2 </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein,_A1916" class="citation journal cs1">Einstein, A (22 June 1916). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte">"Näherungsweise Integration der Feldgleichungen der Gravitation"</a>. <a href="/wiki/Prussian_Academy_of_Sciences" title="Prussian Academy of Sciences">Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin</a> (part 1): 688–696. <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/1916SPAW.......688E">1916SPAW.......688E</a>. 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Retrieved 12 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften+Berlin&rft.atitle=N%C3%A4herungsweise+Integration+der+Feldgleichungen+der+Gravitation&rft.issue=part+1&rft.pages=688-696&rft.date=1916-06-22&rft_id=info%3Abibcode%2F1916SPAW.......688E&rft.au=Einstein%2C+A&rft_id=http%3A%2F%2Feinstein-annalen.mpiwg-berlin.mpg.de%2Frelated_texts%2Fsitzungsberichte&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein,_A1918" class="citation journal cs1">Einstein, A (31 January 1918). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte">"Über Gravitationswellen"</a>. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin (part 1): 154–167. <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/1918SPAW.......154E">1918SPAW.......154E</a>. Archived from <a rel="nofollow" class="external text" href="http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte">the original</a> on 21 March 2019. 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Nature News. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature.2016.19361">10.1038/nature.2016.19361</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:182916902">182916902</a>. Retrieved 11 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+News&rft.atitle=Einstein%27s+gravitational+waves+found+at+last&rft.date=2016-02-11&rft_id=info%3Adoi%2F10.1038%2Fnature.2016.19361&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A182916902%23id-name%3DS2CID&rft.aulast=Castelvecchi&rft.aufirst=Davide&rft.au=Witze%2C+Witze&rft_id=http%3A%2F%2Fwww.nature.com%2Fnews%2Feinstein-s-gravitational-waves-found-at-last-1.19361&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-Abbot-82">^ <a href="#cite_ref-Abbot_82-0">a</a> <a href="#cite_ref-Abbot_82-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._P._Abbott2016" class="citation journal cs1">B. P. Abbott; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.03837">1602.03837</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/2016PhRvL.116f1102A">2016PhRvL.116f1102A</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%2FPhysRevLett.116.061102">10.1103/PhysRevLett.116.061102</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/26918975">26918975</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:124959784">124959784</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+Letters&rft.atitle=Observation+of+Gravitational+Waves+from+a+Binary+Black+Hole+Merger&rft.volume=116&rft.issue=6&rft.pages=061102&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124959784%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016PhRvL.116f1102A&rft_id=info%3Aarxiv%2F1602.03837&rft_id=info%3Apmid%2F26918975&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.116.061102&rft.au=B.+P.+Abbott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-NSF-83">^ <a href="#cite_ref-NSF_83-0">a</a> <a href="#cite_ref-NSF_83-1">b</a> <a href="#cite_ref-NSF_83-2">c</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://www.nsf.gov/news/news_summ.jsp?cntn_id=137628">"Gravitational waves detected 100 years after Einstein's prediction"</a>. NSF – National Science Foundation. 11 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NSF+%E2%80%93+National+Science+Foundation&rft.atitle=Gravitational+waves+detected+100+years+after+Einstein%27s+prediction&rft.date=2016-02-11&rft_id=https%3A%2F%2Fwww.nsf.gov%2Fnews%2Fnews_summ.jsp%3Fcntn_id%3D137628&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> Most advanced textbooks on general relativity contain a description of these properties, e.g. <a href="#CITEREFSchutz1985">Schutz 1985</a>, ch. 9 </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> For example <a href="#CITEREFJaranowskiKrólak2005">Jaranowski & Królak 2005</a> </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, ch. 13 </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <a href="#CITEREFGowdy1971">Gowdy 1971</a>, <a href="#CITEREFGowdy1974">Gowdy 1974</a> </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> See <a href="#CITEREFLehner2002">Lehner 2002</a> for a brief introduction to the methods of numerical relativity, and <a href="#CITEREFSeidel1998">Seidel 1998</a> for the connection with gravitational wave astronomy </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <a href="#CITEREFSchutz2003">Schutz 2003</a>, pp. 48–49, <a href="#CITEREFPais1982">Pais 1982</a>, pp. 253–254 </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <a href="#CITEREFRindler2001">Rindler 2001</a>, sec. 11.9 </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <a href="#CITEREFWill1993">Will 1993</a>, pp. 177–181 </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. <a href="#CITEREFWill2006">Will 2006</a>, sec. 3.5 and <a href="#CITEREFWill1993">Will 1993</a>, sec. 7.3 </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> The most precise measurements are <a href="/wiki/VLBI" class="mw-redirect" title="VLBI">VLBI</a> measurements of planetary positions; see <a href="#CITEREFWill1993">Will 1993</a>, ch. 5, <a href="#CITEREFWill2006">Will 2006</a>, sec. 3.5, <a href="#CITEREFAndersonCampbellJurgensLau1992">Anderson et al. 1992</a>; for an overview, <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, pp. 406–407 </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <a href="#CITEREFKramerStairsManchesterMcLaughlin2006">Kramer et al. 2006</a> </li> <li id="cite_note-FOOTNOTEDediuMagdalenaMartín-Vide2015[httpsbooksgooglecombooksidXmwiCwAAQBAJpgPA141_141]-95"><a href="#cite_ref-FOOTNOTEDediuMagdalenaMartín-Vide2015[httpsbooksgooglecombooksidXmwiCwAAQBAJpgPA141_141]_95-0">^</a> <a href="#CITEREFDediuMagdalenaMartín-Vide2015">Dediu, Magdalena & Martín-Vide 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XmwiCwAAQBAJ&pg=PA141">141</a>. </li> <li id="cite_note-:1-96">^ <a href="#cite_ref-:1_96-0">a</a> <a href="#cite_ref-:1_96-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKramerStairsManchesterWex2021" class="citation journal cs1">Kramer, M.; Stairs, I. H.; Manchester, R. N.; Wex, N.; Deller, A. T.; Coles, W. A.; Ali, M.; Burgay, M.; Camilo, F.; Cognard, I.; Damour, T. (13 December 2021). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevX.11.041050">"Strong-Field Gravity Tests with the Double Pulsar"</a>. Physical Review X. 11 (4): 041050. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2112.06795">2112.06795</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/2021PhRvX..11d1050K">2021PhRvX..11d1050K</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%2FPhysRevX.11.041050">10.1103/PhysRevX.11.041050</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2160-3308">2160-3308</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:245124502">245124502</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+X&rft.atitle=Strong-Field+Gravity+Tests+with+the+Double+Pulsar&rft.volume=11&rft.issue=4&rft.pages=041050&rft.date=2021-12-13&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A245124502%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2021PhRvX..11d1050K&rft_id=info%3Aarxiv%2F2112.06795&rft.issn=2160-3308&rft_id=info%3Adoi%2F10.1103%2FPhysRevX.11.041050&rft.aulast=Kramer&rft.aufirst=M.&rft.au=Stairs%2C+I.+H.&rft.au=Manchester%2C+R.+N.&rft.au=Wex%2C+N.&rft.au=Deller%2C+A.+T.&rft.au=Coles%2C+W.+A.&rft.au=Ali%2C+M.&rft.au=Burgay%2C+M.&rft.au=Camilo%2C+F.&rft.au=Cognard%2C+I.&rft.au=Damour%2C+T.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevX.11.041050&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> <a href="#CITEREFStairs2003">Stairs 2003</a>, <a href="#CITEREFSchutz2003">Schutz 2003</a>, pp. 317–321, <a href="#CITEREFBartusiak2000">Bartusiak 2000</a>, pp. 70–86 </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <a href="#CITEREFWeisbergTaylor2003">Weisberg & Taylor 2003</a>; for the pulsar discovery, see <a href="#CITEREFHulseTaylor1975">Hulse & Taylor 1975</a>; for the initial evidence for gravitational radiation, see <a href="#CITEREFTaylor1994">Taylor 1994</a> </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <a href="#CITEREFKramer2004">Kramer 2004</a> </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <a href="#CITEREFPenrose2004">Penrose 2004</a>, § 14.5, <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, § 11.4 </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, sec. 9.6, <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, sec. 7.8 </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <a href="#CITEREFBertottiCiufoliniBender1987">Bertotti, Ciufolini & Bender 1987</a>, <a href="#CITEREFNordtvedt2003">Nordtvedt 2003</a> </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <a href="#CITEREFKahn2007">Kahn 2007</a> </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> A mission description can be found in <a href="#CITEREFEverittBuchmanDeBraKeiser2001">Everitt et al. 2001</a>; a first post-flight evaluation is given in <a href="#CITEREFEverittParkinsonKahn2007">Everitt, Parkinson & Kahn 2007</a>; further updates will be available on the mission website <a href="#CITEREFKahn1996–2012">Kahn 1996–2012</a>. </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <a href="#CITEREFTownsend1997">Townsend 1997</a>, sec. 4.2.1, <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, pp. 469–471 </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <a href="#CITEREFOhanianRuffini1994">Ohanian & Ruffini 1994</a>, sec. 4.7, <a href="#CITEREFWeinberg1972">Weinberg 1972</a>, sec. 9.7; for a more recent review, see <a href="#CITEREFSchäfer2004">Schäfer 2004</a> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> <a href="#CITEREFCiufoliniPavlis2004">Ciufolini & Pavlis 2004</a>, <a href="#CITEREFCiufoliniPavlisPeron2006">Ciufolini, Pavlis & Peron 2006</a>, <a href="#CITEREFIorio2009">Iorio 2009</a> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <a href="#CITEREFIorio2006">Iorio 2006</a>, <a href="#CITEREFIorio2010">Iorio 2010</a> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> For overviews of gravitational lensing and its applications, see <a href="#CITEREFEhlersFalcoSchneider1992">Ehlers, Falco & Schneider 1992</a> and <a href="#CITEREFWambsganss1998">Wambsganss 1998</a> </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> For a simple derivation, see <a href="#CITEREFSchutz2003">Schutz 2003</a>, ch. 23; cf. <a href="#CITEREFNarayanBartelmann1997">Narayan & Bartelmann 1997</a>, sec. 3 </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> <a href="#CITEREFWalshCarswellWeymann1979">Walsh, Carswell & Weymann 1979</a> </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> Images of all the known lenses can be found on the pages of the CASTLES project, <a href="#CITEREFKochanekFalcoImpeyLehar2007">Kochanek et al. 2007</a> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> <a href="#CITEREFRouletMollerach1997">Roulet & Mollerach 1997</a> </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <a href="#CITEREFNarayanBartelmann1997">Narayan & Bartelmann 1997</a>, sec. 3.7 </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> <a href="#CITEREFBarish2005">Barish 2005</a>, <a href="#CITEREFBartusiak2000">Bartusiak 2000</a>, <a href="#CITEREFBlairMcNamara1997">Blair & McNamara 1997</a> </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> <a href="#CITEREFHoughRowan2000">Hough & Rowan 2000</a> </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHobbsArchibaldArzoumanianBacker2010" class="citation cs2">Hobbs, George; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N. D. R.; Burgay, M.; Burke-Spolaor, S.; et al. (2010), "The international pulsar timing array project: using pulsars as a gravitational wave detector", Classical and Quantum Gravity, 27 (8): 084013, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.5206">0911.5206</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/2010CQGra..27h4013H">2010CQGra..27h4013H</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%2F0264-9381%2F27%2F8%2F084013">10.1088/0264-9381/27/8/084013</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:56073764">56073764</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=The+international+pulsar+timing+array+project%3A+using+pulsars+as+a+gravitational+wave+detector&rft.volume=27&rft.issue=8&rft.pages=084013&rft.date=2010&rft_id=info%3Aarxiv%2F0911.5206&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56073764%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F27%2F8%2F084013&rft_id=info%3Abibcode%2F2010CQGra..27h4013H&rft.aulast=Hobbs&rft.aufirst=George&rft.au=Archibald%2C+A.&rft.au=Arzoumanian%2C+Z.&rft.au=Backer%2C+D.&rft.au=Bailes%2C+M.&rft.au=Bhat%2C+N.+D.+R.&rft.au=Burgay%2C+M.&rft.au=Burke-Spolaor%2C+S.&rft.au=Champion%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> <a href="#CITEREFDanzmannRüdiger2003">Danzmann & Rüdiger 2003</a> </li> <li id="cite_note-119"><a href="#cite_ref-119">^</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.esa.int/esaSC/120397_index_0_m.html">"LISA pathfinder overview"</a>. ESA. Retrieved 23 April 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=LISA+pathfinder+overview&rft.pub=ESA&rft_id=http%3A%2F%2Fwww.esa.int%2FesaSC%2F120397_index_0_m.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-120"><a href="#cite_ref-120">^</a> <a href="#CITEREFThorne1995">Thorne 1995</a> </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> <a href="#CITEREFCutlerThorne2002">Cutler & Thorne 2002</a> </li> <li id="cite_note-122"><a href="#cite_ref-122">^</a> <a href="#CITEREFMiller2002">Miller 2002</a>, lectures 19 and 21 </li> <li id="cite_note-123"><a href="#cite_ref-123">^</a> <a href="#CITEREFCelottiMillerSciama1999">Celotti, Miller & Sciama 1999</a>, sec. 3 </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> <a href="#CITEREFSpringelWhiteJenkinsFrenk2005">Springel et al. 2005</a> and the accompanying summary <a href="#CITEREFGnedin2005">Gnedin 2005</a> </li> <li id="cite_note-125"><a href="#cite_ref-125">^</a> <a href="#CITEREFBlandford1987">Blandford 1987</a>, sec. 8.2.4 </li> <li id="cite_note-126"><a href="#cite_ref-126">^</a> For the basic mechanism, see <a href="#CITEREFCarrollOstlie1996">Carroll & Ostlie 1996</a>, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. <a href="#CITEREFRobson1996">Robson 1996</a> </li> <li id="cite_note-127"><a href="#cite_ref-127">^</a> For a review, see <a href="#CITEREFBegelmanBlandfordRees1984">Begelman, Blandford & Rees 1984</a>. To a distant observer, some of these jets even appear to move <a href="/wiki/Superluminal_motion" title="Superluminal motion">faster than light</a>; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see <a href="#CITEREFRees1966">Rees 1966</a> </li> <li id="cite_note-128"><a href="#cite_ref-128">^</a> For stellar end states, cf. <a href="#CITEREFOppenheimerSnyder1939">Oppenheimer & Snyder 1939</a> or, for more recent numerical work, <a href="#CITEREFFont2003">Font 2003</a>, sec. 4.1; for supernovae, there are still major problems to be solved, cf. <a href="#CITEREFBurasRamppJankaKifonidis2003">Buras et al. 2003</a>; for simulating accretion and the formation of jets, cf. <a href="#CITEREFFont2003">Font 2003</a>, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from <a href="/wiki/X-ray_pulsar" title="X-ray pulsar">X-ray pulsars</a>, cf. <a href="#CITEREFKraus1998">Kraus 1998</a> </li> <li id="cite_note-129"><a href="#cite_ref-129">^</a> The evidence includes limits on compactness from the observation of accretion-driven phenomena ("<a href="/wiki/Eddington_luminosity" title="Eddington luminosity">Eddington luminosity</a>"), see <a href="#CITEREFCelottiMillerSciama1999">Celotti, Miller & Sciama 1999</a>, observations of stellar dynamics in the center of our own <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> galaxy, cf. <a href="#CITEREFSchödelOttGenzelEckart2003">Schödel et al. 2003</a>, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of <a href="/wiki/X-ray_burst" class="mw-redirect" title="X-ray burst">X-ray bursts</a> for which the central compact object is either a neutron star or a black hole; cf. <a href="#CITEREFRemillardLinCooperNarayan2006">Remillard et al. 2006</a> for an overview, <a href="#CITEREFNarayan2006">Narayan 2006</a>, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. <a href="#CITEREFFalckeMeliaAgol2000">Falcke, Melia & Agol 2000</a> </li> <li id="cite_note-130"><a href="#cite_ref-130">^</a> <a href="#CITEREFDalalHolzHughesJain2006">Dalal et al. 2006</a> </li> <li id="cite_note-131"><a href="#cite_ref-131">^</a> <a href="#CITEREFBarackCutler2004">Barack & Cutler 2004</a> </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> <a href="#CITEREFEinstein1917">Einstein 1917</a>; cf. <a href="#CITEREFPais1982">Pais 1982</a>, pp. 285–288 </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> <a href="#CITEREFCarroll2001">Carroll 2001</a>, ch. 2 </li> <li id="cite_note-134"><a href="#cite_ref-134">^</a> <a href="#CITEREFBergströmGoobar2003">Bergström & Goobar 2003</a>, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million <a href="/wiki/Light-year" title="Light-year">light-years</a> and more, our own universe indeed appears to be isotropic and homogeneous, cf. <a href="#CITEREFPeeblesSchrammTurnerKron1991">Peebles et al. 1991</a> </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> E.g. with <a href="/wiki/WMAP" class="mw-redirect" title="WMAP">WMAP</a> data, see <a href="#CITEREFSpergelVerdePeirisKomatsu2003">Spergel et al. 2003</a> </li> <li id="cite_note-136"><a href="#cite_ref-136">^</a> These tests involve the separate observations detailed further on, see, e.g., fig. 2 in <a href="#CITEREFBridleLahavOstrikerSteinhardt2003">Bridle et al. 2003</a> </li> <li id="cite_note-137"><a href="#cite_ref-137">^</a> <a href="#CITEREFPeebles1966">Peebles 1966</a>; for a recent account of predictions, see <a href="#CITEREFCoc,_Vangioni‐Flam_et_al.2004">Coc, Vangioni‐Flam et al. 2004</a>; an accessible account can be found in <a href="#CITEREFWeiss2006">Weiss 2006</a>; compare with the observations in <a href="#CITEREFOliveSkillman2004">Olive & Skillman 2004</a>, <a href="#CITEREFBaniaRoodBalser2002">Bania, Rood & Balser 2002</a>, <a href="#CITEREFO'MearaTytlerKirkmanSuzuki2001">O'Meara et al. 2001</a>, and <a href="#CITEREFCharbonnelPrimas2005">Charbonnel & Primas 2005</a> </li> <li id="cite_note-138"><a href="#cite_ref-138">^</a> <a href="#CITEREFLahavSuto2004">Lahav & Suto 2004</a>, <a href="#CITEREFBertschinger1998">Bertschinger 1998</a>, <a href="#CITEREFSpringelWhiteJenkinsFrenk2005">Springel et al. 2005</a> </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> <a href="#CITEREFAlpherHerman1948">Alpher & Herman 1948</a>, for a pedagogical introduction, see <a href="#CITEREFBergströmGoobar2003">Bergström & Goobar 2003</a>, ch. 11; for the initial detection, see <a href="#CITEREFPenziasWilson1965">Penzias & Wilson 1965</a> and, for precision measurements by satellite observatories, <a href="#CITEREFMatherChengCottinghamEplee1994">Mather et al. 1994</a> (<a href="/wiki/Cosmic_Background_Explorer" title="Cosmic Background Explorer">COBE</a>) and <a href="#CITEREFBennettHalpernHinshawJarosik2003">Bennett et al. 2003</a> (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's <a href="/wiki/Polarized_light" class="mw-redirect" title="Polarized light">polarization</a>, cf. <a href="#CITEREFKamionkowskiKosowskyStebbins1997">Kamionkowski, Kosowsky & Stebbins 1997</a> and <a href="#CITEREFSeljakZaldarriaga1997">Seljak & Zaldarriaga 1997</a> </li> <li id="cite_note-140"><a href="#cite_ref-140">^</a> Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. <a href="#CITEREFPeebles1993">Peebles 1993</a>, ch. 18, evidence from gravitational lensing, cf. <a href="#CITEREFPeacock1999">Peacock 1999</a>, sec. 4.6, and simulations of large-scale structure formation, see <a href="#CITEREFSpringelWhiteJenkinsFrenk2005">Springel et al. 2005</a> </li> <li id="cite_note-141"><a href="#cite_ref-141">^</a> <a href="#CITEREFPeacock1999">Peacock 1999</a>, ch. 12, <a href="#CITEREFPeskin2007">Peskin 2007</a>; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> ("non-<a href="/wiki/Baryon" title="Baryon">baryonic</a> matter"), cf. <a href="#CITEREFPeacock1999">Peacock 1999</a>, ch. 12 </li> <li id="cite_note-142"><a href="#cite_ref-142">^</a> Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in <a href="#CITEREFMannheim2006">Mannheim 2006</a>, sec. 9 </li> <li id="cite_note-143"><a href="#cite_ref-143">^</a> <a href="#CITEREFCarroll2001">Carroll 2001</a>; an accessible overview is given in <a href="#CITEREFCaldwell2004">Caldwell 2004</a>. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. <a href="#CITEREFMannheim2006">Mannheim 2006</a>, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. <a href="#CITEREFBuchert2008">Buchert 2008</a> </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> A good introduction is <a href="#CITEREFLinde2005">Linde 2005</a>; for a more recent review, see <a href="#CITEREFLinde2006">Linde 2006</a> </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> More precisely, these are the <a href="/wiki/Flatness_problem" title="Flatness problem">flatness problem</a>, the <a href="/wiki/Horizon_problem" title="Horizon problem">horizon problem</a>, and the <a href="/wiki/Monopole_problem" class="mw-redirect" title="Monopole problem">monopole problem</a>; a pedagogical introduction can be found in <a href="#CITEREFNarlikar1993">Narlikar 1993</a>, sec. 6.4, see also <a href="#CITEREFBörner1993">Börner 1993</a>, sec. 9.1 </li> <li id="cite_note-146"><a href="#cite_ref-146">^</a> <a href="#CITEREFSpergelBeanDoréNolta2007">Spergel et al. 2007</a>, sec. 5,6 </li> <li id="cite_note-147"><a href="#cite_ref-147">^</a> More concretely, the <a href="/wiki/Potential" title="Potential">potential</a> function that is crucial to determining the dynamics of the <a href="/wiki/Inflaton" title="Inflaton">inflaton</a> is simply postulated, but not derived from an underlying physical theory </li> <li id="cite_note-148"><a href="#cite_ref-148">^</a> <a href="#CITEREFBrandenberger2008">Brandenberger 2008</a>, sec. 2 </li> <li id="cite_note-149"><a href="#cite_ref-149">^</a> <a href="#CITEREFGödel1949">Gödel 1949</a> </li> <li id="cite_note-150"><a href="#cite_ref-150">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFinazziLiberatiBarceló2009" class="citation journal cs1">Finazzi, Stefano; Liberati, Stefano; Barceló, Carlos (15 June 2009). "Semiclassical instability of dynamical warp drives". Physical Review D. 79 (12): 124017. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0904.0141">0904.0141</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/2009PhRvD..79l4017F">2009PhRvD..79l4017F</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%2FPhysRevD.79.124017">10.1103/PhysRevD.79.124017</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:59575856">59575856</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+D&rft.atitle=Semiclassical+instability+of+dynamical+warp+drives&rft.volume=79&rft.issue=12&rft.pages=124017&rft.date=2009-06-15&rft_id=info%3Aarxiv%2F0904.0141&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59575856%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.79.124017&rft_id=info%3Abibcode%2F2009PhRvD..79l4017F&rft.aulast=Finazzi&rft.aufirst=Stefano&rft.au=Liberati%2C+Stefano&rft.au=Barcel%C3%B3%2C+Carlos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-bondi_etal_1962-151"><a href="#cite_ref-bondi_etal_1962_151-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBondiVan_der_BurgMetzner1962" class="citation journal cs1">Bondi, H.; Van der Burg, M.G.J.; Metzner, A. (1962). "Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 269 (1336): 21–52. <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/1962RSPSA.269...21B">1962RSPSA.269...21B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1962.0161">10.1098/rspa.1962.0161</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:120125096">120125096</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+Series+A.+Mathematical+and+Physical+Sciences&rft.atitle=Gravitational+waves+in+general+relativity%3A+VII.+Waves+from+axisymmetric+isolated+systems&rft.volume=269&rft.issue=1336&rft.pages=21-52&rft.date=1962&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120125096%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1962.0161&rft_id=info%3Abibcode%2F1962RSPSA.269...21B&rft.aulast=Bondi&rft.aufirst=H.&rft.au=Van+der+Burg%2C+M.G.J.&rft.au=Metzner%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-sachs1962-152"><a href="#cite_ref-sachs1962_152-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSachs1962" class="citation journal cs1">Sachs, R. (1962). "Asymptotic symmetries in gravitational theory". Physical Review. 128 (6): 2851–2864. <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/1962PhRv..128.2851S">1962PhRv..128.2851S</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.128.2851">10.1103/PhysRev.128.2851</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&rft.atitle=Asymptotic+symmetries+in+gravitational+theory&rft.volume=128&rft.issue=6&rft.pages=2851-2864&rft.date=1962&rft_id=info%3Adoi%2F10.1103%2FPhysRev.128.2851&rft_id=info%3Abibcode%2F1962PhRv..128.2851S&rft.aulast=Sachs&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-strominger2017-153"><a href="#cite_ref-strominger2017_153-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrominger2017" class="citation arxiv cs1">Strominger, Andrew (2017). "Lectures on the Infrared Structure of Gravity and Gauge Theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1703.05448">1703.05448</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/hep-th">hep-th</a>]. <q>...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Lectures+on+the+Infrared+Structure+of+Gravity+and+Gauge+Theory&rft.date=2017&rft_id=info%3Aarxiv%2F1703.05448&rft.aulast=Strominger&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> </li> <li id="cite_note-154"><a href="#cite_ref-154">^</a> <a href="#CITEREFFrauendiener2004">Frauendiener 2004</a>, <a href="#CITEREFWald1984">Wald 1984</a>, sec. 11.1, <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, sec. 6.8, 6.9 </li> <li id="cite_note-155"><a href="#cite_ref-155">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, sec. 9.2–9.4 and <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, ch. 6 </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> <a href="#CITEREFThorne1972">Thorne 1972</a>; for more recent numerical studies, see <a href="#CITEREFBerger2002">Berger 2002</a>, sec. 2.1 </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <a href="#CITEREFIsrael1987">Israel 1987</a>. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and <a href="/wiki/Apparent_horizon" title="Apparent horizon">apparent horizons</a> cf. <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, pp. 312–320 or <a href="#CITEREFWald1984">Wald 1984</a>, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. <a href="#CITEREFAshtekarKrishnan2004">Ashtekar & Krishnan 2004</a> </li> <li id="cite_note-158"><a href="#cite_ref-158">^</a> For first steps, cf. <a href="#CITEREFIsrael1971">Israel 1971</a>; see <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, sec. 9.3 or <a href="#CITEREFHeusler1996">Heusler 1996</a>, ch. 9 and 10 for a derivation, and <a href="#CITEREFHeusler1998">Heusler 1998</a> as well as <a href="#CITEREFBeigChruściel2006">Beig & Chruściel 2006</a> as overviews of more recent results </li> <li id="cite_note-159"><a href="#cite_ref-159">^</a> The laws of black hole mechanics were first described in <a href="#CITEREFBardeenCarterHawking1973">Bardeen, Carter & Hawking 1973</a>; a more pedagogical presentation can be found in <a href="#CITEREFCarter1979">Carter 1979</a>; for a more recent review, see <a href="#CITEREFWald2001">Wald 2001</a>, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics <a href="#CITEREFPoisson2004">Poisson 2004</a>. For the Penrose process, see <a href="#CITEREFPenrose1969">Penrose 1969</a> </li> <li id="cite_note-160"><a href="#cite_ref-160">^</a> <a href="#CITEREFBekenstein1973">Bekenstein 1973</a>, <a href="#CITEREFBekenstein1974">Bekenstein 1974</a> </li> <li id="cite_note-161"><a href="#cite_ref-161">^</a> The fact that black holes radiate, quantum mechanically, was first derived in <a href="#CITEREFHawking1975">Hawking 1975</a>; a more thorough derivation can be found in <a href="#CITEREFWald1975">Wald 1975</a>. A review is given in <a href="#CITEREFWald2001">Wald 2001</a>, ch. 3 </li> <li id="cite_note-162"><a href="#cite_ref-162">^</a> <a href="#CITEREFNarlikar1993">Narlikar 1993</a>, sec. 4.4.4, 4.4.5 </li> <li id="cite_note-163"><a href="#cite_ref-163">^</a> Horizons: cf. <a href="#CITEREFRindler2001">Rindler 2001</a>, sec. 12.4. Unruh effect: <a href="#CITEREFUnruh1976">Unruh 1976</a>, cf. <a href="#CITEREFWald2001">Wald 2001</a>, ch. 3 </li> <li id="cite_note-164"><a href="#cite_ref-164">^</a> <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, sec. 8.1, <a href="#CITEREFWald1984">Wald 1984</a>, sec. 9.1 </li> <li id="cite_note-165"><a href="#cite_ref-165">^</a> <a href="#CITEREFTownsend1997">Townsend 1997</a>, ch. 2; a more extensive treatment of this solution can be found in <a href="#CITEREFChandrasekhar1983">Chandrasekhar 1983</a>, ch. 3 </li> <li id="cite_note-166"><a href="#cite_ref-166">^</a> <a href="#CITEREFTownsend1997">Townsend 1997</a>, ch. 4; for a more extensive treatment, cf. <a href="#CITEREFChandrasekhar1983">Chandrasekhar 1983</a>, ch. 6 </li> <li id="cite_note-167"><a href="#cite_ref-167">^</a> <a href="#CITEREFEllisVan_Elst1999">Ellis & Van Elst 1999</a>; a closer look at the singularity itself is taken in <a href="#CITEREFBörner1993">Börner 1993</a>, sec. 1.2 </li> <li id="cite_note-168"><a href="#cite_ref-168">^</a> Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called <a href="/wiki/Eikonal_approximation" title="Eikonal approximation">eikonal approximations</a> of many wave equations, namely the "<a href="/wiki/Caustic_(mathematics)" title="Caustic (mathematics)">caustics</a>", are resolved into finite peaks beyond that approximation. </li> <li id="cite_note-169"><a href="#cite_ref-169">^</a> Namely when there are <a href="/wiki/Trapped_null_surface" class="mw-redirect" title="Trapped null surface">trapped null surfaces</a>, cf. <a href="#CITEREFPenrose1965">Penrose 1965</a> </li> <li id="cite_note-170"><a href="#cite_ref-170">^</a> <a href="#CITEREFHawking1966">Hawking 1966</a> </li> <li id="cite_note-171"><a href="#cite_ref-171">^</a> The conjecture was made in <a href="#CITEREFBelinskiiKhalatnikovLifschitz1971">Belinskii, Khalatnikov & Lifschitz 1971</a>; for a more recent review, see <a href="#CITEREFBerger2002">Berger 2002</a>. An accessible exposition is given by <a href="#CITEREFGarfinkle2007">Garfinkle 2007</a> </li> <li id="cite_note-172"><a href="#cite_ref-172">^</a> The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in <a href="#CITEREFPenrose1969">Penrose 1969</a>; a textbook-level account is given in <a href="#CITEREFWald1984">Wald 1984</a>, pp. 302–305. For numerical results, see the review <a href="#CITEREFBerger2002">Berger 2002</a>, sec. 2.1 </li> <li id="cite_note-173"><a href="#cite_ref-173">^</a> <a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, sec. 7.1 </li> <li id="cite_note-174"><a href="#cite_ref-174">^</a> <a href="#CITEREFArnowittDeserMisner1962">Arnowitt, Deser & Misner 1962</a>; for a pedagogical introduction, see <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, § 21.4–§ 21.7 </li> <li id="cite_note-175"><a href="#cite_ref-175">^</a> <a href="#CITEREFFourès-Bruhat1952">Fourès-Bruhat 1952</a> and <a href="#CITEREFBruhat1962">Bruhat 1962</a>; for a pedagogical introduction, see <a href="#CITEREFWald1984">Wald 1984</a>, ch. 10; an online review can be found in <a href="#CITEREFReula1998">Reula 1998</a> </li> <li id="cite_note-176"><a href="#cite_ref-176">^</a> <a href="#CITEREFGourgoulhon2007">Gourgoulhon 2007</a>; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see <a href="#CITEREFLehner2001">Lehner 2001</a> </li> <li id="cite_note-177"><a href="#cite_ref-177">^</a> <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, § 20.4 </li> <li id="cite_note-178"><a href="#cite_ref-178">^</a> <a href="#CITEREFArnowittDeserMisner1962">Arnowitt, Deser & Misner 1962</a> </li> <li id="cite_note-179"><a href="#cite_ref-179">^</a> <a href="#CITEREFKomar1959">Komar 1959</a>; for a pedagogical introduction, see <a href="#CITEREFWald1984">Wald 1984</a>, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. <a href="#CITEREFAshtekarMagnon-Ashtekar1979">Ashtekar & Magnon-Ashtekar 1979</a> </li> <li id="cite_note-180"><a href="#cite_ref-180">^</a> For a pedagogical introduction, see <a href="#CITEREFWald1984">Wald 1984</a>, sec. 11.2 </li> <li id="cite_note-181"><a href="#cite_ref-181">^</a> <a href="#CITEREFWald1984">Wald 1984</a>, p. 295 and refs therein; this is important for questions of stability—if there were <a href="/wiki/Negative_mass" title="Negative mass">negative mass</a> states, then flat, empty Minkowski space, which has mass zero, could evolve into these states </li> <li id="cite_note-182"><a href="#cite_ref-182">^</a> <a href="#CITEREFTownsend1997">Townsend 1997</a>, ch. 5 </li> <li id="cite_note-183"><a href="#cite_ref-183">^</a> Such quasi-local mass–energy definitions are the <a href="/wiki/Hawking_energy" title="Hawking energy">Hawking energy</a>, <a href="/wiki/Geroch_energy" title="Geroch energy">Geroch energy</a>, or Penrose's quasi-local energy–momentum based on <a href="/wiki/Twistor_theory" title="Twistor theory">twistor</a> methods; cf. the review article <a href="#CITEREFSzabados2004">Szabados 2004</a> </li> <li id="cite_note-184"><a href="#cite_ref-184">^</a> An overview of quantum theory can be found in standard textbooks such as <a href="#CITEREFMessiah1999">Messiah 1999</a>; a more elementary account is given in <a href="#CITEREFHeyWalters2003">Hey & Walters 2003</a> </li> <li id="cite_note-185"><a href="#cite_ref-185">^</a> <a href="#CITEREFRamond1990">Ramond 1990</a>, <a href="#CITEREFWeinberg1995">Weinberg 1995</a>, <a href="#CITEREFPeskinSchroeder1995">Peskin & Schroeder 1995</a>; a more accessible overview is <a href="#CITEREFAuyang1995">Auyang 1995</a> </li> <li id="cite_note-186"><a href="#cite_ref-186">^</a> <a href="#CITEREFWald1994">Wald 1994</a>, <a href="#CITEREFBirrellDavies1984">Birrell & Davies 1984</a> </li> <li id="cite_note-187"><a href="#cite_ref-187">^</a> For Hawking radiation <a href="#CITEREFHawking1975">Hawking 1975</a>, <a href="#CITEREFWald1975">Wald 1975</a>; an accessible introduction to black hole evaporation can be found in <a href="#CITEREFTraschen2000">Traschen 2000</a> </li> <li id="cite_note-188"><a href="#cite_ref-188">^</a> <a href="#CITEREFWald2001">Wald 2001</a>, ch. 3 </li> <li id="cite_note-189"><a href="#cite_ref-189">^</a> Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. <a href="#CITEREFCarlip2001">Carlip 2001</a>, sec. 2 </li> <li id="cite_note-190"><a href="#cite_ref-190">^</a> <a href="#CITEREFSchutz2003">Schutz 2003</a>, p. 407 </li> <li id="cite_note-Hamber_2009-191">^ <a href="#cite_ref-Hamber_2009_191-0">a</a> <a href="#cite_ref-Hamber_2009_191-1">b</a> <a href="#CITEREFHamber2009">Hamber 2009</a> </li> <li id="cite_note-192"><a href="#cite_ref-192">^</a> A timeline and overview can be found in <a href="#CITEREFRovelli2000">Rovelli 2000</a> </li> <li id="cite_note-193"><a href="#cite_ref-193">^</a> <a href="#CITEREF't_HooftVeltman1974">'t Hooft & Veltman 1974</a> </li> <li id="cite_note-194"><a href="#cite_ref-194">^</a> <a href="#CITEREFDonoghue1995">Donoghue 1995</a> </li> <li id="cite_note-195"><a href="#cite_ref-195">^</a> In particular, a perturbative technique known as <a href="/wiki/Renormalization" title="Renormalization">renormalization</a>, an integral part of deriving predictions which take into account higher-energy contributions, cf. <a href="#CITEREFWeinberg1996">Weinberg 1996</a>, ch. 17, 18, fails in this case; cf. <a href="#CITEREFVeltman1975">Veltman 1975</a>, <a href="#CITEREFGoroffSagnotti1985">Goroff & Sagnotti 1985</a>; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see <a href="#CITEREFHamber2009">Hamber 2009</a> </li> <li id="cite_note-196"><a href="#cite_ref-196">^</a> An accessible introduction at the undergraduate level can be found in <a href="#CITEREFZwiebach2004">Zwiebach 2004</a>; more complete overviews can be found in <a href="#CITEREFPolchinski1998a">Polchinski 1998a</a> and <a href="#CITEREFPolchinski1998b">Polchinski 1998b</a> </li> <li id="cite_note-197"><a href="#cite_ref-197">^</a> At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different <a href="/wiki/Normal_mode" title="Normal mode">modes</a> of oscillation of one and the same type of fundamental string appear as particles with different (<a href="/wiki/Electric_charge" title="Electric charge">electric</a> and other) <a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a>, e.g. <a href="#CITEREFIbanez2000">Ibanez 2000</a>. The theory is successful in that one mode will always correspond to a <a href="/wiki/Graviton" title="Graviton">graviton</a>, the <a href="/wiki/Messenger_particle" class="mw-redirect" title="Messenger particle">messenger particle</a> of gravity, e.g. <a href="#CITEREFGreenSchwarzWitten1987">Green, Schwarz & Witten 1987</a>, sec. 2.3, 5.3 </li> <li id="cite_note-198"><a href="#cite_ref-198">^</a> <a href="#CITEREFGreenSchwarzWitten1987">Green, Schwarz & Witten 1987</a>, sec. 4.2 </li> <li id="cite_note-199"><a href="#cite_ref-199">^</a> <a href="#CITEREFWeinberg2000">Weinberg 2000</a>, ch. 31 </li> <li id="cite_note-200"><a href="#cite_ref-200">^</a> <a href="#CITEREFTownsend1996">Townsend 1996</a>, <a href="#CITEREFDuff1996">Duff 1996</a> </li> <li id="cite_note-201"><a href="#cite_ref-201">^</a> <a href="#CITEREFKuchař1973">Kuchař 1973</a>, sec. 3 </li> <li id="cite_note-202"><a href="#cite_ref-202">^</a> These variables represent geometric gravity using mathematical analogues of <a href="/wiki/Electric_field" title="Electric field">electric</a> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a>; cf. <a href="#CITEREFAshtekar1986">Ashtekar 1986</a>, <a href="#CITEREFAshtekar1987">Ashtekar 1987</a> </li> <li id="cite_note-203"><a href="#cite_ref-203">^</a> For a review, see <a href="#CITEREFThiemann2007">Thiemann 2007</a>; more extensive accounts can be found in <a href="#CITEREFRovelli1998">Rovelli 1998</a>, <a href="#CITEREFAshtekarLewandowski2004">Ashtekar & Lewandowski 2004</a> as well as in the lecture notes <a href="#CITEREFThiemann2003">Thiemann 2003</a> </li> <li id="cite_note-204"><a href="#cite_ref-204">^</a> <a href="#CITEREFIsham1994">Isham 1994</a>, <a href="#CITEREFSorkin1997">Sorkin 1997</a> </li> <li id="cite_note-205"><a href="#cite_ref-205">^</a> <a href="#CITEREFLoll1998">Loll 1998</a> </li> <li id="cite_note-206"><a href="#cite_ref-206">^</a> <a href="#CITEREFSorkin2005">Sorkin 2005</a> </li> <li id="cite_note-207"><a href="#cite_ref-207">^</a> <a href="#CITEREFPenrose2004">Penrose 2004</a>, ch. 33 and refs therein </li> <li id="cite_note-208"><a href="#cite_ref-208">^</a> <a href="#CITEREFHawking1987">Hawking 1987</a> </li> <li id="cite_note-209"><a href="#cite_ref-209">^</a> <a href="#CITEREFAshtekar2007">Ashtekar 2007</a>, <a href="#CITEREFSchwarz2007">Schwarz 2007</a> </li> <li id="cite_note-210"><a href="#cite_ref-210">^</a> <a href="#CITEREFMaddox1998">Maddox 1998</a>, pp. 52–59, 98–122; <a href="#CITEREFPenrose2004">Penrose 2004</a>, sec. 34.1, ch. 30 </li> <li id="cite_note-211"><a href="#cite_ref-211">^</a> section <a href="#Quantum_gravity">Quantum gravity</a>, above </li> <li id="cite_note-212"><a href="#cite_ref-212">^</a> section <a href="#Cosmology">Cosmology</a>, above </li> <li id="cite_note-213"><a href="#cite_ref-213">^</a> <a href="#CITEREFFriedrich2005">Friedrich 2005</a> </li> <li id="cite_note-214"><a href="#cite_ref-214">^</a> A review of the various problems and the techniques being developed to overcome them, see <a href="#CITEREFLehner2002">Lehner 2002</a> </li> <li id="cite_note-215"><a href="#cite_ref-215">^</a> See <a href="#CITEREFBartusiak2000">Bartusiak 2000</a> for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as <a href="/wiki/GEO600" title="GEO600">GEO600</a> and <a href="/wiki/LIGO" title="LIGO">LIGO</a> </li> <li id="cite_note-216"><a href="#cite_ref-216">^</a> For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see <a href="#CITEREFBlanchetFayeIyerSinha2008">Blanchet et al. 2008</a>, and <a href="#CITEREFArunBlanchetIyerQusailah2008">Arun et al. 2008</a>; for a review of work on compact binaries, see <a href="#CITEREFBlanchet2006">Blanchet 2006</a> and <a href="#CITEREFFutamaseItoh2006">Futamase & Itoh 2006</a>; for a general review of experimental tests of general relativity, see <a href="#CITEREFWill2006">Will 2006</a> </li> <li id="cite_note-217"><a href="#cite_ref-217">^</a> See, e.g., the <a href="/wiki/Living_Reviews_in_Relativity" title="Living Reviews in Relativity">Living Reviews in Relativity</a> journal. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=38" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlpherHerman1948" class="citation cs2"><a href="/wiki/Ralph_Asher_Alpher" class="mw-redirect" title="Ralph Asher Alpher">Alpher, R. A.</a>; Herman, R. C. (1948), "Evolution of the universe", Nature, 162 (4124): 774–775, <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/1948Natur.162..774A">1948Natur.162..774A</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F162774b0">10.1038/162774b0</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:4113488">4113488</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Evolution+of+the+universe&rft.volume=162&rft.issue=4124&rft.pages=774-775&rft.date=1948&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4113488%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F162774b0&rft_id=info%3Abibcode%2F1948Natur.162..774A&rft.aulast=Alpher&rft.aufirst=R.+A.&rft.au=Herman%2C+R.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonCampbellJurgensLau1992" class="citation cs2">Anderson, J. D.; Campbell, J. K.; Jurgens, R. F.; Lau, E. L. (1992), "Recent developments in solar-system tests of general relativity", in Sato, H.; Nakamura, T. (eds.), Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, World Scientific, pp. 353–355, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-0950-6" title="Special:BookSources/978-981-02-0950-6"><bdi>978-981-02-0950-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Recent+developments+in+solar-system+tests+of+general+relativity&rft.btitle=Proceedings+of+the+Sixth+Marcel+Grossmann+Meeting+on+General+Relativity&rft.pages=353-355&rft.pub=World+Scientific&rft.date=1992&rft.isbn=978-981-02-0950-6&rft.aulast=Anderson&rft.aufirst=J.+D.&rft.au=Campbell%2C+J.+K.&rft.au=Jurgens%2C+R.+F.&rft.au=Lau%2C+E.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold1989" class="citation cs2"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, V. 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Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-26841-5" title="Special:BookSources/978-3-319-26841-5"><bdi>978-3-319-26841-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Practice+of+Natural+Computing%3A+Fourth+International+Conference%2C+TPNC+2015%2C+Mieres%2C+Spain%2C+December+15%E2%80%9316%2C+2015.+Proceedings&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-319-26841-5&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXmwiCwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuff1996" class="citation cs2"><a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff, Michael</a> (1996), "M-Theory (the Theory Formerly Known as Strings)", Int. J. 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A, 11 (32): 5623–5641, <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-th/9608117">hep-th/9608117</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/1996IJMPA..11.5623D">1996IJMPA..11.5623D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X96002583">10.1142/S0217751X96002583</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:17432791">17432791</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Int.+J.+Mod.+Phys.+A&rft.atitle=M-Theory+%28the+Theory+Formerly+Known+as+Strings%29&rft.volume=11&rft.issue=32&rft.pages=5623-5641&rft.date=1996&rft_id=info%3Aarxiv%2Fhep-th%2F9608117&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17432791%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0217751X96002583&rft_id=info%3Abibcode%2F1996IJMPA..11.5623D&rft.aulast=Duff&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhlers1973" class="citation cs2"><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers, Jürgen</a> (1973), "Survey of general relativity theory", in Israel, Werner (ed.), Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-277-0369-9" title="Special:BookSources/978-90-277-0369-9"><bdi>978-90-277-0369-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Survey+of+general+relativity+theory&rft.btitle=Relativity%2C+Astrophysics+and+Cosmology&rft.pages=1-125&rft.pub=D.+Reidel&rft.date=1973&rft.isbn=978-90-277-0369-9&rft.aulast=Ehlers&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhlersFalcoSchneider1992" class="citation cs2">Ehlers, Jürgen; Falco, Emilio E.; Schneider, Peter (1992), Gravitational lenses, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66506-9" title="Special:BookSources/978-3-540-66506-9"><bdi>978-3-540-66506-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitational+lenses&rft.pub=Springer&rft.date=1992&rft.isbn=978-3-540-66506-9&rft.aulast=Ehlers&rft.aufirst=J%C3%BCrgen&rft.au=Falco%2C+Emilio+E.&rft.au=Schneider%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhlersLämmerzahl2006" class="citation cs2">Ehlers, Jürgen; Lämmerzahl, Claus, eds. (2006), Special Relativity—Will it Survive the Next 101 Years?, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-34522-0" title="Special:BookSources/978-3-540-34522-0"><bdi>978-3-540-34522-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+Relativity%E2%80%94Will+it+Survive+the+Next+101+Years%3F&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-34522-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhlersRindler1997" class="citation cs2">Ehlers, Jürgen; <a href="/wiki/Wolfgang_Rindler" title="Wolfgang Rindler">Rindler, Wolfgang</a> (1997), "Local and Global Light Bending in Einstein's and other Gravitational Theories", General Relativity and Gravitation, 29 (4): 519–529, <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/1997GReGr..29..519E">1997GReGr..29..519E</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1018843001842">10.1023/A:1018843001842</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11858%2F00-001M-0000-0013-5AB5-4">11858/00-001M-0000-0013-5AB5-4</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:118162303">118162303</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=Local+and+Global+Light+Bending+in+Einstein%27s+and+other+Gravitational+Theories&rft.volume=29&rft.issue=4&rft.pages=519-529&rft.date=1997&rft_id=info%3Ahdl%2F11858%2F00-001M-0000-0013-5AB5-4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118162303%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1018843001842&rft_id=info%3Abibcode%2F1997GReGr..29..519E&rft.aulast=Ehlers&rft.aufirst=J%C3%BCrgen&rft.au=Rindler%2C+Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1907" class="citation cs2"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1907), <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol2-doc/468">"Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen"</a>, Jahrbuch der Radioaktivität und Elektronik, 4: 411</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahrbuch+der+Radioaktivit%C3%A4t+und+Elektronik&rft.atitle=%C3%9Cber+das+Relativit%C3%A4tsprinzip+und+die+aus+demselben+gezogene+Folgerungen&rft.volume=4&rft.pages=411&rft.date=1907&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Feinsteinpapers.press.princeton.edu%2Fvol2-doc%2F468&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> See also <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol2-trans/266">English translation at Einstein Papers Project</a></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1915" class="citation cs2"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1915), <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-doc/272">"Die Feldgleichungen der Gravitation"</a>, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+Preussischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=Die+Feldgleichungen+der+Gravitation&rft.pages=844-847&rft.date=1915&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Feinsteinpapers.press.princeton.edu%2Fvol6-doc%2F272&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> See also <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-trans/129">English translation at Einstein Papers Project</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1917" class="citation cs2"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1917), <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-doc/568">"Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie"</a>, Sitzungsberichte der Preußischen Akademie der Wissenschaften: 142</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+Preu%C3%9Fischen+Akademie+der+Wissenschaften&rft.atitle=Kosmologische+Betrachtungen+zur+allgemeinen+Relativit%C3%A4tstheorie&rft.pages=142&rft.date=1917&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Feinsteinpapers.press.princeton.edu%2Fvol6-doc%2F568&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> See also <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-trans/433">English translation at Einstein Papers Project</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEllisVan_Elst1999" class="citation cs2"><a href="/wiki/George_Francis_Rayner_Ellis" class="mw-redirect" title="George Francis Rayner Ellis">Ellis, George F R</a>; Van Elst, Henk (1999), "Cosmological Models", in Lachièze-Rey, Marc (ed.), Theoretical and Observational Cosmology, vol. 541, pp. 1–116, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9812046">gr-qc/9812046</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/1999ASIC..541....1E">1999ASIC..541....1E</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%2F978-94-011-4455-1_1">10.1007/978-94-011-4455-1_1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-5946-3" title="Special:BookSources/978-0-7923-5946-3"><bdi>978-0-7923-5946-3</bdi></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:122994560">122994560</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cosmological+Models&rft.btitle=Theoretical+and+Observational+Cosmology&rft.pages=1-116&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122994560%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1999ASIC..541....1E&rft_id=info%3Aarxiv%2Fgr-qc%2F9812046&rft_id=info%3Adoi%2F10.1007%2F978-94-011-4455-1_1&rft.isbn=978-0-7923-5946-3&rft.aulast=Ellis&rft.aufirst=George+F+R&rft.au=Van+Elst%2C+Henk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngler2002" class="citation cs2">Engler, Gideon (2002), "Einstein and the most beautiful theories in physics", International Studies in the Philosophy of Science, 16 (1): 27–37, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external 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W. F.; Buchman, S.; DeBra, D. B.; Keiser, G. M. (2001), "Gravity Probe B: Countdown to launch", in Lämmerzahl, C.; Everitt, C. W. F.; Hehl, F. W. 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Living Reviews in Relativity. 7 (1). 6. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0306052">gr-qc/0306052</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/2004LRR.....7....6P">2004LRR.....7....6P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12942%2Flrr-2004-6">10.12942/lrr-2004-6</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256043">5256043</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/28179866">28179866</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Reviews+in+Relativity&rft.atitle=The+Motion+of+Point+Particles+in+Curved+Spacetime&rft.volume=7&rft.issue=1&rft.pages=6&rft.date=2004-05-27&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256043%23id-name%3DPMC&rft_id=info%3Abibcode%2F2004LRR.....7....6P&rft_id=info%3Aarxiv%2Fgr-qc%2F0306052&rft_id=info%3Apmid%2F28179866&rft_id=info%3Adoi%2F10.12942%2Flrr-2004-6&rft.aulast=Poisson&rft.aufirst=Eric&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256043&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoisson2004" class="citation cs2">Poisson, Eric (2004), A Relativist's Toolkit. 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title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-43973-2" title="Special:BookSources/978-0-521-43973-2"><bdi>978-0-521-43973-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+experiment+in+gravitational+physics&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=978-0-521-43973-2&rft.aulast=Will&rft.aufirst=Clifford+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWill2006" class="citation cs2"><a href="/wiki/Clifford_Will" class="mw-redirect" title="Clifford Will">Will, Clifford M.</a> (2006), "The Confrontation between General Relativity and Experiment", Living Reviews in Relativity, 9 (1): 3, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0510072">gr-qc/0510072</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/2006LRR.....9....3W">2006LRR.....9....3W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12942%2Flrr-2006-3">10.12942/lrr-2006-3</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256066">5256066</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/28179873">28179873</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Reviews+in+Relativity&rft.atitle=The+Confrontation+between+General+Relativity+and+Experiment&rft.volume=9&rft.issue=1&rft.pages=3&rft.date=2006&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256066%23id-name%3DPMC&rft_id=info%3Abibcode%2F2006LRR.....9....3W&rft_id=info%3Aarxiv%2Fgr-qc%2F0510072&rft_id=info%3Apmid%2F28179873&rft_id=info%3Adoi%2F10.12942%2Flrr-2006-3&rft.aulast=Will&rft.aufirst=Clifford+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZwiebach2004" class="citation cs2"><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach, Barton</a> (2004), A First Course in String Theory, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83143-7" title="Special:BookSources/978-0-521-83143-7"><bdi>978-0-521-83143-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+String+Theory&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-83143-7&rft.aulast=Zwiebach&rft.aufirst=Barton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=General_relativity&action=edit&section=39" title="Edit section: Further reading">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Popular_books">Popular books</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=40" title="Edit section: Popular books">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, A.</a> (1916), <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>, Berlin, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-528-06059-6" title="Special:BookSources/978-3-528-06059-6"><bdi>978-3-528-06059-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+The+Special+and+the+General+Theory&rft.place=Berlin&rft.date=1916&rft.isbn=978-3-528-06059-6&rft.aulast=Einstein&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch, R.</a> (1981), General Relativity from A to B, Chicago: University of Chicago Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-28864-2" title="Special:BookSources/978-0-226-28864-2"><bdi>978-0-226-28864-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity+from+A+to+B&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1981&rft.isbn=978-0-226-28864-2&rft.aulast=Geroch&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Lillian_Lieber" class="mw-redirect" title="Lillian Lieber">Lieber, Lillian</a> (2008), The Einstein Theory of Relativity: A Trip to the Fourth Dimension, Philadelphia: Paul Dry Books, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58988-044-3" title="Special:BookSources/978-1-58988-044-3"><bdi>978-1-58988-044-3</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+Einstein+Theory+of+Relativity%3A+A+Trip+to+the+Fourth+Dimension&rft.place=Philadelphia&rft.pub=Paul+Dry+Books%2C+Inc.&rft.date=2008&rft.isbn=978-1-58988-044-3&rft.au=Lieber%2C+Lillian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Bernard_F._Schutz" title="Bernard F. Schutz">Schutz, Bernard F.</a> (2001), "Gravitational radiation", in Murdin, Paul (ed.), Encyclopedia of Astronomy and Astrophysics, Institute of Physics Pub., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56159-268-5" title="Special:BookSources/978-1-56159-268-5"><bdi>978-1-56159-268-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Gravitational+radiation&rft.btitle=Encyclopedia+of+Astronomy+and+Astrophysics&rft.pub=Institute+of+Physics+Pub.&rft.date=2001&rft.isbn=978-1-56159-268-5&rft.aulast=Schutz&rft.aufirst=Bernard+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip</a>; Hawking, Stephen (1994). <a href="/wiki/Black_Holes_and_Time_Warps" title="Black Holes and Time Warps">Black Holes and Time Warps: Einstein's Outrageous Legacy</a>. New York: W. W. Norton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-03505-0" title="Special:BookSources/0-393-03505-0"><bdi>0-393-03505-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Black+Holes+and+Time+Warps%3A+Einstein%27s+Outrageous+Legacy&rft.place=New+York&rft.pub=W.+W.+Norton&rft.date=1994&rft.isbn=0-393-03505-0&rft.aulast=Thorne&rft.aufirst=Kip&rft.au=Hawking%2C+Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, Robert M.</a> (1992), Space, Time, and Gravity: the Theory of the Big Bang and Black Holes, Chicago: University of Chicago Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-87029-8" title="Special:BookSources/978-0-226-87029-8"><bdi>978-0-226-87029-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space%2C+Time%2C+and+Gravity%3A+the+Theory+of+the+Big+Bang+and+Black+Holes&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1992&rft.isbn=978-0-226-87029-8&rft.au=Wald%2C+Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John</a>; Ford, Kenneth (1998), Geons, Black Holes, & Quantum Foam: a life in physics, New York: W. W. Norton, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-31991-0" title="Special:BookSources/978-0-393-31991-0"><bdi>978-0-393-31991-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geons%2C+Black+Holes%2C+%26+Quantum+Foam%3A+a+life+in+physics&rft.place=New+York&rft.pub=W.+W.+Norton&rft.date=1998&rft.isbn=978-0-393-31991-0&rft.aulast=Wheeler&rft.aufirst=John&rft.au=Ford%2C+Kenneth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Beginning_undergraduate_textbooks">Beginning undergraduate textbooks</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=41" title="Edit section: Beginning undergraduate textbooks">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Callahan, James J. (2000), 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/978-0-387-98641-8" title="Special:BookSources/978-0-387-98641-8"><bdi>978-0-387-98641-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Spacetime%3A+an+Introduction+to+Special+and+General+Relativity&rft.place=New+York&rft.pub=Springer&rft.date=2000&rft.isbn=978-0-387-98641-8&rft.au=Callahan%2C+James+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-38423-9" title="Special:BookSources/978-0-201-38423-9"><bdi>978-0-201-38423-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exploring+Black+Holes%3A+Introduction+to+General+Relativity&rft.pub=Addison+Wesley&rft.date=2000&rft.isbn=978-0-201-38423-9&rft.au=Taylor%2C+Edwin+F.&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Advanced_undergraduate_textbooks">Advanced undergraduate textbooks</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=42" title="Edit section: Advanced undergraduate textbooks">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Cheng, Ta-Pei (2005), Relativity, Gravitation and Cosmology: a Basic Introduction, Oxford and New York: Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852957-6" title="Special:BookSources/978-0-19-852957-6"><bdi>978-0-19-852957-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%2C+Gravitation+and+Cosmology%3A+a+Basic+Introduction&rft.place=Oxford+and+New+York&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-852957-6&rft.aulast=Cheng&rft.aufirst=Ta-Pei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac, Paul</a> (1996), General Theory of Relativity, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-01146-2" title="Special:BookSources/978-0-691-01146-2"><bdi>978-0-691-01146-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Theory+of+Relativity&rft.pub=Princeton+University+Press&rft.date=1996&rft.isbn=978-0-691-01146-2&rft.aulast=Dirac&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, <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&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-69199-2&rft.aulast=Gron&rft.aufirst=O.&rft.au=Hervik%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/James_Hartle" title="James Hartle">Hartle, James B.</a> (2003), Gravity: an Introduction to Einstein's General Relativity, San Francisco: Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-8662-2" title="Special:BookSources/978-0-8053-8662-2"><bdi>978-0-8053-8662-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravity%3A+an+Introduction+to+Einstein%27s+General+Relativity&rft.place=San+Francisco&rft.pub=Addison-Wesley&rft.date=2003&rft.isbn=978-0-8053-8662-2&rft.au=Hartle%2C+James+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Lane_P._Hughston" title="Lane P. Hughston">Hughston, L. P.</a>; Tod, K. P. (1991), Introduction to General Relativity, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-33943-8" title="Special:BookSources/978-0-521-33943-8"><bdi>978-0-521-33943-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+General+Relativity&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1991&rft.isbn=978-0-521-33943-8&rft.au=Hughston%2C+L.+P.&rft.au=Tod%2C+K.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">d'Inverno, Ray (1992), <a rel="nofollow" class="external text" href="https://archive.org/details/introducingeinst0000dinv">Introducing Einstein's Relativity</a>, Oxford: Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-859686-8" title="Special:BookSources/978-0-19-859686-8"><bdi>978-0-19-859686-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Einstein%27s+Relativity&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=1992&rft.isbn=978-0-19-859686-8&rft.au=d%27Inverno%2C+Ray&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroducingeinst0000dinv&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-35797-8" title="Special:BookSources/978-3-642-35797-8"><bdi>978-3-642-35797-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein+in+Matrix+Form&rft.place=Berlin&rft.edition=1st&rft.pub=Springer&rft.date=2013&rft.isbn=978-3-642-35797-8&rft.aulast=Ludyk&rft.aufirst=G%C3%BCnter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Møller, Christian (1955) [1952], <a rel="nofollow" class="external text" href="https://archive.org/details/theoryofrelativi029229mbp">The Theory of Relativity</a>, Oxford University Press, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/7644624">7644624</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Relativity&rft.pub=Oxford+University+Press&rft.date=1955&rft_id=info%3Aoclcnum%2F7644624&rft.aulast=M%C3%B8ller&rft.aufirst=Christian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryofrelativi029229mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Moore, Thomas A (2012), A General Relativity Workbook, University Science Books, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891389-82-5" title="Special:BookSources/978-1-891389-82-5"><bdi>978-1-891389-82-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+General+Relativity+Workbook&rft.pub=University+Science+Books&rft.date=2012&rft.isbn=978-1-891389-82-5&rft.au=Moore%2C+Thomas+A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Bernard_F._Schutz" title="Bernard F. Schutz">Schutz, B. F.</a> (2009), <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseingen00bern_0">A First Course in General Relativity</a> (Second ed.), Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88705-2" title="Special:BookSources/978-0-521-88705-2"><bdi>978-0-521-88705-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+General+Relativity&rft.edition=Second&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-88705-2&rft.au=Schutz%2C+B.+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffirstcourseingen00bern_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graduate_textbooks">Graduate textbooks</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=43" title="Edit section: Graduate textbooks">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Sean_M._Carroll" title="Sean M. Carroll">Carroll, Sean M.</a> (2004), Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison-Wesley, <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.place=San+Francisco&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-8053-8732-2&rft.au=Carroll%2C+Sean+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/%C3%98yvind_Gr%C3%B8n" title="Øyvind Grøn">Grøn, Øyvind</a>; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, <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&rft.place=New+York&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-69199-2&rft.aulast=Gr%C3%B8n&rft.aufirst=%C3%98yvind&rft.au=Hervik%2C+Sigbj%C3%B8rn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evgeny F.</a> (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2768-9" title="Special:BookSources/978-0-7506-2768-9"><bdi>978-0-7506-2768-9</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+Classical+Theory+of+Fields+%284th+ed.%29&rft.place=London&rft.pub=Butterworth-Heinemann&rft.date=1980&rft.isbn=978-0-7506-2768-9&rft.au=Landau%2C+Lev+D.&rft.au=Lifshitz%2C+Evgeny+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Stephani, Hans (1990), General Relativity: An Introduction to the Theory of the Gravitational Field, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-37941-0" title="Special:BookSources/978-0-521-37941-0"><bdi>978-0-521-37941-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity%3A+An+Introduction+to+the+Theory+of+the+Gravitational+Field&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1990&rft.isbn=978-0-521-37941-0&rft.au=Stephani%2C+Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Clifford_M._Will" class="mw-redirect" title="Clifford M. Will">Will, Clifford</a>; Poisson, Eric (2014). Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-03286-6" title="Special:BookSources/978-1-107-03286-6"><bdi>978-1-107-03286-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravity%3A+Newtonian%2C+Post-Newtonian%2C+Relativistic&rft.pub=Cambridge+University+Press&rft.date=2014&rft.isbn=978-1-107-03286-6&rft.aulast=Will&rft.aufirst=Clifford&rft.au=Poisson%2C+Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><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_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> (1973), <a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Gravitation</a>, W. H. Freeman, Princeton University Press, <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"><bdi>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.pub=W.+H.+Freeman%2C+Princeton+University+Press&rft.date=1973&rft.isbn=0-7167-0344-0&rft.au=Charles+W.+Misner&rft.au=Kip+S.+Thorne&rft.au=John+Archibald+Wheeler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Rainer_K._Sachs" title="Rainer K. Sachs">R.K. Sachs</a>; H. Wu (1977), General Relativity for Mathematicians, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1461299055" title="Special:BookSources/1461299055"><bdi>1461299055</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity+for+Mathematicians&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=1461299055&rft.au=R.K.+Sachs&rft.au=H.+Wu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, Robert M.</a> (1984). <a href="/wiki/General_Relativity_(book)" title="General Relativity (book)">General Relativity</a>. Chicago: University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-87032-4" title="Special:BookSources/0-226-87032-4"><bdi>0-226-87032-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/10018614">10018614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1984&rft_id=info%3Aoclcnum%2F10018614&rft.isbn=0-226-87032-4&rft.aulast=Wald&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Specialists'_books">Specialists' books</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=44" title="Edit section: Specialists' books">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking, Stephen</a>; <a href="/wiki/George_F._R._Ellis" title="George F. R. Ellis">Ellis, George</a> (1975). <a href="/wiki/The_Large_Scale_Structure_of_Space-Time" class="mw-redirect" title="The Large Scale Structure of Space-Time">The Large Scale Structure of Space-time</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09906-6" title="Special:BookSources/978-0-521-09906-6"><bdi>978-0-521-09906-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Large+Scale+Structure+of+Space-time&rft.pub=Cambridge+University+Press&rft.date=1975&rft.isbn=978-0-521-09906-6&rft.aulast=Hawking&rft.aufirst=Stephen&rft.au=Ellis%2C+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Eric_Poisson" title="Eric Poisson">Poisson, Eric</a> (2007). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53780-3" title="Special:BookSources/978-0-521-53780-3"><bdi>978-0-521-53780-3</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+Relativist%27s+Toolkit%3A+The+Mathematics+of+Black-Hole+Mechanics&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-53780-3&rft.aulast=Poisson&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Journal_articles">Journal articles</h3>[<a href="/w/index.php?title=General_relativity&action=edit&section=45" title="Edit section: Journal articles">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1916), <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-doc/311">"Die Grundlage der allgemeinen Relativitätstheorie"</a>, Annalen der Physik, 49 (7): 769–822, <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/1916AnP...354..769E">1916AnP...354..769E</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19163540702">10.1002/andp.19163540702</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Die+Grundlage+der+allgemeinen+Relativit%C3%A4tstheorie&rft.volume=49&rft.issue=7&rft.pages=769-822&rft.date=1916&rft_id=info%3Adoi%2F10.1002%2Fandp.19163540702&rft_id=info%3Abibcode%2F1916AnP...354..769E&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Feinsteinpapers.press.princeton.edu%2Fvol6-doc%2F311&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"> See also <a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol6-trans/158">English translation at Einstein Papers Project</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J. Phys., 7 (1): 204, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0501041">gr-qc/0501041</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/2005NJPh....7..204F">2005NJPh....7..204F</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%2F1367-2630%2F7%2F1%2F204">10.1088/1367-2630/7/1/204</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+J.+Phys.&rft.atitle=The+basics+of+gravitational+wave+theory&rft.volume=7&rft.issue=1&rft.pages=204&rft.date=2005&rft_id=info%3Aarxiv%2Fgr-qc%2F0501041&rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F7%2F1%2F204&rft_id=info%3Abibcode%2F2005NJPh....7..204F&rft.aulast=Flanagan&rft.aufirst=%C3%89anna+%C3%89.&rft.au=Hughes%2C+Scott+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. 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"Lecture Notes on General Relativity". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9712019">gr-qc/9712019</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Lecture+Notes+on+General+Relativity&rft.date=1997&rft_id=info%3Aarxiv%2Fgr-qc%2F9712019&rft.au=Carroll%2C+Sean+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1">Moor, Rafi. <a rel="nofollow" class="external text" href="http://www.rafimoor.com/english/GRE.htm">"Understanding General Relativity"</a>. Retrieved 11 July 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Understanding+General+Relativity&rft.au=Moor%2C+Rafi&rft_id=http%3A%2F%2Fwww.rafimoor.com%2Fenglish%2FGRE.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1">Waner, Stefan. <a rel="nofollow" class="external text" href="http://www.zweigmedia.com/diff_geom/tc.html">"Introduction to Differential Geometry and General Relativity"</a>. Retrieved 5 April 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+to+Differential+Geometry+and+General+Relativity&rft.au=Waner%2C+Stefan&rft_id=http%3A%2F%2Fwww.zweigmedia.com%2Fdiff_geom%2Ftc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+relativity" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_42.html">The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space</a></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 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></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 href="/wiki/Special_relativity" title="Special relativity">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 href="/wiki/Special_relativity" title="Special relativity">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 class="mw-selflink selflink">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"></div><div role="navigation" class="navbox" aria-labelledby="Theories_of_gravitation" 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:Theories_of_gravitation" title="Template:Theories of gravitation"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Theories_of_gravitation" title="Template talk:Theories of gravitation"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Theories_of_gravitation" title="Special:EditPage/Template:Theories of gravitation"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Theories_of_gravitation" style="font-size:114%;margin:0 4em"><a href="/wiki/Gravity" title="Gravity">Theories of gravitation</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standard</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;">Newtonian gravity (NG)</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/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li> <li><a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a></li> <li><a href="/wiki/Poisson%27s_equation#Newtonian_gravity" title="Poisson's equation">Poisson's equation for gravity</a></li> <li><a href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">History of gravitational theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a class="mw-selflink selflink">General relativity (GR)</a></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/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics</a></li> <li><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Exact solutions</a></li> <li><a class="mw-selflink-fragment" href="#Further_reading">Resources</a></li> <li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/Gibbons%E2%80%93Hawking%E2%80%93York_boundary_term" title="Gibbons–Hawking–York boundary term">Gibbons–Hawking–York boundary term</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;line-height:1.2em;"><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Alternatives to general relativity</a></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;">Paradigms</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/Alternatives_to_general_relativity" title="Alternatives to general relativity">Classical theories of gravitation</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Classical</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="/w/index.php?title=Poincar%C3%A9_gauge_theory&action=edit&redlink=1" class="new" title="Poincaré gauge theory (page does not exist)">Poincaré gauge theory</a> <ul><li><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan</a></li> <li><a href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</a></li></ul></li> <li><a href="/wiki/Bimetric_gravity" title="Bimetric gravity">Bimetric theories</a></li> <li><a href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</a></li> <li><a href="/wiki/Composite_gravity" title="Composite gravity">Composite gravity</a></li> <li><a href="/wiki/F(R)_gravity" title="F(R) gravity">f(R) gravity</a></li> <li><a href="/wiki/Infinite_derivative_gravity" title="Infinite derivative gravity">Infinite derivative gravity</a></li> <li><a href="/wiki/Massive_gravity" title="Massive gravity">Massive gravity</a></li> <li><a href="/wiki/Modified_Newtonian_dynamics" title="Modified Newtonian dynamics">Modified Newtonian dynamics, MOND</a> <ul><li><a href="/wiki/AQUAL" title="AQUAL">AQUAL</a></li> <li><a href="/wiki/Tensor%E2%80%93vector%E2%80%93scalar_gravity" title="Tensor–vector–scalar gravity">Tensor–vector–scalar</a></li></ul></li> <li><a href="/wiki/Nonsymmetric_gravitational_theory" title="Nonsymmetric gravitational theory">Nonsymmetric gravitation</a></li> <li><a href="/wiki/Scalar%E2%80%93tensor_theory" title="Scalar–tensor theory">Scalar–tensor theories</a> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke</a></li></ul></li> <li><a href="/wiki/Scalar%E2%80%93tensor%E2%80%93vector_gravity" title="Scalar–tensor–vector gravity">Scalar–tensor–vector</a></li> <li><a href="/wiki/Conformal_gravity" title="Conformal gravity">Conformal gravity</a></li> <li><a href="/wiki/Scalar_theories_of_gravitation" title="Scalar theories of gravitation">Scalar theories</a> <ul><li><a href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström</a></li></ul></li> <li><a href="/wiki/Whitehead%27s_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead</a></li> <li><a href="/wiki/Geometrodynamics" title="Geometrodynamics">Geometrodynamics</a></li> <li><a href="/wiki/Induced_gravity" title="Induced gravity">Induced gravity</a></li> <li><a href="/wiki/Degenerate_Higher-Order_Scalar-Tensor_theories" title="Degenerate Higher-Order Scalar-Tensor theories">Degenerate Higher-Order Scalar-Tensor theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Quantum-mechanical</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/Euclidean_quantum_gravity" title="Euclidean quantum gravity">Euclidean quantum gravity</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a> <ul><li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Spin_foam" title="Spin foam">Spin foam</a></li></ul></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Asymptotic_safety_in_quantum_gravity" title="Asymptotic safety in quantum gravity">Asymptotic safety in quantum gravity</a></li> <li><a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></li> <li><a href="/wiki/DGP_model" title="DGP model">DGP model</a></li> <li><a href="/wiki/Rainbow_gravity_theory" title="Rainbow gravity theory">Rainbow gravity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric</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/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unified-field-theoric and quantum-mechanical</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/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative geometry</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a> <ul><li><a href="/wiki/Superfluid_vacuum_theory#Logarithmic_BEC_vacuum_theory" title="Superfluid vacuum theory">Logarithmic BEC vacuum</a></li></ul></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/M-theory" title="M-theory">M-theory</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string theory</a></li> <li><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string theory</a></li> <li><a href="/wiki/Type_0_string_theory" title="Type 0 string theory">Type 0 string theory</a></li> <li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string theory</a></li> <li><a href="/wiki/Little_string_theory" title="Little string theory">Little string theory</a></li></ul></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a> <ul><li><a href="/wiki/Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Generalisations / extensions of GR</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/Liouville_gravity" class="mw-redirect" title="Liouville gravity">Liouville gravity</a></li> <li><a href="/wiki/Lovelock_theory_of_gravity" title="Lovelock theory of gravity">Lovelock theory</a></li> <li><a href="/wiki/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</a></li> <li><a href="/wiki/Gauss%E2%80%93Bonnet_gravity" title="Gauss–Bonnet gravity">Gauss–Bonnet gravity</a></li> <li><a href="/wiki/Jackiw%E2%80%93Teitelboim_gravity" title="Jackiw–Teitelboim gravity">Jackiw–Teitelboim gravity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pre-Newtonian theories and <a href="/wiki/Toy_model" title="Toy model">toy models</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/Aristotelian_physics" title="Aristotelian physics">Aristotelian physics</a></li> <li><a href="/wiki/CGHS_model" title="CGHS model">CGHS model</a></li> <li><a href="/wiki/RST_model" title="RST model">RST model</a></li> <li><a href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">Mechanical explanations</a> <ul><li><a href="/wiki/Le_Sage%27s_theory_of_gravitation" title="Le Sage's theory of gravitation">Fatio–Le Sage</a></li> <li><a href="/wiki/Entropic_gravity" title="Entropic gravity">Entropic gravity</a></li></ul></li> <li><a href="/wiki/Gravitational_interaction_of_antimatter" title="Gravitational interaction of antimatter">Gravitational interaction of antimatter</a></li> <li><a href="/wiki/Physics_in_the_medieval_Islamic_world" title="Physics in the medieval Islamic world">Physics in the medieval Islamic world</a></li> <li><a href="/wiki/Theory_of_impetus" title="Theory of impetus">Theory of impetus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</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/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</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 href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a class="mw-selflink selflink">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="Major_branches_of_physics" 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:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</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/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</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/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</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/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" 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 href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a class="mw-selflink selflink">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="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 class="mw-selflink selflink">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 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[\"CITEREFAshtekarLewandowski2004\"] = 1,\n [\"CITEREFAshtekarMagnon-Ashtekar1979\"] = 1,\n [\"CITEREFAuyang1995\"] = 1,\n [\"CITEREFB._P._Abbott2016\"] = 1,\n [\"CITEREFBaniaRoodBalser2002\"] = 1,\n [\"CITEREFBarackCutler2004\"] = 1,\n [\"CITEREFBardeenCarterHawking1973\"] = 1,\n [\"CITEREFBarish2005\"] = 1,\n [\"CITEREFBarstow,_Bond_et_al.2005\"] = 1,\n [\"CITEREFBartusiak2000\"] = 1,\n [\"CITEREFBegelmanBlandfordRees1984\"] = 1,\n [\"CITEREFBeigChruściel2006\"] = 1,\n [\"CITEREFBekenstein1973\"] = 1,\n [\"CITEREFBekenstein1974\"] = 1,\n [\"CITEREFBelinskiiKhalatnikovLifschitz1971\"] = 1,\n [\"CITEREFBennettHalpernHinshawJarosik2003\"] = 1,\n [\"CITEREFBerger2002\"] = 1,\n [\"CITEREFBergströmGoobar2003\"] = 1,\n [\"CITEREFBertottiCiufoliniBender1987\"] = 1,\n [\"CITEREFBertottiIessTortora2003\"] = 1,\n [\"CITEREFBertschinger1998\"] = 1,\n [\"CITEREFBirrellDavies1984\"] = 1,\n [\"CITEREFBlairMcNamara1997\"] = 1,\n [\"CITEREFBlanchet2006\"] = 1,\n [\"CITEREFBlanchetFayeIyerSinha2008\"] = 1,\n [\"CITEREFBlandford1987\"] = 1,\n [\"CITEREFBondiVan_der_BurgMetzner1962\"] = 1,\n [\"CITEREFBrandenberger2008\"] = 1,\n [\"CITEREFBransDicke1961\"] = 1,\n [\"CITEREFBridleLahavOstrikerSteinhardt2003\"] = 1,\n [\"CITEREFBruhat1962\"] = 1,\n [\"CITEREFBuchert2008\"] = 1,\n [\"CITEREFBurasRamppJankaKifonidis2003\"] = 1,\n [\"CITEREFBörner1993\"] = 1,\n [\"CITEREFCaldwell2004\"] = 1,\n [\"CITEREFCarlip2001\"] = 1,\n [\"CITEREFCarroll2001\"] = 1,\n [\"CITEREFCarrollOstlie1996\"] = 1,\n [\"CITEREFCarter1979\"] = 1,\n [\"CITEREFCastelvecchiWitze2016\"] = 1,\n [\"CITEREFCelottiMillerSciama1999\"] = 1,\n [\"CITEREFChandrasekhar1983\"] = 1,\n [\"CITEREFChandrasekhar1984\"] = 1,\n [\"CITEREFCharbonnelPrimas2005\"] = 1,\n [\"CITEREFCheng,_Ta-Pei2005\"] = 1,\n [\"CITEREFCiufoliniPavlis2004\"] = 1,\n [\"CITEREFCiufoliniPavlisPeron2006\"] = 1,\n [\"CITEREFCoc,_Vangioni‐Flam_et_al.2004\"] = 1,\n [\"CITEREFCutlerThorne2002\"] = 1,\n [\"CITEREFDalalHolzHughesJain2006\"] = 1,\n [\"CITEREFDanzmannRüdiger2003\"] = 1,\n [\"CITEREFDediuMagdalenaMartín-Vide2015\"] = 1,\n [\"CITEREFDonoghue1995\"] = 1,\n [\"CITEREFDuff1996\"] = 1,\n [\"CITEREFEhlers1973\"] = 1,\n [\"CITEREFEhlersFalcoSchneider1992\"] = 1,\n [\"CITEREFEhlersLämmerzahl2006\"] = 1,\n [\"CITEREFEhlersRindler1997\"] = 1,\n [\"CITEREFEinstein,_A1916\"] = 1,\n [\"CITEREFEinstein,_A1918\"] = 1,\n [\"CITEREFEinstein1907\"] = 1,\n [\"CITEREFEinstein1915\"] = 1,\n [\"CITEREFEinstein1917\"] = 1,\n [\"CITEREFEllisVan_Elst1999\"] = 1,\n [\"CITEREFEngler2002\"] = 1,\n [\"CITEREFEverittBuchmanDeBraKeiser2001\"] = 1,\n [\"CITEREFEverittParkinsonKahn2007\"] = 1,\n [\"CITEREFFalckeMeliaAgol2000\"] = 1,\n [\"CITEREFFinazziLiberatiBarceló2009\"] = 1,\n [\"CITEREFFont2003\"] = 1,\n [\"CITEREFFourès-Bruhat1952\"] = 1,\n [\"CITEREFFrauendiener2004\"] = 1,\n [\"CITEREFFriedrich2005\"] = 1,\n [\"CITEREFFutamaseItoh2006\"] = 1,\n [\"CITEREFGamow1970\"] = 1,\n [\"CITEREFGarfinkle2007\"] = 1,\n [\"CITEREFGeroch1996\"] = 1,\n [\"CITEREFGiulini2005\"] = 1,\n [\"CITEREFGiulini2006\"] = 1,\n [\"CITEREFGiulini2007\"] = 1,\n [\"CITEREFGnedin2005\"] = 1,\n [\"CITEREFGoenner2004\"] = 1,\n [\"CITEREFGoroffSagnotti1985\"] = 1,\n [\"CITEREFGourgoulhon2007\"] = 1,\n [\"CITEREFGowdy1971\"] = 1,\n [\"CITEREFGowdy1974\"] = 1,\n [\"CITEREFGreenSchwarzWitten1987\"] = 1,\n [\"CITEREFGreensteinOkeShipman1971\"] = 1,\n [\"CITEREFGödel1949\"] = 1,\n [\"CITEREFHafeleKeating1972a\"] = 1,\n [\"CITEREFHafeleKeating1972b\"] = 1,\n [\"CITEREFHamber2009\"] = 1,\n [\"CITEREFHavas1964\"] = 1,\n [\"CITEREFHawking1966\"] = 1,\n [\"CITEREFHawking1975\"] = 1,\n [\"CITEREFHawking1987\"] = 1,\n [\"CITEREFHawkingEllis1973\"] = 1,\n [\"CITEREFHeckmannSchücking1959\"] = 1,\n [\"CITEREFHeusler1996\"] = 1,\n [\"CITEREFHeusler1998\"] = 1,\n [\"CITEREFHeyWalters2003\"] = 1,\n [\"CITEREFHobbsArchibaldArzoumanianBacker2010\"] = 1,\n [\"CITEREFHoughRowan2000\"] = 1,\n [\"CITEREFHubble1929\"] = 1,\n [\"CITEREFHulseTaylor1975\"] = 1,\n [\"CITEREFIbanez2000\"] = 1,\n [\"CITEREFIorio2006\"] = 1,\n [\"CITEREFIorio2009\"] = 1,\n [\"CITEREFIorio2010\"] = 1,\n [\"CITEREFIsham1994\"] = 1,\n [\"CITEREFIsrael1971\"] = 1,\n [\"CITEREFIsrael1987\"] = 1,\n [\"CITEREFJanssen2005\"] = 1,\n [\"CITEREFJaranowskiKrólak2005\"] = 1,\n [\"CITEREFKahn1996–2012\"] = 1,\n [\"CITEREFKahn2007\"] = 1,\n [\"CITEREFKamionkowskiKosowskyStebbins1997\"] = 1,\n [\"CITEREFKennefick2005\"] = 1,\n [\"CITEREFKennefick2007\"] = 1,\n [\"CITEREFKenyon1990\"] = 1,\n [\"CITEREFKochanekFalcoImpeyLehar2007\"] = 1,\n [\"CITEREFKomar1959\"] = 1,\n [\"CITEREFKramer2004\"] = 1,\n [\"CITEREFKramerStairsManchesterMcLaughlin2006\"] = 1,\n [\"CITEREFKramerStairsManchesterWex2021\"] = 1,\n [\"CITEREFKraus1998\"] = 1,\n [\"CITEREFKuchař1973\"] = 1,\n [\"CITEREFKünzle1972\"] = 1,\n [\"CITEREFLahavSuto2004\"] = 1,\n [\"CITEREFLandauLifshitz1975\"] = 1,\n [\"CITEREFLehner2001\"] = 1,\n [\"CITEREFLehner2002\"] = 1,\n [\"CITEREFLinde2005\"] = 1,\n [\"CITEREFLinde2006\"] = 1,\n [\"CITEREFLoll1998\"] = 1,\n [\"CITEREFLovelock1972\"] = 1,\n [\"CITEREFMacCallum2006\"] = 1,\n [\"CITEREFMaddox1998\"] = 1,\n [\"CITEREFMannheim2006\"] = 1,\n [\"CITEREFMatherChengCottinghamEplee1994\"] = 1,\n [\"CITEREFMermin2005\"] = 1,\n [\"CITEREFMessiah1999\"] = 1,\n [\"CITEREFMiller2002\"] = 1,\n [\"CITEREFMisnerThorneWheeler1973\"] = 1,\n [\"CITEREFNarayan2006\"] = 1,\n [\"CITEREFNarayanBartelmann1997\"] = 1,\n [\"CITEREFNarlikar1993\"] = 1,\n [\"CITEREFNordström1918\"] = 1,\n [\"CITEREFNordtvedt2003\"] = 1,\n [\"CITEREFNorton1985\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson1996\"] = 1,\n [\"CITEREFO\u0026#039;MearaTytlerKirkmanSuzuki2001\"] = 1,\n [\"CITEREFOhanianRuffini1994\"] = 1,\n [\"CITEREFOliveSkillman2004\"] = 1,\n [\"CITEREFOppenheimerSnyder1939\"] = 1,\n [\"CITEREFOverbye1999\"] = 1,\n [\"CITEREFPais1982\"] = 1,\n [\"CITEREFPeacock1999\"] = 1,\n [\"CITEREFPeebles1966\"] = 1,\n [\"CITEREFPeebles1993\"] = 1,\n [\"CITEREFPeeblesSchrammTurnerKron1991\"] = 1,\n [\"CITEREFPenrose1965\"] = 1,\n [\"CITEREFPenrose1969\"] = 1,\n [\"CITEREFPenrose2004\"] = 1,\n [\"CITEREFPenziasWilson1965\"] = 1,\n [\"CITEREFPeskin2007\"] = 1,\n [\"CITEREFPeskinSchroeder1995\"] = 1,\n [\"CITEREFPoincaré1905\"] = 1,\n [\"CITEREFPoisson2004\"] = 1,\n [\"CITEREFPoisson2004a\"] = 1,\n [\"CITEREFPolchinski1998a\"] = 1,\n [\"CITEREFPolchinski1998b\"] = 1,\n [\"CITEREFPoundRebka1959\"] = 1,\n [\"CITEREFPoundRebka1960\"] = 1,\n [\"CITEREFPoundSnider1964\"] = 1,\n [\"CITEREFRamond1990\"] = 1,\n [\"CITEREFRees1966\"] = 1,\n [\"CITEREFReissner1916\"] = 1,\n [\"CITEREFRemillardLinCooperNarayan2006\"] = 1,\n [\"CITEREFRenn2005\"] = 1,\n [\"CITEREFRenn2007\"] = 1,\n [\"CITEREFReula1998\"] = 1,\n [\"CITEREFRindler1991\"] = 1,\n [\"CITEREFRindler2001\"] = 1,\n [\"CITEREFRobson1996\"] = 1,\n [\"CITEREFRouletMollerach1997\"] = 1,\n [\"CITEREFRovelli1998\"] = 1,\n [\"CITEREFRovelli2000\"] = 1,\n [\"CITEREFRovelli2015\"] = 1,\n [\"CITEREFSachs1962\"] = 1,\n [\"CITEREFSchutz1985\"] = 1,\n [\"CITEREFSchutz2003\"] = 1,\n [\"CITEREFSchwarz2007\"] = 1,\n [\"CITEREFSchwarzschild1916a\"] = 1,\n [\"CITEREFSchwarzschild1916b\"] = 1,\n [\"CITEREFSchäfer2004\"] = 1,\n [\"CITEREFSchödelOttGenzelEckart2003\"] = 1,\n [\"CITEREFSeidel1998\"] = 1,\n [\"CITEREFSeljakZaldarriaga1997\"] = 1,\n [\"CITEREFShapiro1964\"] = 1,\n [\"CITEREFShapiroDavisLebachGregory2004\"] = 1,\n [\"CITEREFSingh2004\"] = 1,\n [\"CITEREFSorkin1997\"] = 1,\n [\"CITEREFSorkin2005\"] = 1,\n [\"CITEREFSpergelBeanDoréNolta2007\"] = 1,\n [\"CITEREFSpergelVerdePeirisKomatsu2003\"] = 1,\n [\"CITEREFSpringelWhiteJenkinsFrenk2005\"] = 1,\n [\"CITEREFStairs2003\"] = 1,\n [\"CITEREFStephaniKramerMacCallumHoenselaers2003\"] = 1,\n [\"CITEREFStrominger2017\"] = 1,\n [\"CITEREFSynge1972\"] = 1,\n [\"CITEREFSzabados2004\"] = 1,\n [\"CITEREFTaylor1994\"] = 1,\n [\"CITEREFThiemann2003\"] = 1,\n [\"CITEREFThiemann2007\"] = 1,\n [\"CITEREFThorne1972\"] = 1,\n [\"CITEREFThorne1994\"] = 1,\n [\"CITEREFThorne1995\"] = 1,\n [\"CITEREFThorne2003\"] = 1,\n [\"CITEREFTownsend1996\"] = 1,\n [\"CITEREFTownsend1997\"] = 1,\n [\"CITEREFTraschen2000\"] = 1,\n [\"CITEREFTrautman2006\"] = 1,\n [\"CITEREFUnruh1976\"] = 1,\n [\"CITEREFVeltman1975\"] = 1,\n [\"CITEREFWald1975\"] = 1,\n [\"CITEREFWald1984\"] = 1,\n [\"CITEREFWald1994\"] = 1,\n [\"CITEREFWald2001\"] = 1,\n [\"CITEREFWalshCarswellWeymann1979\"] = 1,\n [\"CITEREFWambsganss1998\"] = 1,\n [\"CITEREFWeinberg,_Steven1972\"] = 1,\n [\"CITEREFWeinberg1972\"] = 1,\n [\"CITEREFWeinberg1995\"] = 1,\n [\"CITEREFWeinberg1996\"] = 1,\n [\"CITEREFWeinberg2000\"] = 1,\n [\"CITEREFWeisbergTaylor2003\"] = 1,\n [\"CITEREFWeiss2006\"] = 1,\n [\"CITEREFWheeler1990\"] = 1,\n [\"CITEREFWill1993\"] = 1,\n [\"CITEREFWill2006\"] = 1,\n [\"CITEREFZwiebach2004\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Annotated link\"] = 13,\n [\"Authority control\"] = 1,\n [\"Branches of physics\"] = 1,\n [\"Citation\"] = 257,\n [\"Cite arXiv\"] = 7,\n [\"Cite book\"] = 13,\n [\"Cite conference\"] = 1,\n [\"Cite journal\"] = 14,\n [\"Cite web\"] = 8,\n [\"Clear\"] = 2,\n [\"Commons category\"] = 1,\n [\"Einstein\"] = 1,\n [\"Equation box 1\"] = 1,\n [\"Featured article\"] = 1,\n [\"For\"] = 1,\n [\"For introduction\"] = 1,\n [\"General relativity sidebar\"] = 1,\n [\"Harvid\"] = 2,\n [\"Harvnb\"] = 370,\n [\"Hlist\"] = 1,\n [\"Main\"] = 21,\n [\"Math\"] = 4,\n [\"Portal bar\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Relativity\"] = 1,\n [\"See also\"] = 3,\n [\"Sfn\"] = 1,\n [\"SfnRef\"] = 2,\n [\"Short description\"] = 1,\n [\"TOC limit\"] = 1,\n [\"Tensors\"] = 1,\n [\"Theories of gravitation\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 1,\n [\"Wikibooks\"] = 1,\n [\"Wikiquote\"] = 1,\n [\"Wikisource\"] = 1,\n [\"Wikisource portal\"] = 1,\n [\"Wikiversity\"] = 1,\n [\"YouTube\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["dataWrapper 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