History of Lorentz transformations

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<ul id="toc-Heaviside_(1888),_Thomson_(1889),_Searle_(1896)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_(1892,_1895)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_(1892,_1895)"> <div class="vector-toc-text"> 2.4 Lorentz (1892, 1895) </div> </a> <ul id="toc-Lorentz_(1892,_1895)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Larmor_(1897,_1900)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Larmor_(1897,_1900)"> <div class="vector-toc-text"> 2.5 Larmor (1897, 1900) </div> </a> <ul id="toc-Larmor_(1897,_1900)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_(1899,_1904)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_(1899,_1904)"> <div class="vector-toc-text"> 2.6 Lorentz (1899, 1904) </div> </a> <ul id="toc-Lorentz_(1899,_1904)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poincaré_(1900,_1905)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poincaré_(1900,_1905)"> <div class="vector-toc-text"> 2.7 Poincaré (1900, 1905) </div> </a> <ul id="toc-Poincaré_(1900,_1905)-sublist" class="vector-toc-list"> <li id="toc-Local_time" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Local_time"> <div class="vector-toc-text"> 2.7.1 Local time </div> </a> <ul id="toc-Local_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_transformation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lorentz_transformation"> <div class="vector-toc-text"> 2.7.2 Lorentz transformation </div> </a> <ul id="toc-Lorentz_transformation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Einstein_(1905)_–_Special_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein_(1905)_–_Special_relativity"> <div class="vector-toc-text"> 2.8 Einstein (1905) – Special relativity </div> </a> <ul id="toc-Einstein_(1905)_–_Special_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minkowski_(1907–1908)_–_Spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minkowski_(1907–1908)_–_Spacetime"> <div class="vector-toc-text"> 2.9 Minkowski (1907–1908) – Spacetime </div> </a> <ul id="toc-Minkowski_(1907–1908)_–_Spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sommerfeld_(1909)_–_Spherical_trigonometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sommerfeld_(1909)_–_Spherical_trigonometry"> <div class="vector-toc-text"> 2.10 Sommerfeld (1909) – Spherical trigonometry </div> </a> <ul id="toc-Sommerfeld_(1909)_–_Spherical_trigonometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frank_(1909)_–_Hyperbolic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frank_(1909)_–_Hyperbolic_functions"> <div class="vector-toc-text"> 2.11 Frank (1909) – Hyperbolic functions </div> </a> <ul id="toc-Frank_(1909)_–_Hyperbolic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bateman_and_Cunningham_(1909–1910)_–_Spherical_wave_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bateman_and_Cunningham_(1909–1910)_–_Spherical_wave_transformation"> <div class="vector-toc-text"> 2.12 Bateman and Cunningham (1909–1910) – Spherical wave transformation </div> </a> <ul id="toc-Bateman_and_Cunningham_(1909–1910)_–_Spherical_wave_transformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Herglotz_(1909/10)_–_Möbius_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Herglotz_(1909/10)_–_Möbius_transformation"> <div class="vector-toc-text"> 2.13 Herglotz (1909/10) – Möbius transformation </div> </a> <ul id="toc-Herglotz_(1909/10)_–_Möbius_transformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Varićak_(1910)_–_Hyperbolic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Varićak_(1910)_–_Hyperbolic_functions"> <div class="vector-toc-text"> 2.14 Varićak (1910) – Hyperbolic functions </div> </a> <ul id="toc-Varićak_(1910)_–_Hyperbolic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plummer_(1910)_–_Trigonometric_Lorentz_boosts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plummer_(1910)_–_Trigonometric_Lorentz_boosts"> <div class="vector-toc-text"> 2.15 Plummer (1910) – Trigonometric Lorentz boosts </div> </a> <ul id="toc-Plummer_(1910)_–_Trigonometric_Lorentz_boosts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ignatowski_(1910)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ignatowski_(1910)"> <div class="vector-toc-text"> 2.16 Ignatowski (1910) </div> </a> <ul id="toc-Ignatowski_(1910)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noether_(1910),_Klein_(1910)_–_Quaternions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noether_(1910),_Klein_(1910)_–_Quaternions"> <div class="vector-toc-text"> 2.17 Noether (1910), Klein (1910) – Quaternions </div> </a> <ul id="toc-Noether_(1910),_Klein_(1910)_–_Quaternions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conway_(1911),_Silberstein_(1911)_–_Quaternions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conway_(1911),_Silberstein_(1911)_–_Quaternions"> <div class="vector-toc-text"> 2.18 Conway (1911), Silberstein (1911) – Quaternions </div> </a> <ul id="toc-Conway_(1911),_Silberstein_(1911)_–_Quaternions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ignatowski_(1910/11),_Herglotz_(1911),_and_others_–_Vector_transformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ignatowski_(1910/11),_Herglotz_(1911),_and_others_–_Vector_transformation"> <div class="vector-toc-text"> 2.19 Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation </div> </a> <ul id="toc-Ignatowski_(1910/11),_Herglotz_(1911),_and_others_–_Vector_transformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Borel_(1913–14)_–_Cayley–Hermite_parameter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Borel_(1913–14)_–_Cayley–Hermite_parameter"> <div class="vector-toc-text"> 2.20 Borel (1913–14) – Cayley–Hermite parameter </div> </a> <ul id="toc-Borel_(1913–14)_–_Cayley–Hermite_parameter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gruner_(1921)_–_Trigonometric_Lorentz_boosts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gruner_(1921)_–_Trigonometric_Lorentz_boosts"> <div class="vector-toc-text"> 2.21 Gruner (1921) – Trigonometric Lorentz boosts </div> </a> <ul id="toc-Gruner_(1921)_–_Trigonometric_Lorentz_boosts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 3 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"> 4 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Historical_mathematical_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_mathematical_sources"> <div class="vector-toc-text"> 4.1 Historical mathematical sources </div> </a> <ul id="toc-Historical_mathematical_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_relativity_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_relativity_sources"> <div class="vector-toc-text"> 4.2 Historical relativity sources </div> </a> <ul id="toc-Historical_relativity_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secondary_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Secondary_sources"> <div class="vector-toc-text"> 4.3 Secondary sources </div> </a> <ul id="toc-Secondary_sources-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"> 5 External links </div> 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">The history of <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a> comprises the development of <a href="/wiki/Linear_map" title="Linear map">linear transformations</a> forming the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> or <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> preserving the <a href="/wiki/Lorentz_interval" class="mw-redirect" title="Lorentz interval">Lorentz interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeaaf19e649447ee32f924033e4c859955174c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.144ex; height:3.176ex;" alt="{\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}}"> and the <a href="/wiki/Minkowski_inner_product" class="mw-redirect" title="Minkowski inner product">Minkowski inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43fcbe32f1674bd50b4621db988d3de905a8f269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.695ex; height:2.343ex;" alt="{\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}">. In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a>, <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, <a href="/wiki/M%C3%B6bius_geometry" class="mw-redirect" title="Möbius geometry">Möbius geometry</a>, and <a href="/wiki/Spherical_wave_transformation" title="Spherical wave transformation">sphere geometry</a>, which is connected to the fact that the group of <a href="/wiki/Hyperbolic_motion" title="Hyperbolic motion">motions in hyperbolic space</a>, the <a href="/wiki/M%C3%B6bius_group" class="mw-redirect" title="Möbius group">Möbius group</a> or <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a>, and the <a href="/wiki/Spherical_wave_transformation#Transformation_by_reciprocal_directions" title="Spherical wave transformation">Laguerre group</a> are <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>. In <a href="/wiki/Physics" title="Physics">physics</a>, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. Subsequently, they became fundamental to all of physics, because they formed the basis of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> in which they exhibit the symmetry of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>, making the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frames of reference</a> with constant relative speed v. In one frame, the position of an event is given by x,y,z and time t, while in the other frame the same event has coordinates x′,y′,z′ and t′. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_prehistory">Mathematical prehistory</h2>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=1" title="Edit section: Mathematical prehistory">edit</a>]</div> Using the coefficients of a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> A, the associated <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>, and a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> in terms of <a href="/wiki/Transformation_matrix" title="Transformation matrix">transformation matrix</a> g, the Lorentz transformation is given if the following conditions are satisfied: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline {\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{matrix}}\\\hline {\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \end{aligned}}\end{matrix}}\\\hline \mathbf {A} ={\rm {diag}}(-1,1,\dots ,1)\\\det \mathbf {g} =\pm 1\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid solid solid none"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline {\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{matrix}}\\\hline {\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \end{aligned}}\end{matrix}}\\\hline \mathbf {A} ={\rm {diag}}(-1,1,\dots ,1)\\\det \mathbf {g} =\pm 1\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1774f8b3f66a305a3158bad7e0b0854e22ea7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.671ex; width:45.85ex; height:30.509ex;" alt="{\displaystyle {\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline {\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{matrix}}\\\hline {\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \end{aligned}}\end{matrix}}\\\hline \mathbf {A} ={\rm {diag}}(-1,1,\dots ,1)\\\det \mathbf {g} =\pm 1\end{matrix}}}"></dd></dl> It forms an <a href="/wiki/Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal group</a> called the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> O(1,n), while the case det g=+1 forms the restricted <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> SO(1,n). The quadratic form becomes the <a href="/wiki/Lorentz_interval" class="mw-redirect" title="Lorentz interval">Lorentz interval</a> in terms of an <a href="/wiki/Indefinite_quadratic_form" class="mw-redirect" title="Indefinite quadratic form">indefinite quadratic form</a> of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> (being a special case of <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>), and the associated bilinear form becomes the <a href="/wiki/Minkowski_inner_product" class="mw-redirect" title="Minkowski inner product">Minkowski inner product</a>.<a href="#cite_note-ratcliffe-1">[1]</a><a href="#cite_note-2">[2]</a> Long before the advent of special relativity it was used in topics such as the <a href="/wiki/Cayley%E2%80%93Klein_metric" title="Cayley–Klein metric">Cayley–Klein metric</a>, <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> and other models of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, computations of <a href="/wiki/Elliptic_function" title="Elliptic function">elliptic functions</a> and integrals, transformation of <a href="/wiki/Indefinite_quadratic_form" class="mw-redirect" title="Indefinite quadratic form">indefinite quadratic forms</a>, <a href="/wiki/Squeeze_mapping" title="Squeeze mapping">squeeze mappings</a> of the hyperbola, <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>, <a href="/wiki/Spherical_wave_transformation" title="Spherical wave transformation">spherical wave transformation</a>, transformation of the <a href="/wiki/Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon equation</a>, <a href="/wiki/Biquaternion" title="Biquaternion">Biquaternion</a> algebra, <a href="/wiki/Split-complex_numbers" class="mw-redirect" title="Split-complex numbers">split-complex numbers</a>, <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, and others. <div style="border:1px solid black"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> Learning materials from Wikiversity: <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(general)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (general)">Wikiversity: History of most general Lorentz transformations</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1818), <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a> (1827, 1833/34), <a href="/wiki/Michel_Chasles" title="Michel Chasles">Michel Chasles</a> (1829), <a href="/wiki/Victor-Am%C3%A9d%C3%A9e_Lebesgue" title="Victor-Amédée Lebesgue">Victor-Amédée Lebesgue</a> (1837), <a href="/wiki/Thomas_Weddle" title="Thomas Weddle">Thomas Weddle</a> (1847), <a href="/wiki/Edmond_Bour" title="Edmond Bour">Edmond Bour</a> (1856), <a href="/wiki/Osip_Ivanovich_Somov" class="mw-redirect" title="Osip Ivanovich Somov">Osip Ivanovich Somov</a> (1863), <a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a> (1878–1893), <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> (1881), <a href="/wiki/Homersham_Cox_(mathematician)" title="Homersham Cox (mathematician)">Homersham Cox</a> (1881–1883), <a href="/wiki/George_William_Hill" title="George William Hill">George William Hill</a> (1882), <a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a> (1882-1884), <a href="/wiki/Octave_Callandreau" title="Octave Callandreau">Octave Callandreau</a> (1885), <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> (1885-1890), <a href="/w/index.php?title=Louis_G%C3%A9rard&action=edit&redlink=1" class="new" title="Louis Gérard (page does not exist)">Louis Gérard</a> (1892), <a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a> (1899), <a href="/wiki/Frederick_S._Woods" title="Frederick S. Woods">Frederick S. Woods</a> (1901-05), <a href="/wiki/Heinrich_Liebmann" title="Heinrich Liebmann">Heinrich Liebmann</a> (1904/05).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(imaginary)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (imaginary)">Wikiversity: History of Lorentz transformations via imaginary orthogonal transformation</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> (1871), <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> (1907–1908), <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> (1909).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(hyperbolic)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (hyperbolic)">Wikiversity: History of Lorentz transformations via hyperbolic functions</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Vincenzo_Riccati" title="Vincenzo Riccati">Vincenzo Riccati</a> (1757), <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a> (1768–1770), <a href="/wiki/Franz_Taurinus" title="Franz Taurinus">Franz Taurinus</a> (1826), <a href="/wiki/Eugenio_Beltrami" title="Eugenio Beltrami">Eugenio Beltrami</a> (1868), <a href="/wiki/Charles-Ange_Laisant" title="Charles-Ange Laisant">Charles-Ange Laisant</a> (1874), <a href="/wiki/Gustav_von_Escherich" title="Gustav von Escherich">Gustav von Escherich</a> (1874), <a href="/wiki/James_Whitbread_Lee_Glaisher" title="James Whitbread Lee Glaisher">James Whitbread Lee Glaisher</a> (1878), <a href="/wiki/Siegmund_G%C3%BCnther" title="Siegmund Günther">Siegmund Günther</a> (1880/81), <a href="/wiki/Homersham_Cox_(mathematician)" title="Homersham Cox (mathematician)">Homersham Cox</a> (1881/82), <a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a> (1885/86), <a href="/wiki/Friedrich_Schur" title="Friedrich Schur">Friedrich Schur</a> (1885-1902), <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Ferdinand von Lindemann</a> (1890–91), <a href="/w/index.php?title=Louis_G%C3%A9rard&action=edit&redlink=1" class="new" title="Louis Gérard (page does not exist)">Louis Gérard</a> (1892), <a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a> (1893-97), <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> (1897/98), <a href="/wiki/Edwin_Bailey_Elliott" title="Edwin Bailey Elliott">Edwin Bailey Elliott</a> (1903), <a href="/wiki/Frederick_S._Woods" title="Frederick S. Woods">Frederick S. Woods</a> (1903), <a href="/wiki/Heinrich_Liebmann" title="Heinrich Liebmann">Heinrich Liebmann</a> (1904/05), <a href="/wiki/Philipp_Frank" title="Philipp Frank">Philipp Frank</a> (1909), <a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Gustav Herglotz</a> (1909/10), <a href="/wiki/Vladimir_Vari%C4%87ak" title="Vladimir Varićak">Vladimir Varićak</a> (1910).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(conformal)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (conformal)">Wikiversity: History of Lorentz transformations via sphere transformation</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Pierre_Ossian_Bonnet" title="Pierre Ossian Bonnet">Pierre Ossian Bonnet</a> (1856), <a href="/wiki/Albert_Ribaucour" title="Albert Ribaucour">Albert Ribaucour</a> (1870), <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> (1871a), <a href="/wiki/Gaston_Darboux" class="mw-redirect" title="Gaston Darboux">Gaston Darboux</a> (1873-87), <a href="/wiki/Edmond_Laguerre" title="Edmond Laguerre">Edmond Laguerre</a> (1880), <a href="/wiki/Cyparissos_Stephanos" title="Cyparissos Stephanos">Cyparissos Stephanos</a> (1883), <a href="/wiki/Georg_Scheffers" title="Georg Scheffers">Georg Scheffers</a> (1899), <a href="/wiki/Percey_F._Smith" title="Percey F. Smith">Percey F. Smith</a> (1900), <a href="/wiki/Harry_Bateman" title="Harry Bateman">Harry Bateman</a> and <a href="/wiki/Ebenezer_Cunningham" title="Ebenezer Cunningham">Ebenezer Cunningham</a> (1909–1910).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Cayley-Hermite)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)">Wikiversity: History of Lorentz transformations via Cayley–Hermite transformation</a></li></ul> <dl><dd><dl><dd>was used by <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> (1846–1855), <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a> (1853, 1854), <a href="/wiki/Paul_Gustav_Heinrich_Bachmann" title="Paul Gustav Heinrich Bachmann">Paul Gustav Heinrich Bachmann</a> (1869), <a href="/wiki/Edmond_Laguerre" title="Edmond Laguerre">Edmond Laguerre</a> (1882), <a href="/wiki/Gaston_Darboux" class="mw-redirect" title="Gaston Darboux">Gaston Darboux</a> (1887), <a href="/wiki/Percey_F._Smith" title="Percey F. Smith">Percey F. Smith</a> (1900), <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a> (1913).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(M%C3%B6bius)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (Möbius)">Wikiversity: History of Lorentz transformations via Cayley–Klein parameters, Möbius and spin transformations</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1801/63), <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (1871–97), <a href="/wiki/Eduard_Selling" title="Eduard Selling">Eduard Selling</a> (1873–74), <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> (1881), <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Luigi Bianchi</a> (1888-93), <a href="/wiki/Robert_Fricke" title="Robert Fricke">Robert Fricke</a> (1891–97), <a href="/wiki/Frederick_S._Woods" title="Frederick S. Woods">Frederick S. Woods</a> (1895), <a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Gustav Herglotz</a> (1909/10).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Quaternions)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (Quaternions)">Wikiversity: History of Lorentz transformations via quaternions and hyperbolic numbers</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/James_Cockle" title="James Cockle">James Cockle</a> (1848), <a href="/wiki/Homersham_Cox_(mathematician)" title="Homersham Cox (mathematician)">Homersham Cox</a> (1882/83), <a href="/wiki/Cyparissos_Stephanos" title="Cyparissos Stephanos">Cyparissos Stephanos</a> (1883), <a href="/wiki/Arthur_Buchheim" title="Arthur Buchheim">Arthur Buchheim</a> (1884), <a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a> (1885/86), <a href="/wiki/Theodor_Vahlen" title="Theodor Vahlen">Theodor Vahlen</a> (1901/02), <a href="/wiki/Fritz_Noether" title="Fritz Noether">Fritz Noether</a> (1910), <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (1910), <a href="/wiki/Arthur_W._Conway" title="Arthur W. Conway">Arthur W. Conway</a> (1911), <a href="/wiki/Ludwik_Silberstein" title="Ludwik Silberstein">Ludwik Silberstein</a> (1911).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(trigonometric)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (trigonometric)">Wikiversity: Lorentz transformation via trigonometric functions</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Luigi Bianchi</a> (1886), <a href="/wiki/Gaston_Darboux" class="mw-redirect" title="Gaston Darboux">Gaston Darboux</a> (1891/94), <a href="/wiki/Georg_Scheffers" title="Georg Scheffers">Georg Scheffers</a> (1899), <a href="/wiki/Luther_Pfahler_Eisenhart" class="mw-redirect" title="Luther Pfahler Eisenhart">Luther Pfahler Eisenhart</a> (1905), <a href="/wiki/Vladimir_Vari%C4%87ak" title="Vladimir Varićak">Vladimir Varićak</a> (1910), <a href="/wiki/Henry_Crozier_Keating_Plummer" title="Henry Crozier Keating Plummer">Henry Crozier Keating Plummer</a> (1910), <a href="/wiki/Paul_Gruner" title="Paul Gruner">Paul Gruner</a> (1921).</dd></dl></dd></dl> <ul><li>The <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(squeeze)" class="extiw" title="v:History of Topics in Special Relativity/Lorentz transformation (squeeze)">Wikiversity: History of Lorentz transformations via squeeze mappings</a></li></ul> <dl><dd><dl><dd>includes contributions of <a href="/wiki/Antoine_Andr%C3%A9_Louis_Reynaud" title="Antoine André Louis Reynaud">Antoine André Louis Reynaud</a> (1819), <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (1871), <a href="/wiki/Charles-Ange_Laisant" title="Charles-Ange Laisant">Charles-Ange Laisant</a> (1874), <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> (1879-84), <a href="/wiki/Siegmund_G%C3%BCnther" title="Siegmund Günther">Siegmund Günther</a> (1880/81), <a href="/wiki/Edmond_Laguerre" title="Edmond Laguerre">Edmond Laguerre</a> (1882), <a href="/wiki/Gaston_Darboux" class="mw-redirect" title="Gaston Darboux">Gaston Darboux</a> (1883–1891), <a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a> (1885/86), <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Luigi Bianchi</a> (1886–1894), <a href="/wiki/Ferdinand_von_Lindemann" title="Ferdinand von Lindemann">Ferdinand von Lindemann</a> (1890/91), <a href="/wiki/Mellen_W._Haskell" class="mw-redirect" title="Mellen W. Haskell">Mellen W. Haskell</a> (1895), <a href="/wiki/Percey_F._Smith" title="Percey F. Smith">Percey F. Smith</a> (1900), <a href="/wiki/Edwin_Bailey_Elliott" title="Edwin Bailey Elliott">Edwin Bailey Elliott</a> (1903), <a href="/wiki/Luther_Pfahler_Eisenhart" class="mw-redirect" title="Luther Pfahler Eisenhart">Luther Pfahler Eisenhart</a> (1905).</dd></dl></dd></dl></div> <div class="mw-heading mw-heading2"><h2 id="Electrodynamics_and_special_relativity">Electrodynamics and special relativity</h2>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=2" title="Edit section: Electrodynamics and special relativity">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Overview">Overview</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=3" title="Edit section: Overview">edit</a>]</div> In the <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, Lorentz transformations exhibit the symmetry of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a> by using a constant c as the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, and a parameter v as the relative <a href="/wiki/Velocity" title="Velocity">velocity</a> between two <a href="/wiki/Inertial_reference_frames" class="mw-redirect" title="Inertial reference frames">inertial reference frames</a>. Using the above conditions, the Lorentz transformation in 3+1 dimensions assume the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\\hline \left.{\begin{aligned}t'&=\gamma \left(t-x{\frac {v}{c^{2}}}\right)\\x'&=\gamma (x-vt)\\y'&=y\\z'&=z\end{aligned}}\right|{\begin{aligned}t&=\gamma \left(t'+x{\frac {v}{c^{2}}}\right)\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'\end{aligned}}\end{matrix}}\Rightarrow {\begin{aligned}(ct'+x')&=(ct+x){\sqrt {\frac {c+v}{c-v}}}\\(ct'-x')&=(ct-x){\sqrt {\frac {c-v}{c+v}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>c</mi> <mo>+</mo> <mi>v</mi> </mrow> <mrow> <mi>c</mi> <mo>−</mo> <mi>v</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>c</mi> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>v</mi> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\\hline \left.{\begin{aligned}t'&=\gamma \left(t-x{\frac {v}{c^{2}}}\right)\\x'&=\gamma (x-vt)\\y'&=y\\z'&=z\end{aligned}}\right|{\begin{aligned}t&=\gamma \left(t'+x{\frac {v}{c^{2}}}\right)\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'\end{aligned}}\end{matrix}}\Rightarrow {\begin{aligned}(ct'+x')&=(ct+x){\sqrt {\frac {c+v}{c-v}}}\\(ct'-x')&=(ct-x){\sqrt {\frac {c-v}{c+v}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ca20811b7bae58fc90314c7380db586cbae058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:82.193ex; height:19.843ex;" alt="{\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\\hline \left.{\begin{aligned}t'&=\gamma \left(t-x{\frac {v}{c^{2}}}\right)\\x'&=\gamma (x-vt)\\y'&=y\\z'&=z\end{aligned}}\right|{\begin{aligned}t&=\gamma \left(t'+x{\frac {v}{c^{2}}}\right)\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'\end{aligned}}\end{matrix}}\Rightarrow {\begin{aligned}(ct'+x')&=(ct+x){\sqrt {\frac {c+v}{c-v}}}\\(ct'-x')&=(ct-x){\sqrt {\frac {c-v}{c+v}}}\end{aligned}}}"></dd></dl> In physics, analogous transformations have been introduced by <a href="#Voigt">Voigt (1887)</a> related to an incompressible medium, and by <a href="#Heaviside">Heaviside (1888), Thomson (1889), Searle (1896)</a> and <a href="#Lorentz1">Lorentz (1892, 1895)</a> who analyzed <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. They were completed by <a href="#Larmor">Larmor (1897, 1900)</a> and <a href="#Lorentz2">Lorentz (1899, 1904)</a>, and brought into their modern form by <a href="#Poincare3">Poincaré (1905)</a> who gave the transformation the name of Lorentz.<a href="#cite_note-3">[3]</a> Eventually, <a href="#Einstein">Einstein (1905)</a> showed in his development of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> that the transformations follow from the <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a> and constant light speed alone by modifying the traditional concepts of space and time, without requiring a <a href="/wiki/Lorentz_ether_theory" title="Lorentz ether theory">mechanical aether</a> in contradistinction to Lorentz and Poincaré.<a href="#cite_note-4">[4]</a> <a href="#Minkowski">Minkowski (1907–1908)</a> used them to argue that space and time are inseparably connected as <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. Regarding special representations of the Lorentz transformations: <a href="#Minkowski">Minkowski (1907–1908)</a> and <a href="#Sommerfeld">Sommerfeld (1909)</a> used imaginary trigonometric functions, <a href="#Frank">Frank (1909)</a> and <a href="#Varicak">Varićak (1910)</a> used <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>, <a href="#Bateman">Bateman and Cunningham (1909–1910)</a> used <a href="/wiki/Spherical_wave_transformation" title="Spherical wave transformation">spherical wave transformations</a>, <a href="#Herglotz1">Herglotz (1909–10)</a> used Möbius transformations, <a href="#Plummer">Plummer (1910)</a> and <a href="#Gruner">Gruner (1921)</a> used trigonometric Lorentz boosts, <a href="#Ignatowski">Ignatowski (1910)</a> derived the transformations without light speed postulate, <a href="#Noether">Noether (1910) and Klein (1910)</a> as well <a href="#Conway">Conway (1911) and Silberstein (1911)</a> used Biquaternions, <a href="#Herglotz2">Ignatowski (1910/11), Herglotz (1911), and others</a> used vector transformations valid in arbitrary directions, <a href="#Borel">Borel (1913–14)</a> used Cayley–Hermite parameter, <div class="mw-heading mw-heading3"><h3 id="Voigt_(1887)"> Voigt (1887)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=4" title="Edit section: Voigt (1887)">edit</a>]</div> <a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a> (1887)<a href="#cite_note-5">[R 1]</a> developed a transformation in connection with the <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a> and an incompressible medium, being in modern notation:<a href="#cite_note-6">[5]</a><a href="#cite_note-pais-7">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}\\q&={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&={\frac {y}{\gamma }}\\z^{\prime }&={\frac {z}{\gamma }}\\t^{\prime }&=t-{\frac {vx}{c^{2}}}\\{\frac {1}{\gamma }}&={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>ϰ</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϰ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}\\q&={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&={\frac {y}{\gamma }}\\z^{\prime }&={\frac {z}{\gamma }}\\t^{\prime }&=t-{\frac {vx}{c^{2}}}\\{\frac {1}{\gamma }}&={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24680a99bce9e167c14aaf42f68b875b18ee6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.838ex; width:36.901ex; height:30.843ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}\\q&={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&={\frac {y}{\gamma }}\\z^{\prime }&={\frac {z}{\gamma }}\\t^{\prime }&=t-{\frac {vx}{c^{2}}}\\{\frac {1}{\gamma }}&={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}"></dd></dl> If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation. In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are <a href="/wiki/Scale_invariance" title="Scale invariance">scale</a>, <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>, and <a href="/wiki/Lorentz_covariance" title="Lorentz covariance">Lorentz invariant</a>, so the combination is invariant too.<a href="#cite_note-pais-7">[6]</a> For instance, Lorentz transformations can be extended by using factor <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/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}">:<a href="#cite_note-8">[R 2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>l</mi> <mi>y</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>l</mi> <mi>z</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e66258e6a0efab322774ea0afa2bf9f8c0fb69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.6ex; height:6.176ex;" alt="{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)}">.</dd></dl> l=1/γ gives the Voigt transformation, l=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a> in general. It was demonstrated by Poincaré and Einstein that one has to set l=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice. Voigt sent his 1887 paper to Lorentz in 1908,<a href="#cite_note-9">[7]</a> and that was acknowledged in 1909: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">In a paper "Über das Doppler'sche Princip", published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \Psi -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}\Psi }{\partial t^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Psi -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}\Psi }{\partial t^{2}}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f1513902f84ff6436d33cd8b8a9cdfe141f417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:17.121ex; height:4.343ex;" alt="{\displaystyle \Delta \Psi -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}\Psi }{\partial t^{2}}}=0}">] a transformation equivalent to the formulae (287) and (288) [namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\ y^{\prime }=ly,\ z^{\prime }=lz,\ t^{\prime }=\gamma l\left(t-{\tfrac {v}{c^{2}}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>l</mi> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>l</mi> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\ y^{\prime }=ly,\ z^{\prime }=lz,\ t^{\prime }=\gamma l\left(t-{\tfrac {v}{c^{2}}}x\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42fc2b1cdafcdd936f2a43d7eae674ed205c0aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.211ex; height:4.843ex;" alt="{\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\ y^{\prime }=ly,\ z^{\prime }=lz,\ t^{\prime }=\gamma l\left(t-{\tfrac {v}{c^{2}}}x\right)}">]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper.<a href="#cite_note-10">[R 3]</a></blockquote> Also <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.<a href="#cite_note-11">[R 4]</a> <div class="mw-heading mw-heading3"><h3 id="Heaviside_(1888),_Thomson_(1889),_Searle_(1896)"> Heaviside (1888), Thomson (1889), Searle (1896)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=5" title="Edit section: Heaviside (1888), Thomson (1889), Searle (1896)">edit</a>]</div> In 1888, <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a><a href="#cite_note-12">[R 5]</a> investigated the properties of <a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">charges in motion</a> according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:<a href="#cite_note-13">[8]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} =\left({\frac {q\mathrm {r} }{r^{2}}}\right)\left(1-{\frac {v^{2}\sin ^{2}\theta }{c^{2}}}\right)^{-3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} =\left({\frac {q\mathrm {r} }{r^{2}}}\right)\left(1-{\frac {v^{2}\sin ^{2}\theta }{c^{2}}}\right)^{-3/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e9fcbd14a230da5a97b80a6fd86d84cd32f593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.622ex; height:7.009ex;" alt="{\displaystyle \mathrm {E} =\left({\frac {q\mathrm {r} }{r^{2}}}\right)\left(1-{\frac {v^{2}\sin ^{2}\theta }{c^{2}}}\right)^{-3/2}}">.</dd></dl> Consequently, <a href="/wiki/Joseph_John_Thomson" class="mw-redirect" title="Joseph John Thomson">Joseph John Thomson</a> (1889)<a href="#cite_note-14">[R 6]</a> found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> z-vt in his equation<a href="#cite_note-mil-15">[9]</a>): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}z&=\left\{1-{\frac {\omega ^{2}}{v^{2}}}\right\}^{\frac {1}{2}}z'\end{aligned}}\right|&{\begin{aligned}z^{\ast }=z-vt&={\frac {z'}{\gamma }}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mo>′</mo> </msup> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}z&=\left\{1-{\frac {\omega ^{2}}{v^{2}}}\right\}^{\frac {1}{2}}z'\end{aligned}}\right|&{\begin{aligned}z^{\ast }=z-vt&={\frac {z'}{\gamma }}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eedecd26da05a9ce29fed0286c3b0bc409717c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:42.453ex; height:11.509ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}z&=\left\{1-{\frac {\omega ^{2}}{v^{2}}}\right\}^{\frac {1}{2}}z'\end{aligned}}\right|&{\begin{aligned}z^{\ast }=z-vt&={\frac {z'}{\gamma }}\end{aligned}}\end{matrix}}}"></dd></dl> Thereby, <a href="/wiki/Inhomogeneous_electromagnetic_wave_equation" title="Inhomogeneous electromagnetic wave equation">inhomogeneous electromagnetic wave equations</a> are transformed into a <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a>.<a href="#cite_note-mil-15">[9]</a> Eventually, <a href="/wiki/George_Frederick_Charles_Searle" title="George Frederick Charles Searle">George Frederick Charles Searle</a><a href="#cite_note-16">[R 7]</a> noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of <a href="/wiki/Axial_ratio" title="Axial ratio">axial ratio</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}&{\sqrt {\alpha }}:1:1\\\alpha =&1-{\frac {u^{2}}{v^{2}}}\end{aligned}}\right|&{\begin{aligned}&{\frac {1}{\gamma }}:1:1\\{\frac {1}{\gamma ^{2}}}&=1-{\frac {v^{2}}{c^{2}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> <mo>=</mo> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mo>:</mo> <mn>1</mn> <mo>:</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}&{\sqrt {\alpha }}:1:1\\\alpha =&1-{\frac {u^{2}}{v^{2}}}\end{aligned}}\right|&{\begin{aligned}&{\frac {1}{\gamma }}:1:1\\{\frac {1}{\gamma ^{2}}}&=1-{\frac {v^{2}}{c^{2}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f3552d590c3e8b76552177951f464a309c1b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:33.935ex; height:16.509ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}&{\sqrt {\alpha }}:1:1\\\alpha =&1-{\frac {u^{2}}{v^{2}}}\end{aligned}}\right|&{\begin{aligned}&{\frac {1}{\gamma }}:1:1\\{\frac {1}{\gamma ^{2}}}&=1-{\frac {v^{2}}{c^{2}}}\end{aligned}}\end{matrix}}}"><a href="#cite_note-mil-15">[9]</a></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Lorentz_(1892,_1895)"> Lorentz (1892, 1895)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=6" title="Edit section: Lorentz (1892, 1895)">edit</a>]</div> In order to explain the <a href="/wiki/Aberration_of_light" class="mw-redirect" title="Aberration of light">aberration of light</a> and the result of the <a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a> in accordance with <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, Lorentz in 1892 developed a model ("<a href="/wiki/Lorentz_ether_theory" title="Lorentz ether theory">Lorentz ether theory</a>") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)<a href="#cite_note-17">[R 8]</a><a href="#cite_note-milf-18">[10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}{\mathfrak {x}}&={\frac {V}{\sqrt {V^{2}-p^{2}}}}x\\t'&=t-{\frac {\varepsilon }{V}}{\mathfrak {x}}\\\varepsilon &={\frac {p}{\sqrt {V^{2}-p^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\\\gamma {\frac {v}{c}}&={\frac {v}{\sqrt {c^{2}-v^{2}}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">x</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msqrt> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>V</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">x</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <msqrt> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}{\mathfrak {x}}&={\frac {V}{\sqrt {V^{2}-p^{2}}}}x\\t'&=t-{\frac {\varepsilon }{V}}{\mathfrak {x}}\\\varepsilon &={\frac {p}{\sqrt {V^{2}-p^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\\\gamma {\frac {v}{c}}&={\frac {v}{\sqrt {c^{2}-v^{2}}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c728d05117951ad354759065eeb78fbcd09e7a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:57.13ex; height:22.176ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}{\mathfrak {x}}&={\frac {V}{\sqrt {V^{2}-p^{2}}}}x\\t'&=t-{\frac {\varepsilon }{V}}{\mathfrak {x}}\\\varepsilon &={\frac {p}{\sqrt {V^{2}-p^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\\\gamma {\frac {v}{c}}&={\frac {v}{\sqrt {c^{2}-v^{2}}}}\end{aligned}}\end{matrix}}}"></dd></dl> where x* is the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a> x-vt. Except the additional γ in the time transformation, this is the complete Lorentz transformation.<a href="#cite_note-milf-18">[10]</a> While t is the "true" time for observers resting in the aether, t′ is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a>, he (1892b)<a href="#cite_note-19">[R 9]</a> introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a> in his theory (without proof as he admitted). The same hypothesis had been made previously by <a href="/wiki/George_Francis_FitzGerald" title="George Francis FitzGerald">George FitzGerald</a> in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation. In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his "fictitious" field makes the same observations as a resting observers in his "real" field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:<a href="#cite_note-20">[R 10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=x^{\prime }{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {x^{\prime }}{\gamma }}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=x^{\prime }{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {x^{\prime }}{\gamma }}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b578499966b24c77068a79a6e909977e860682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:41.227ex; height:21.176ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=x^{\prime }{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {x^{\prime }}{\gamma }}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\end{matrix}}}"></dd></dl> For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (<a href="/wiki/German_language" title="German language">German</a>: Ortszeit) by him:<a href="#cite_note-21">[R 11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=\mathrm {x} -{\mathfrak {p}}_{x}t\\y&=\mathrm {y} -{\mathfrak {p}}_{y}t\\z&=\mathrm {z} -{\mathfrak {p}}_{z}t\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}x-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}y-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}z\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-v_{x}t\\y^{\prime }&=y-v_{y}t\\z^{\prime }&=z-v_{z}t\\t^{\prime }&=t-{\frac {v_{x}}{c^{2}}}x'-{\frac {v_{y}}{c^{2}}}y'-{\frac {v_{z}}{c^{2}}}z'\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">z</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=\mathrm {x} -{\mathfrak {p}}_{x}t\\y&=\mathrm {y} -{\mathfrak {p}}_{y}t\\z&=\mathrm {z} -{\mathfrak {p}}_{z}t\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}x-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}y-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}z\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-v_{x}t\\y^{\prime }&=y-v_{y}t\\z^{\prime }&=z-v_{z}t\\t^{\prime }&=t-{\frac {v_{x}}{c^{2}}}x'-{\frac {v_{y}}{c^{2}}}y'-{\frac {v_{z}}{c^{2}}}z'\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7eafe4171180f77b9652763ffd04912851124e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:65.284ex; height:19.843ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=\mathrm {x} -{\mathfrak {p}}_{x}t\\y&=\mathrm {y} -{\mathfrak {p}}_{y}t\\z&=\mathrm {z} -{\mathfrak {p}}_{z}t\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}x-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}y-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}z\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-v_{x}t\\y^{\prime }&=y-v_{y}t\\z^{\prime }&=z-v_{z}t\\t^{\prime }&=t-{\frac {v_{x}}{c^{2}}}x'-{\frac {v_{y}}{c^{2}}}y'-{\frac {v_{z}}{c^{2}}}z'\end{aligned}}\end{matrix}}}"></dd></dl> With this concept Lorentz could explain the <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a>, the <a href="/wiki/Aberration_of_light" class="mw-redirect" title="Aberration of light">aberration of light</a>, and the <a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a>.<a href="#cite_note-22">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Larmor_(1897,_1900)"> Larmor (1897, 1900)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=7" title="Edit section: Larmor (1897, 1900)">edit</a>]</div> In 1897, Larmor extended the work of Lorentz and derived the following transformation<a href="#cite_note-23">[R 12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=x\varepsilon ^{\frac {1}{2}}\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-vx/c^{2}\\dt_{1}&=dt^{\prime }\varepsilon ^{-{\frac {1}{2}}}\\\varepsilon &=\left(1-v^{2}/c^{2}\right)^{-1}\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\ast }=\gamma (x-vt)\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-{\frac {vx^{\ast }}{c^{2}}}=t-{\frac {v(x-vt)}{c^{2}}}\\dt_{1}&={\frac {dt^{\prime }}{\gamma }}\\\gamma ^{2}&={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" 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<mtd> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right 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<msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mi>γ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=x\varepsilon ^{\frac {1}{2}}\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-vx/c^{2}\\dt_{1}&=dt^{\prime }\varepsilon ^{-{\frac {1}{2}}}\\\varepsilon &=\left(1-v^{2}/c^{2}\right)^{-1}\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\ast }=\gamma (x-vt)\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-{\frac {vx^{\ast }}{c^{2}}}=t-{\frac {v(x-vt)}{c^{2}}}\\dt_{1}&={\frac {dt^{\prime }}{\gamma }}\\\gamma ^{2}&={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e650cac6d7c4a7b431e401ac3023aaa04d188b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.005ex; width:58.086ex; height:33.009ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=x\varepsilon ^{\frac {1}{2}}\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-vx/c^{2}\\dt_{1}&=dt^{\prime }\varepsilon ^{-{\frac {1}{2}}}\\\varepsilon &=\left(1-v^{2}/c^{2}\right)^{-1}\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\ast }=\gamma (x-vt)\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-{\frac {vx^{\ast }}{c^{2}}}=t-{\frac {v(x-vt)}{c^{2}}}\\dt_{1}&={\frac {dt^{\prime }}{\gamma }}\\\gamma ^{2}&={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}}"></dd></dl> Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a>. It's notable that Larmor was the first who recognized that some sort of <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> is a consequence of this transformation as well, because "individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ".<a href="#cite_note-24">[12]</a><a href="#cite_note-25">[13]</a> Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than (v/c)2 – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of v/c:<a href="#cite_note-26">[R 13]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Nothing need be neglected: the transformation is exact if v/c2 is replaced by εv/c2 in the equations and also in the change following from t to t′, as is worked out in Aether and Matter (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.</blockquote> In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time t″=t′-εvx′/c2 instead of the 1897 expression t′=t-vx/c2 by replacing v/c2 with εv/c2, so that t″ is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the x′, y′, z′, t′ coordinates:<a href="#cite_note-27">[R 14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }&=t^{\prime }-\varepsilon vx^{\prime }/c^{2}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }=t^{\prime }-{\frac {\gamma ^{2}vx^{\prime }}{c^{2}}}&=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mi>ε</mi> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }&=t^{\prime }-\varepsilon vx^{\prime }/c^{2}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }=t^{\prime }-{\frac {\gamma ^{2}vx^{\prime }}{c^{2}}}&=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7227d2e50c18836dfd4d70febe00e3afcb64630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:55.975ex; height:22.843ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }&=t^{\prime }-\varepsilon vx^{\prime }/c^{2}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }=t^{\prime }-{\frac {\gamma ^{2}vx^{\prime }}{c^{2}}}&=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"></dd></dl> Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor (v/c)2, and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x′=x-vt and t″ as given above) as:<a href="#cite_note-28">[R 15]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=\varepsilon ^{\frac {1}{2}}x^{\prime }\\y_{1}&=y^{\prime }\\z_{1}&=z^{\prime }\\dt_{1}&=\varepsilon ^{-{\frac {1}{2}}}dt^{\prime \prime }=\varepsilon ^{-{\frac {1}{2}}}\left(dt^{\prime }-{\frac {v}{c^{2}}}\varepsilon dx^{\prime }\right)\\t_{1}&=\varepsilon ^{-{\frac {1}{2}}}t^{\prime }-{\frac {v}{c^{2}}}\varepsilon ^{\frac {1}{2}}x^{\prime }\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\prime }=\gamma (x-vt)\\y_{1}&=y'=y\\z_{1}&=z'=z\\dt_{1}&={\frac {dt^{\prime \prime }}{\gamma }}={\frac {1}{\gamma }}\left(dt^{\prime }-{\frac {\gamma ^{2}vdx^{\prime }}{c^{2}}}\right)=\gamma \left(dt-{\frac {vdx}{c^{2}}}\right)\\t_{1}&={\frac {t^{\prime }}{\gamma }}-{\frac {\gamma vx^{\prime }}{c^{2}}}=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>ε</mi> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mi>γ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>d</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mi>γ</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=\varepsilon ^{\frac {1}{2}}x^{\prime }\\y_{1}&=y^{\prime }\\z_{1}&=z^{\prime }\\dt_{1}&=\varepsilon ^{-{\frac {1}{2}}}dt^{\prime \prime }=\varepsilon ^{-{\frac {1}{2}}}\left(dt^{\prime }-{\frac {v}{c^{2}}}\varepsilon dx^{\prime }\right)\\t_{1}&=\varepsilon ^{-{\frac {1}{2}}}t^{\prime }-{\frac {v}{c^{2}}}\varepsilon ^{\frac {1}{2}}x^{\prime }\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\prime }=\gamma (x-vt)\\y_{1}&=y'=y\\z_{1}&=z'=z\\dt_{1}&={\frac {dt^{\prime \prime }}{\gamma }}={\frac {1}{\gamma }}\left(dt^{\prime }-{\frac {\gamma ^{2}vdx^{\prime }}{c^{2}}}\right)=\gamma \left(dt-{\frac {vdx}{c^{2}}}\right)\\t_{1}&={\frac {t^{\prime }}{\gamma }}-{\frac {\gamma vx^{\prime }}{c^{2}}}=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122256e127b2213c117a432d0c9149425938fafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:94.302ex; height:26.176ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=\varepsilon ^{\frac {1}{2}}x^{\prime }\\y_{1}&=y^{\prime }\\z_{1}&=z^{\prime }\\dt_{1}&=\varepsilon ^{-{\frac {1}{2}}}dt^{\prime \prime }=\varepsilon ^{-{\frac {1}{2}}}\left(dt^{\prime }-{\frac {v}{c^{2}}}\varepsilon dx^{\prime }\right)\\t_{1}&=\varepsilon ^{-{\frac {1}{2}}}t^{\prime }-{\frac {v}{c^{2}}}\varepsilon ^{\frac {1}{2}}x^{\prime }\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\prime }=\gamma (x-vt)\\y_{1}&=y'=y\\z_{1}&=z'=z\\dt_{1}&={\frac {dt^{\prime \prime }}{\gamma }}={\frac {1}{\gamma }}\left(dt^{\prime }-{\frac {\gamma ^{2}vdx^{\prime }}{c^{2}}}\right)=\gamma \left(dt-{\frac {vdx}{c^{2}}}\right)\\t_{1}&={\frac {t^{\prime }}{\gamma }}-{\frac {\gamma vx^{\prime }}{c^{2}}}=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"></dd></dl> by which he arrived at the complete Lorentz transformation. Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c" – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in v/c. Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether. p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]<a href="#cite_note-29">[R 16]</a> p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its "local time" in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.<a href="#cite_note-30">[R 17]</a></blockquote> <div class="mw-heading mw-heading3"><h3 id="Lorentz_(1899,_1904)"> Lorentz (1899, 1904)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=8" title="Edit section: Lorentz (1899, 1904)">edit</a>]</div> Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, x* must be replaced by x-vt):<a href="#cite_note-31">[R 18]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}x\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}-{\mathfrak {p}}_{x}^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msqrt> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}x\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}-{\mathfrak {p}}_{x}^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ade6b59a25e4f6601bb27173bcc77ac60f9d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:57.876ex; height:23.176ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}x\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}-{\mathfrak {p}}_{x}^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"></dd></dl> Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of t′ has to be inserted):<a href="#cite_note-32">[R 19]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&={\frac {\varepsilon }{k}}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon x^{\prime \prime }\\t^{\prime }&=k\varepsilon t^{\prime \prime }\\k&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {\varepsilon }{\gamma }}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon z^{\prime \prime }\\t^{\prime }=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)&=\gamma \varepsilon t^{\prime \prime }\\\gamma &={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>k</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <mi>ε</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msqrt> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>γ</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mi>ε</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&={\frac {\varepsilon }{k}}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon x^{\prime \prime }\\t^{\prime }&=k\varepsilon t^{\prime \prime }\\k&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {\varepsilon }{\gamma }}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon z^{\prime \prime }\\t^{\prime }=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)&=\gamma \varepsilon t^{\prime \prime }\\\gamma &={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9160842cf2e8d25a9551c1f3efba2a95bc6d8520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.505ex; width:53.476ex; height:30.176ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&={\frac {\varepsilon }{k}}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon x^{\prime \prime }\\t^{\prime }&=k\varepsilon t^{\prime \prime }\\k&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {\varepsilon }{\gamma }}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon z^{\prime \prime }\\t^{\prime }=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)&=\gamma \varepsilon t^{\prime \prime }\\\gamma &={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}\end{matrix}}}"></dd></dl> This is equivalent to the complete Lorentz transformation when solved for x″ and t″ and with ε=1. Like Larmor, Lorentz noticed in 1899<a href="#cite_note-33">[R 20]</a> also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in S the time of vibrations be kε times as great as in S0", where S0 is the aether frame.<a href="#cite_note-34">[14]</a> In 1904 he rewrote the equations in the following form by setting l=1/ε (again, x* must be replaced by x-vt):<a href="#cite_note-35">[R 21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=klx\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&={\frac {l}{k}}t-kl{\frac {w}{c^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma lx^{\ast }=\gamma l(x-vt)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t^{\prime }&={\frac {lt}{\gamma }}-{\frac {\gamma lvx^{\ast }}{c^{2}}}=\gamma l\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <mi>l</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mi>k</mi> </mfrac> </mrow> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mi>t</mi> </mrow> <mi>γ</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>l</mi> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=klx\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&={\frac {l}{k}}t-kl{\frac {w}{c^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma lx^{\ast }=\gamma l(x-vt)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t^{\prime }&={\frac {lt}{\gamma }}-{\frac {\gamma lvx^{\ast }}{c^{2}}}=\gamma l\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc59798d652a3fc2d33e4d81fa92b4b11d50297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:56.515ex; height:19.509ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=klx\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&={\frac {l}{k}}t-kl{\frac {w}{c^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma lx^{\ast }=\gamma l(x-vt)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t^{\prime }&={\frac {lt}{\gamma }}-{\frac {\gamma lvx^{\ast }}{c^{2}}}=\gamma l\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}}"></dd></dl> Under the assumption that l=1 when v=0, he demonstrated that l=1 must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor l to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in v/c. He also derived the correct formulas for the velocity dependence of <a href="/wiki/Electromagnetic_mass" title="Electromagnetic mass">electromagnetic mass</a>, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.<a href="#cite_note-36">[R 22]</a> However, he didn't achieve full covariance of the transformation equations for charge density and velocity.<a href="#cite_note-37">[15]</a> When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:<a href="#cite_note-38">[16]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.</blockquote> Lorentz's 1904 transformation was cited and used by <a href="/wiki/Alfred_Bucherer" title="Alfred Bucherer">Alfred Bucherer</a> in July 1904:<a href="#cite_note-39">[R 23]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }={\sqrt {s}}x,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{\sqrt {s}}}-{\sqrt {s}}{\frac {u}{v^{2}}}x,\quad s=1-{\frac {u^{2}}{v^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> </msqrt> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>s</mi> </msqrt> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime }={\sqrt {s}}x,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{\sqrt {s}}}-{\sqrt {s}}{\frac {u}{v^{2}}}x,\quad s=1-{\frac {u^{2}}{v^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1096ae78b7c3d42c6dcd1789b2995316bf4a4d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:65.067ex; height:6.676ex;" alt="{\displaystyle x^{\prime }={\sqrt {s}}x,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{\sqrt {s}}}-{\sqrt {s}}{\frac {u}{v^{2}}}x,\quad s=1-{\frac {u^{2}}{v^{2}}}}"></dd></dl> or by <a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wilhelm Wien</a> in July 1904:<a href="#cite_note-40">[R 24]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=kx',\quad y=y',\quad z=z',\quad t'=kt-{\frac {v}{kc^{2}}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>k</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mrow> <mi>k</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=kx',\quad y=y',\quad z=z',\quad t'=kt-{\frac {v}{kc^{2}}}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b48a75983639547637edac7f607dfaf4c728dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:44.736ex; height:5.009ex;" alt="{\displaystyle x=kx',\quad y=y',\quad z=z',\quad t'=kt-{\frac {v}{kc^{2}}}x}"></dd></dl> or by <a href="/wiki/Emil_Cohn" title="Emil Cohn">Emil Cohn</a> in November 1904 (setting the speed of light to unity):<a href="#cite_note-41">[R 25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {x_{0}}{k}},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}={\frac {1}{1-w^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>=</mo> <mi>k</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>w</mi> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {x_{0}}{k}},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}={\frac {1}{1-w^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88147ce8eb68d49668899baf1f94b6832194bad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:72.337ex; height:5.676ex;" alt="{\displaystyle x={\frac {x_{0}}{k}},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}={\frac {1}{1-w^{2}}}}"></dd></dl> or by <a href="/wiki/Richard_Gans" title="Richard Gans">Richard Gans</a> in February 1905:<a href="#cite_note-42">[R 26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }=kx,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{k}}-{\frac {kwx}{c^{2}}},\quad k^{2}={\frac {c^{2}}{c^{2}-w^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>k</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>w</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime }=kx,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{k}}-{\frac {kwx}{c^{2}}},\quad k^{2}={\frac {c^{2}}{c^{2}-w^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a739e9a8d8db304539c72e1cb9a5f5584b5417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.512ex; height:6.176ex;" alt="{\displaystyle x^{\prime }=kx,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{k}}-{\frac {kwx}{c^{2}}},\quad k^{2}={\frac {c^{2}}{c^{2}-w^{2}}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Poincaré_(1900,_1905)"> Poincaré (1900, 1905)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=9" title="Edit section: Poincaré (1900, 1905)">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Local_time">Local time</h4>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=10" title="Edit section: Local time">edit</a>]</div> Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> in 1900 commented on the origin of Lorentz's "wonderful invention" of local time.<a href="#cite_note-43">[17]</a> He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> in both directions, which lead to what is nowadays called <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>, although Poincaré's calculation does not involve length contraction or time dilation.<a href="#cite_note-44">[R 27]</a> In order to synchronise the clocks here on Earth (the x*, t* frame) a light signal from one clock (at the origin) is sent to another (at x*), and is sent back. It's supposed that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x, t) (i.e. the <a href="/wiki/Luminiferous_aether" title="Luminiferous aether">luminiferous aether</a> system for Lorentz and Larmor). The time of flight outwards is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta t_{a}={\frac {x^{\ast }}{\left(c-v\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>−</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta t_{a}={\frac {x^{\ast }}{\left(c-v\right)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5103b7fb49e84f95f3c04259dd37c54dae49021f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.709ex; height:6.176ex;" alt="{\displaystyle \delta t_{a}={\frac {x^{\ast }}{\left(c-v\right)}}}"></dd></dl> and the time of flight back is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta t_{b}={\frac {x^{\ast }}{\left(c+v\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>+</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta t_{b}={\frac {x^{\ast }}{\left(c+v\right)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfc36778009287e792df422ce42fdc4fd4da2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.545ex; height:6.176ex;" alt="{\displaystyle \delta t_{b}={\frac {x^{\ast }}{\left(c+v\right)}}}">.</dd></dl> The elapsed time on the clock when the signal is returned is δta+δtb and the time t*=(δta+δtb)/2 is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time t=δta is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{\ast }=t-{\frac {\gamma ^{2}vx^{*}}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{\ast }=t-{\frac {\gamma ^{2}vx^{*}}{c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5555b9fa5761eaacfb946db1060c5a63deda1a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.354ex; height:6.009ex;" alt="{\displaystyle t^{\ast }=t-{\frac {\gamma ^{2}vx^{*}}{c^{2}}}}"></dd></dl> identical to Lorentz (1892). By dropping the factor γ2 under the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {v^{2}}{c^{2}}}\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {v^{2}}{c^{2}}}\ll 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fd33a1ba55784a33cae0ddf05f87ac167715b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.242ex; height:4.343ex;" alt="{\displaystyle {\tfrac {v^{2}}{c^{2}}}\ll 1}">, Poincaré gave the result t*=t-vx*/c2, which is the form used by Lorentz in 1895. Similar physical interpretations of local time were later given by <a href="/wiki/Emil_Cohn" title="Emil Cohn">Emil Cohn</a> (1904)<a href="#cite_note-45">[R 28]</a> and <a href="/wiki/Max_Abraham" title="Max Abraham">Max Abraham</a> (1905).<a href="#cite_note-46">[R 29]</a> <div class="mw-heading mw-heading4"><h4 id="Lorentz_transformation">Lorentz transformation</h4>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=11" title="Edit section: Lorentz transformation">edit</a>]</div> On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form:<a href="#cite_note-47">[R 30]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x^{\prime }&=kl(x+\varepsilon t)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&=kl(t+\varepsilon x)\\k&={\frac {1}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>ε</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>l</mi> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>ε</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{\prime }&=kl(x+\varepsilon t)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&=kl(t+\varepsilon x)\\k&={\frac {1}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0e0f83e9688c531911f4233f6523b57d6a0503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.676ex; margin-bottom: -0.328ex; width:15.671ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}x^{\prime }&=kl(x+\varepsilon t)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&=kl(t+\varepsilon x)\\k&={\frac {1}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}}">.</dd></dl> Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation".<a href="#cite_note-48">[18]</a><a href="#cite_note-49">[19]</a> Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting l=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, i.e. making them fully Lorentz covariant.<a href="#cite_note-50">[20]</a> In July 1905 (published in January 1906)<a href="#cite_note-51">[R 31]</a> Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the <a href="/wiki/Principle_of_least_action" class="mw-redirect" title="Principle of least action">principle of least action</a>; he demonstrated in more detail the group characteristics of the transformation, which he called <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>, and he showed that the combination x2+y2+z2-t2 is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ct{\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ct{\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd666a0ffc00a10a7f354c27f17cbadf8368e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.753ex; height:3.009ex;" alt="{\displaystyle ct{\sqrt {-1}}}"> as a fourth imaginary coordinate, and he used an early form of <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>. He also formulated the velocity addition formula, which he had already derived in unpublished letters to Lorentz from May 1905:<a href="#cite_note-52">[R 32]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi '={\frac {\xi +\varepsilon }{1+\xi \varepsilon }},\ \eta '={\frac {\eta }{k(1+\xi \varepsilon )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ξ</mi> <mo>+</mo> <mi>ε</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ξ</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ξ</mi> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi '={\frac {\xi +\varepsilon }{1+\xi \varepsilon }},\ \eta '={\frac {\eta }{k(1+\xi \varepsilon )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6cce0ed7830a33ab96bfb433ccb03f06e818ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.314ex; height:6.176ex;" alt="{\displaystyle \xi '={\frac {\xi +\varepsilon }{1+\xi \varepsilon }},\ \eta '={\frac {\eta }{k(1+\xi \varepsilon )}}}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Einstein_(1905)_–_Special_relativity"> Einstein (1905) – Special relativity</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=12" title="Edit section: Einstein (1905) – Special relativity">edit</a>]</div> On June 30, 1905 (published September 1905) Einstein published what is now called <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and gave a new derivation of the transformation, which was based only on the principle of relativity and the principle of the constancy of the speed of light. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in v/c this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations applied to the kinematics of moving frames.<a href="#cite_note-53">[21]</a><a href="#cite_note-54">[22]</a><a href="#cite_note-55">[23]</a> The notation for this transformation is equivalent to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:<a href="#cite_note-56">[R 33]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tau &=\beta \left(t-{\frac {v}{V^{2}}}x\right)\\\xi &=\beta (x-vt)\\\eta &=y\\\zeta &=z\\\beta &={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ξ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>η</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>ζ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tau &=\beta \left(t-{\frac {v}{V^{2}}}x\right)\\\xi &=\beta (x-vt)\\\eta &=y\\\zeta &=z\\\beta &={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f2dd647a5086633fddfde0f0a07a3f52ac2cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.612ex; margin-bottom: -0.226ex; width:19.139ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\tau &=\beta \left(t-{\frac {v}{V^{2}}}x\right)\\\xi &=\beta (x-vt)\\\eta &=y\\\zeta &=z\\\beta &={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}\end{aligned}}}"></dd></dl> Einstein also defined the velocity addition formula:<a href="#cite_note-57">[R 34]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x={\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{V^{2}}}}}t,\ y={\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1+{\frac {vw_{\xi }}{V^{2}}}}}w_{\eta }t\\U^{2}=\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2},\ w^{2}=w_{\xi }^{2}+w_{\eta }^{2},\ \alpha =\operatorname {arctg} {\frac {w_{y}}{w_{x}}}\\U={\frac {\sqrt {\left(v^{2}+w^{2}+2vw\cos \alpha \right)-\left({\frac {vw\sin \alpha }{V}}\right)^{2}}}{1+{\frac {vw\cos \alpha }{V^{2}}}}}\end{matrix}}\left|{\begin{matrix}{\frac {u_{x}-v}{1-{\frac {u_{x}v}{V^{2}}}}}=u_{\xi }\\{\frac {u_{y}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\eta }\\{\frac {u_{z}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\zeta }\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo>+</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <mi>α</mi> <mo>=</mo> <mi>arctg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mrow> <mo>(</mo> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>v</mi> <mi>w</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>w</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>w</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x={\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{V^{2}}}}}t,\ y={\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1+{\frac {vw_{\xi }}{V^{2}}}}}w_{\eta }t\\U^{2}=\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2},\ w^{2}=w_{\xi }^{2}+w_{\eta }^{2},\ \alpha =\operatorname {arctg} {\frac {w_{y}}{w_{x}}}\\U={\frac {\sqrt {\left(v^{2}+w^{2}+2vw\cos \alpha \right)-\left({\frac {vw\sin \alpha }{V}}\right)^{2}}}{1+{\frac {vw\cos \alpha }{V^{2}}}}}\end{matrix}}\left|{\begin{matrix}{\frac {u_{x}-v}{1-{\frac {u_{x}v}{V^{2}}}}}=u_{\xi }\\{\frac {u_{y}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\eta }\\{\frac {u_{z}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\zeta }\end{matrix}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7882552d6a0aa90748fa9747c2706039a7539a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:69.523ex; height:23.176ex;" alt="{\displaystyle {\begin{matrix}x={\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{V^{2}}}}}t,\ y={\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1+{\frac {vw_{\xi }}{V^{2}}}}}w_{\eta }t\\U^{2}=\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2},\ w^{2}=w_{\xi }^{2}+w_{\eta }^{2},\ \alpha =\operatorname {arctg} {\frac {w_{y}}{w_{x}}}\\U={\frac {\sqrt {\left(v^{2}+w^{2}+2vw\cos \alpha \right)-\left({\frac {vw\sin \alpha }{V}}\right)^{2}}}{1+{\frac {vw\cos \alpha }{V^{2}}}}}\end{matrix}}\left|{\begin{matrix}{\frac {u_{x}-v}{1-{\frac {u_{x}v}{V^{2}}}}}=u_{\xi }\\{\frac {u_{y}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\eta }\\{\frac {u_{z}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\zeta }\end{matrix}}\right.}"></dd></dl> and the light aberration formula:<a href="#cite_note-58">[R 35]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi '={\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>V</mi> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>V</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi '={\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a23f9e3a1d82e46c760f4f920f587020b5339a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.146ex; height:7.509ex;" alt="{\displaystyle \cos \varphi '={\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Minkowski_(1907–1908)_–_Spacetime"> Minkowski (1907–1908) – Spacetime</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=13" title="Edit section: Minkowski (1907–1908) – Spacetime">edit</a>]</div> The work on the principle of relativity by Lorentz, Einstein, <a href="/wiki/Max_Planck" title="Max Planck">Planck</a>, together with Poincaré's four-dimensional approach, were further elaborated and combined with the <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> by <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> in 1907 and 1908.<a href="#cite_note-59">[R 36]</a><a href="#cite_note-60">[R 37]</a> Minkowski particularly reformulated electrodynamics in a four-dimensional way (<a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>).<a href="#cite_note-61">[24]</a> For instance, he wrote x, y, z, it in the form x1, x2, x3, x4. By defining ψ as the angle of rotation around the z-axis, the Lorentz transformation assumes the form (with c=1):<a href="#cite_note-mink1-62">[R 38]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x'_{1}&=x_{1}\\x'_{2}&=x_{2}\\x'_{3}&=x_{3}\cos i\psi +x_{4}\sin i\psi \\x'_{4}&=-x_{3}\sin i\psi +x_{4}\cos i\psi \\\cos i\psi &={\frac {1}{\sqrt {1-q^{2}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x'_{1}&=x_{1}\\x'_{2}&=x_{2}\\x'_{3}&=x_{3}\cos i\psi +x_{4}\sin i\psi \\x'_{4}&=-x_{3}\sin i\psi +x_{4}\cos i\psi \\\cos i\psi &={\frac {1}{\sqrt {1-q^{2}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f4ae3f891fb0dafe5a3fd9a012076419e1d9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.034ex; margin-bottom: -0.304ex; width:31.227ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}x'_{1}&=x_{1}\\x'_{2}&=x_{2}\\x'_{3}&=x_{3}\cos i\psi +x_{4}\sin i\psi \\x'_{4}&=-x_{3}\sin i\psi +x_{4}\cos i\psi \\\cos i\psi &={\frac {1}{\sqrt {1-q^{2}}}}\end{aligned}}}"></dd></dl> Even though Minkowski used the imaginary number iψ, he for once<a href="#cite_note-mink1-62">[R 38]</a> directly used the <a href="/wiki/Tangens_hyperbolicus" class="mw-redirect" title="Tangens hyperbolicus">tangens hyperbolicus</a> in the equation for velocity <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\tan i\psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>tan</mi> <mo>⁡</mo> <mi>i</mi> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ψ</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ψ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\tan i\psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be88eac93128db69233c66f748fcfc4e93215eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.053ex; height:6.176ex;" alt="{\displaystyle -i\tan i\psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\frac {1}{2}}\ln {\frac {1+q}{1-q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>q</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\frac {1}{2}}\ln {\frac {1+q}{1-q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887b46cb8ea819708e3a3ff0c5741c31c04b42c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.232ex; height:5.843ex;" alt="{\displaystyle \psi ={\frac {1}{2}}\ln {\frac {1+q}{1-q}}}">.</dd></dl> Minkowski's expression can also by written as ψ=atanh(q) and was later called <a href="/wiki/Rapidity" title="Rapidity">rapidity</a>. He also wrote the Lorentz transformation in matrix form:<a href="#cite_note-63">[R 39]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}+x_{4}^{\prime 2}\\\left(x_{1}^{\prime }=x',\ x_{2}^{\prime }=y',\ x_{3}^{\prime }=z',\ x_{4}^{\prime }=it'\right)\\-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime 2}-y^{\prime 2}-z^{\prime 2}+t^{\prime 2}\\\hline x_{h}=\alpha _{h1}x_{1}^{\prime }+\alpha _{h2}x_{2}^{\prime }+\alpha _{h3}x_{3}^{\prime }+\alpha _{h4}x_{4}^{\prime }\\\mathrm {A} =\mathrm {\left|{\begin{matrix}\alpha _{11},&\alpha _{12},&\alpha _{13},&\alpha _{14}\\\alpha _{21},&\alpha _{22},&\alpha _{23},&\alpha _{24}\\\alpha _{31},&\alpha _{32},&\alpha _{33},&\alpha _{34}\\\alpha _{41},&\alpha _{42},&\alpha _{43},&\alpha _{44}\end{matrix}}\right|,\ {\begin{aligned}{\bar {\mathrm {A} }}\mathrm {A} &=1\\\left(\det \mathrm {A} \right)^{2}&=1\\\det \mathrm {A} &=1\\\alpha _{44}&>0\end{aligned}}} \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none none solid none"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mi>i</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mn>1</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mn>2</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mn>3</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mn>4</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> 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</mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}+x_{4}^{\prime 2}\\\left(x_{1}^{\prime }=x',\ x_{2}^{\prime }=y',\ x_{3}^{\prime }=z',\ x_{4}^{\prime }=it'\right)\\-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime 2}-y^{\prime 2}-z^{\prime 2}+t^{\prime 2}\\\hline x_{h}=\alpha _{h1}x_{1}^{\prime }+\alpha _{h2}x_{2}^{\prime }+\alpha _{h3}x_{3}^{\prime }+\alpha _{h4}x_{4}^{\prime }\\\mathrm {A} =\mathrm {\left|{\begin{matrix}\alpha _{11},&\alpha _{12},&\alpha _{13},&\alpha _{14}\\\alpha _{21},&\alpha _{22},&\alpha _{23},&\alpha _{24}\\\alpha _{31},&\alpha _{32},&\alpha _{33},&\alpha _{34}\\\alpha _{41},&\alpha _{42},&\alpha _{43},&\alpha _{44}\end{matrix}}\right|,\ {\begin{aligned}{\bar {\mathrm {A} }}\mathrm {A} &=1\\\left(\det \mathrm {A} \right)^{2}&=1\\\det \mathrm {A} &=1\\\alpha _{44}&>0\end{aligned}}} \end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a9c08b67ae1c546687e65165227f45d0ba2e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.338ex; width:47.097ex; height:27.843ex;" alt="{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}+x_{4}^{\prime 2}\\\left(x_{1}^{\prime }=x',\ x_{2}^{\prime }=y',\ x_{3}^{\prime }=z',\ x_{4}^{\prime }=it'\right)\\-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime 2}-y^{\prime 2}-z^{\prime 2}+t^{\prime 2}\\\hline x_{h}=\alpha _{h1}x_{1}^{\prime }+\alpha _{h2}x_{2}^{\prime }+\alpha _{h3}x_{3}^{\prime }+\alpha _{h4}x_{4}^{\prime }\\\mathrm {A} =\mathrm {\left|{\begin{matrix}\alpha _{11},&\alpha _{12},&\alpha _{13},&\alpha _{14}\\\alpha _{21},&\alpha _{22},&\alpha _{23},&\alpha _{24}\\\alpha _{31},&\alpha _{32},&\alpha _{33},&\alpha _{34}\\\alpha _{41},&\alpha _{42},&\alpha _{43},&\alpha _{44}\end{matrix}}\right|,\ {\begin{aligned}{\bar {\mathrm {A} }}\mathrm {A} &=1\\\left(\det \mathrm {A} \right)^{2}&=1\\\det \mathrm {A} &=1\\\alpha _{44}&>0\end{aligned}}} \end{matrix}}}"></dd></dl> As a graphical representation of the Lorentz transformation he introduced the <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a>, which became a standard tool in textbooks and research articles on relativity:<a href="#cite_note-64">[R 40]</a> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Minkowski1.png/400px-Minkowski1.png" decoding="async" width="400" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Minkowski1.png/600px-Minkowski1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Minkowski1.png/800px-Minkowski1.png 2x" data-file-width="1328" data-file-height="464" /></a><figcaption>Original spacetime diagram by Minkowski in 1908.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Sommerfeld_(1909)_–_Spherical_trigonometry"> Sommerfeld (1909) – Spherical trigonometry</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=14" title="Edit section: Sommerfeld (1909) – Spherical trigonometry">edit</a>]</div> Using an imaginary rapidity such as Minkowski, <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> (1909) formulated the Lorentz boost and the relativistic velocity addition in terms of trigonometric functions and the <a href="/wiki/Spherical_law_of_cosines" title="Spherical law of cosines">spherical law of cosines</a>:<a href="#cite_note-65">[R 41]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\left.{\begin{array}{lrl}x'=&x\ \cos \varphi +l\ \sin \varphi ,&y'=y\\l'=&-x\ \sin \varphi +l\ \cos \varphi ,&z'=z\end{array}}\right\}\\\left(\operatorname {tg} \varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}\right)\\\hline \beta ={\frac {1}{i}}\operatorname {tg} \left(\varphi _{1}+\varphi _{2}\right)={\frac {1}{i}}{\frac {\operatorname {tg} \varphi _{1}+\operatorname {tg} \varphi _{2}}{1-\operatorname {tg} \varphi _{1}\operatorname {tg} \varphi _{2}}}={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}\\\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}\cos \alpha \\v^{2}={\frac {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -{\frac {1}{c^{2}}}v_{1}^{2}v_{2}^{2}\sin ^{2}\alpha }{\left(1+{\frac {1}{c^{2}}}v_{1}v_{2}\cos \alpha \right)^{2}}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid none"> <mtr> <mtd> <mrow> <mo fence="true" 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</mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\left.{\begin{array}{lrl}x'=&x\ \cos \varphi +l\ \sin \varphi ,&y'=y\\l'=&-x\ \sin \varphi +l\ \cos \varphi ,&z'=z\end{array}}\right\}\\\left(\operatorname {tg} \varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}\right)\\\hline \beta ={\frac {1}{i}}\operatorname {tg} \left(\varphi _{1}+\varphi _{2}\right)={\frac {1}{i}}{\frac {\operatorname {tg} \varphi _{1}+\operatorname {tg} \varphi _{2}}{1-\operatorname {tg} \varphi _{1}\operatorname {tg} \varphi _{2}}}={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}\\\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}\cos \alpha \\v^{2}={\frac {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -{\frac {1}{c^{2}}}v_{1}^{2}v_{2}^{2}\sin ^{2}\alpha }{\left(1+{\frac {1}{c^{2}}}v_{1}v_{2}\cos \alpha \right)^{2}}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a78eb05c48ca4d302855f920e7793dcf0ce1e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.671ex; width:48.193ex; height:32.509ex;" alt="{\displaystyle {\begin{matrix}\left.{\begin{array}{lrl}x'=&x\ \cos \varphi +l\ \sin \varphi ,&y'=y\\l'=&-x\ \sin \varphi +l\ \cos \varphi ,&z'=z\end{array}}\right\}\\\left(\operatorname {tg} \varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}\right)\\\hline \beta ={\frac {1}{i}}\operatorname {tg} \left(\varphi _{1}+\varphi _{2}\right)={\frac {1}{i}}{\frac {\operatorname {tg} \varphi _{1}+\operatorname {tg} \varphi _{2}}{1-\operatorname {tg} \varphi _{1}\operatorname {tg} \varphi _{2}}}={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}\\\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}\cos \alpha \\v^{2}={\frac {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -{\frac {1}{c^{2}}}v_{1}^{2}v_{2}^{2}\sin ^{2}\alpha }{\left(1+{\frac {1}{c^{2}}}v_{1}v_{2}\cos \alpha \right)^{2}}}\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Frank_(1909)_–_Hyperbolic_functions"> Frank (1909) – Hyperbolic functions</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=15" title="Edit section: Frank (1909) – Hyperbolic functions">edit</a>]</div> Hyperbolic functions were used by <a href="/wiki/Philipp_Frank" title="Philipp Frank">Philipp Frank</a> (1909), who derived the Lorentz transformation using ψ as <a href="/wiki/Rapidity" title="Rapidity">rapidity</a>:<a href="#cite_note-66">[R 42]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x'=x\varphi (a)\,{\rm {ch}}\,\psi +t\varphi (a)\,{\rm {sh}}\,\psi \\t'=-x\varphi (a)\,{\rm {sh}}\,\psi +t\varphi (a)\,{\rm {ch}}\,\psi \\\hline {\rm {th}}\,\psi =-a,\ {\rm {sh}}\,\psi ={\frac {a}{\sqrt {1-a^{2}}}},\ {\rm {ch}}\,\psi ={\frac {1}{\sqrt {1-a^{2}}}},\ \varphi (a)=1\\\hline x'={\frac {x-at}{\sqrt {1-a^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {-ax+t}{\sqrt {1-a^{2}}}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid solid"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>x</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>+</mo> <mi>t</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>+</mo> <mi>t</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">h</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x'=x\varphi (a)\,{\rm {ch}}\,\psi +t\varphi (a)\,{\rm {sh}}\,\psi \\t'=-x\varphi (a)\,{\rm {sh}}\,\psi +t\varphi (a)\,{\rm {ch}}\,\psi \\\hline {\rm {th}}\,\psi =-a,\ {\rm {sh}}\,\psi ={\frac {a}{\sqrt {1-a^{2}}}},\ {\rm {ch}}\,\psi ={\frac {1}{\sqrt {1-a^{2}}}},\ \varphi (a)=1\\\hline x'={\frac {x-at}{\sqrt {1-a^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {-ax+t}{\sqrt {1-a^{2}}}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516dd77809c5f4c226c35e394156119181b7f336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:53.6ex; height:17.509ex;" alt="{\displaystyle {\begin{matrix}x'=x\varphi (a)\,{\rm {ch}}\,\psi +t\varphi (a)\,{\rm {sh}}\,\psi \\t'=-x\varphi (a)\,{\rm {sh}}\,\psi +t\varphi (a)\,{\rm {ch}}\,\psi \\\hline {\rm {th}}\,\psi =-a,\ {\rm {sh}}\,\psi ={\frac {a}{\sqrt {1-a^{2}}}},\ {\rm {ch}}\,\psi ={\frac {1}{\sqrt {1-a^{2}}}},\ \varphi (a)=1\\\hline x'={\frac {x-at}{\sqrt {1-a^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {-ax+t}{\sqrt {1-a^{2}}}}\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bateman_and_Cunningham_(1909–1910)_–_Spherical_wave_transformation"> Bateman and Cunningham (1909–1910) – Spherical wave transformation</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=16" title="Edit section: Bateman and Cunningham (1909–1910) – Spherical wave transformation">edit</a>]</div> In line with <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>'s (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by <a href="/wiki/Harry_Bateman" title="Harry Bateman">Bateman</a> and <a href="/wiki/Ebenezer_Cunningham" title="Ebenezer Cunningham">Cunningham</a> (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d415907036451775e6b010d5060aa065f0beb19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.384ex; height:3.343ex;" alt="{\displaystyle \lambda \left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)}">, but also <a href="/wiki/Maxwells_equations" class="mw-redirect" title="Maxwells equations">Maxwells equations</a> are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called <a href="/wiki/Spherical_wave_transformation" title="Spherical wave transformation">spherical wave transformations</a> by Bateman.<a href="#cite_note-67">[R 43]</a><a href="#cite_note-68">[R 44]</a> However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>.<a href="#cite_note-69">[R 45]</a> In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(1,3). Bateman (1910–12)<a href="#cite_note-70">[25]</a> also alluded to the identity between the <a href="/wiki/Spherical_wave_transformation" title="Spherical wave transformation">Laguerre inversion</a> and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> (1912, 1915–55),<a href="#cite_note-71">[R 46]</a> <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> (1912–21)<a href="#cite_note-72">[R 47]</a> and others. <div class="mw-heading mw-heading3"><h3 id="Herglotz_(1909/10)_–_Möbius_transformation"> Herglotz (1909/10) – Möbius transformation</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=17" title="Edit section: Herglotz (1909/10) – Möbius transformation">edit</a>]</div> Following <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, <a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Gustav Herglotz</a> (1909–10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) and the hyperbolic case equivalent to Lorentz transformations or squeeze mappings are as follows:<a href="#cite_note-73">[R 48]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\Z={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}}={\frac {x+iy}{t-z}},\ Z'={\frac {x'+iy'}{t'-z'}}\\Z={\frac {\alpha Z'+\beta }{\gamma Z'+\delta }}\end{matrix}}\right|{\begin{matrix}Z=Z'e^{\vartheta }\\{\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>z</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> <mrow> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>i</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mi>γ</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>Z</mi> <mo>=</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϑ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> </mtd> <mtd> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϑ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> </mtd> <mtd> <mi>t</mi> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ϑ</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\Z={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}}={\frac {x+iy}{t-z}},\ Z'={\frac {x'+iy'}{t'-z'}}\\Z={\frac {\alpha Z'+\beta }{\gamma Z'+\delta }}\end{matrix}}\right|{\begin{matrix}Z=Z'e^{\vartheta }\\{\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ea9638a18a72a819bbf483690fea998f8fc336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.444ex; margin-bottom: -0.227ex; width:65.207ex; height:16.509ex;" alt="{\displaystyle \left.{\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\Z={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}}={\frac {x+iy}{t-z}},\ Z'={\frac {x'+iy'}{t'-z'}}\\Z={\frac {\alpha Z'+\beta }{\gamma Z'+\delta }}\end{matrix}}\right|{\begin{matrix}Z=Z'e^{\vartheta }\\{\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Varićak_(1910)_–_Hyperbolic_functions"> Varićak (1910) – Hyperbolic functions</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=18" title="Edit section: Varićak (1910) – Hyperbolic functions">edit</a>]</div> Following <a href="#Sommerfeld">Sommerfeld (1909)</a>, hyperbolic functions were used by <a href="/wiki/Vladimir_Vari%C4%87ak" title="Vladimir Varićak">Vladimir Varićak</a> in several papers starting from 1910, who represented the equations of special relativity on the basis of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> in terms of Weierstrass coordinates. For instance, by setting l=ct and v/c=tanh(u) with u as rapidity he wrote the Lorentz transformation:<a href="#cite_note-var1-74">[R 49]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}l'&=-x\operatorname {sh} u+l\operatorname {ch} u,\\x'&=x\operatorname {ch} u-l\operatorname {sh} u,\\y'&=y,\quad z'=z,\\\operatorname {ch} u&={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>l</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mi>sh</mi> <mo>⁡</mo> <mi>u</mi> <mo>+</mo> <mi>l</mi> <mi>ch</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>ch</mi> <mo>⁡</mo> <mi>u</mi> <mo>−</mo> <mi>l</mi> <mi>sh</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>ch</mi> <mo>⁡</mo> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}l'&=-x\operatorname {sh} u+l\operatorname {ch} u,\\x'&=x\operatorname {ch} u-l\operatorname {sh} u,\\y'&=y,\quad z'=z,\\\operatorname {ch} u&={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94a5a1f9ea78a1ac97f2a9e442146362cf3c10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.734ex; margin-bottom: -0.271ex; width:23.951ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}l'&=-x\operatorname {sh} u+l\operatorname {ch} u,\\x'&=x\operatorname {ch} u-l\operatorname {sh} u,\\y'&=y,\quad z'=z,\\\operatorname {ch} u&={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\end{aligned}}}"></dd></dl> and showed the relation of rapidity to the <a href="/wiki/Gudermannian_function" title="Gudermannian function">Gudermannian function</a> and the <a href="/wiki/Angle_of_parallelism" title="Angle of parallelism">angle of parallelism</a>:<a href="#cite_note-var1-74">[R 49]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v}{c}}=\operatorname {th} u=\operatorname {tg} \psi =\sin \operatorname {gd} (u)=\cos \Pi (u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mi>th</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mi>tg</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>gd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v}{c}}=\operatorname {th} u=\operatorname {tg} \psi =\sin \operatorname {gd} (u)=\cos \Pi (u)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbc728756d1cbe3bfb026c9d0c59f1017e124d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.456ex; height:4.676ex;" alt="{\displaystyle {\frac {v}{c}}=\operatorname {th} u=\operatorname {tg} \psi =\sin \operatorname {gd} (u)=\cos \Pi (u)}"></dd></dl> He also related the velocity addition to the <a href="/wiki/Hyperbolic_law_of_cosines" title="Hyperbolic law of cosines">hyperbolic law of cosines</a>:<a href="#cite_note-75">[R 50]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\operatorname {ch} {u}=\operatorname {ch} {u_{1}}\operatorname {c} h{u_{2}}+\operatorname {sh} {u_{1}}\operatorname {sh} {u_{2}}\cos \alpha \\\operatorname {ch} {u_{i}}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} {u_{i}}={\frac {v_{i}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}}\\v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}\ \left(a={\frac {\pi }{2}}\right)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mo>=</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mi mathvariant="normal">c</mi> <mo>⁡</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\operatorname {ch} {u}=\operatorname {ch} {u_{1}}\operatorname {c} h{u_{2}}+\operatorname {sh} {u_{1}}\operatorname {sh} {u_{2}}\cos \alpha \\\operatorname {ch} {u_{i}}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} {u_{i}}={\frac {v_{i}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}}\\v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}\ \left(a={\frac {\pi }{2}}\right)\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe37d2758b200675b376b7866e69363e25679f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:37.419ex; height:15.176ex;" alt="{\displaystyle {\begin{matrix}\operatorname {ch} {u}=\operatorname {ch} {u_{1}}\operatorname {c} h{u_{2}}+\operatorname {sh} {u_{1}}\operatorname {sh} {u_{2}}\cos \alpha \\\operatorname {ch} {u_{i}}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} {u_{i}}={\frac {v_{i}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}}\\v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}\ \left(a={\frac {\pi }{2}}\right)\end{matrix}}}"></dd></dl> Subsequently, other authors such as <a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a> (1910) or <a href="/wiki/Alfred_Robb" title="Alfred Robb">Alfred Robb</a> (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks. <div class="mw-heading mw-heading3"><h3 id="Plummer_(1910)_–_Trigonometric_Lorentz_boosts"> Plummer (1910) – Trigonometric Lorentz boosts</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=19" title="Edit section: Plummer (1910) – Trigonometric Lorentz boosts">edit</a>]</div> <a href="/wiki/Henry_Crozier_Keating_Plummer" title="Henry Crozier Keating Plummer">w:Henry Crozier Keating Plummer</a> (1910) defined the Lorentz boost in terms of trigonometric functions<a href="#cite_note-76">[R 51]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\tau =t\sec \beta -x\tan \beta /U\\\xi =x\sec \beta -Ut\tan \beta \\\eta =y,\ \zeta =z,\\\hline \sin \beta =v/U\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none none solid"> <mtr> <mtd> <mi>τ</mi> <mo>=</mo> <mi>t</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mo>−</mo> <mi>x</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>U</mi> </mtd> </mtr> <mtr> <mtd> <mi>ξ</mi> <mo>=</mo> <mi>x</mi> <mi>sec</mi> <mo>⁡</mo> <mi>β</mi> <mo>−</mo> <mi>U</mi> <mi>t</mi> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>η</mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mtext> </mtext> <mi>ζ</mi> <mo>=</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>U</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\tau =t\sec \beta -x\tan \beta /U\\\xi =x\sec \beta -Ut\tan \beta \\\eta =y,\ \zeta =z,\\\hline \sin \beta =v/U\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9db28df209a5c085e94d20b3ac8f8b7dd3d599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.418ex; height:13.843ex;" alt="{\displaystyle {\begin{matrix}\tau =t\sec \beta -x\tan \beta /U\\\xi =x\sec \beta -Ut\tan \beta \\\eta =y,\ \zeta =z,\\\hline \sin \beta =v/U\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ignatowski_(1910)"> Ignatowski (1910)</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=20" title="Edit section: Ignatowski (1910)">edit</a>]</div> While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, <a href="/wiki/Vladimir_Ignatowski" title="Vladimir Ignatowski">Vladimir Ignatowski</a> (1910) showed that it is possible to use the principle of relativity (and related <a href="/wiki/Group_theory" title="Group theory">group theoretical</a> principles) alone, in order to derive the following transformation between two inertial frames:<a href="#cite_note-77">[R 52]</a><a href="#cite_note-78">[R 53]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}dx'&=p\ dx-pq\ dt\\dt'&=-pqn\ dx+p\ dt\\p&={\frac {1}{\sqrt {1-q^{2}n}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mi>n</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>p</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}dx'&=p\ dx-pq\ dt\\dt'&=-pqn\ dx+p\ dt\\p&={\frac {1}{\sqrt {1-q^{2}n}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8314622037b919fbc0dbfbc2bf33a4d668daa836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:22.294ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}dx'&=p\ dx-pq\ dt\\dt'&=-pqn\ dx+p\ dt\\p&={\frac {1}{\sqrt {1-q^{2}n}}}\end{aligned}}}"></dd></dl> The variable n can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by x/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when n=1/c2, resulting in p=γ and the Lorentz transformation. With n=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by <a href="/wiki/Philipp_Frank" title="Philipp Frank">Philipp Frank</a> and <a href="/wiki/Hermann_Rothe" title="Hermann Rothe">Hermann Rothe</a> (1911, 1912),<a href="#cite_note-79">[R 54]</a> with various authors developing similar methods in subsequent years.<a href="#cite_note-baccetti-80">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Noether_(1910),_Klein_(1910)_–_Quaternions"> Noether (1910), Klein (1910) – Quaternions</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=21" title="Edit section: Noether (1910), Klein (1910) – Quaternions">edit</a>]</div> <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.<a href="#cite_note-81">[R 55]</a> In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), <a href="/wiki/Fritz_Noether" title="Fritz Noether">Fritz Noether</a> showed how to formulate hyperbolic rotations using biquaternions with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4883f48f6d43ff56da45d4f697ff7339bb424934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.451ex; height:3.009ex;" alt="{\displaystyle \omega ={\sqrt {-1}}}">, which he also related to the speed of light by setting ω2=-c2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:<a href="#cite_note-82">[R 56]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}V={\frac {Q_{1}vQ_{2}}{T_{1}T_{2}}}\\\hline X^{2}+Y^{2}+Z^{2}+\omega ^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega ^{2}s^{2}\\\hline {\begin{aligned}V&=Xi+Yj+Zk+\omega S\\v&=xi+yj+zk+\omega s\\Q_{1}&=(+Ai+Bj+Ck+D)+\omega (A'i+B'j+C'k+D')\\Q_{2}&=(-Ai-Bj-Ck+D)+\omega (A'i+B'j+C'k-D')\\T_{1}T_{2}&=T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega ^{2}\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid solid"> <mtr> <mtd> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>v</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>X</mi> <mi>i</mi> <mo>+</mo> <mi>Y</mi> <mi>j</mi> <mo>+</mo> <mi>Z</mi> <mi>k</mi> <mo>+</mo> <mi>ω</mi> <mi>S</mi> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>i</mi> <mo>+</mo> <mi>y</mi> <mi>j</mi> <mo>+</mo> <mi>z</mi> <mi>k</mi> <mo>+</mo> <mi>ω</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>+</mo> <mi>A</mi> <mi>i</mi> <mo>+</mo> <mi>B</mi> <mi>j</mi> <mo>+</mo> <mi>C</mi> <mi>k</mi> <mo>+</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>i</mi> <mo>+</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mi>j</mi> <mo>+</mo> <msup> <mi>C</mi> <mo>′</mo> </msup> <mi>k</mi> <mo>+</mo> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <mi>i</mi> <mo>−</mo> <mi>B</mi> <mi>j</mi> <mo>−</mo> <mi>C</mi> <mi>k</mi> <mo>+</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>i</mi> <mo>+</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mi>j</mi> <mo>+</mo> <msup> <mi>C</mi> <mo>′</mo> </msup> <mi>k</mi> <mo>−</mo> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}V={\frac {Q_{1}vQ_{2}}{T_{1}T_{2}}}\\\hline X^{2}+Y^{2}+Z^{2}+\omega ^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega ^{2}s^{2}\\\hline {\begin{aligned}V&=Xi+Yj+Zk+\omega S\\v&=xi+yj+zk+\omega s\\Q_{1}&=(+Ai+Bj+Ck+D)+\omega (A'i+B'j+C'k+D')\\Q_{2}&=(-Ai-Bj-Ck+D)+\omega (A'i+B'j+C'k-D')\\T_{1}T_{2}&=T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega ^{2}\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22bab7db4a02745ef02b54889f0f6186339ddf47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:72.63ex; height:24.843ex;" alt="{\displaystyle {\begin{matrix}V={\frac {Q_{1}vQ_{2}}{T_{1}T_{2}}}\\\hline X^{2}+Y^{2}+Z^{2}+\omega ^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega ^{2}s^{2}\\\hline {\begin{aligned}V&=Xi+Yj+Zk+\omega S\\v&=xi+yj+zk+\omega s\\Q_{1}&=(+Ai+Bj+Ck+D)+\omega (A'i+B'j+C'k+D')\\Q_{2}&=(-Ai-Bj-Ck+D)+\omega (A'i+B'j+C'k-D')\\T_{1}T_{2}&=T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega ^{2}\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\end{aligned}}\end{matrix}}}"></dd></dl> Besides citing quaternion related standard works by <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> (1854), Noether referred to the entries in Klein's encyclopedia by <a href="/wiki/Eduard_Study" title="Eduard Study">Eduard Study</a> (1899) and the French version by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> (1908).<a href="#cite_note-83">[27]</a> Cartan's version contains a description of Study's <a href="/wiki/Dual_number" title="Dual number">dual numbers</a>, Clifford's biquaternions (including the choice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4883f48f6d43ff56da45d4f697ff7339bb424934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.451ex; height:3.009ex;" alt="{\displaystyle \omega ={\sqrt {-1}}}"> for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884–85), Vahlen (1901–02) and others. Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:<a href="#cite_note-84">[R 57]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{aligned}&\left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\&\quad -\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)\end{aligned}}={\frac {\left[{\begin{aligned}&\left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\&\quad \cdot \left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\&\quad \quad \cdot \left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)\end{aligned}}\right]}{\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}}\\\hline {\text{where}}\\AA'+BB'+CC'+DD'=0\\A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid none"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>i</mi> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mi>c</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>i</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>i</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo>+</mo> <mi>i</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo>+</mo> <mi>i</mi> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <mi>i</mi> <mi>c</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mspace width="1em" /> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>C</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>D</mi> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{aligned}&\left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\&\quad -\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)\end{aligned}}={\frac {\left[{\begin{aligned}&\left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\&\quad \cdot \left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\&\quad \quad \cdot \left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)\end{aligned}}\right]}{\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}}\\\hline {\text{where}}\\AA'+BB'+CC'+DD'=0\\A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92cea3d609ab839d9ee987b8da876b911afb25af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:81.189ex; height:22.176ex;" alt="{\displaystyle {\begin{matrix}{\begin{aligned}&\left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\&\quad -\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)\end{aligned}}={\frac {\left[{\begin{aligned}&\left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\&\quad \cdot \left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\&\quad \quad \cdot \left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)\end{aligned}}\right]}{\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}}\\\hline {\text{where}}\\AA'+BB'+CC'+DD'=0\\A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{matrix}}}"></dd></dl> or in March 1911<a href="#cite_note-85">[R 58]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}g'={\frac {pg\pi }{M}}\\\hline {\begin{aligned}g&={\sqrt {-1}}ct+ix+jy+kz\\g'&={\sqrt {-1}}ct'+ix'+jy'+kz'\\p&=(D+{\sqrt {-1}}D')+i(A+{\sqrt {-1}}A')+j(B+{\sqrt {-1}}B')+k(C+{\sqrt {-1}}C')\\\pi &=(D-{\sqrt {-1}}D')-i(A-{\sqrt {-1}}A')-j(B-{\sqrt {-1}}B')-k(C-{\sqrt {-1}}C')\\M&=\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\\&AA'+BB'+CC'+DD'=0\\&A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mi>g</mi> <mi>π</mi> </mrow> <mi>M</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mi>c</mi> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mi>x</mi> <mo>+</mo> <mi>j</mi> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>i</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>j</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>k</mi> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>A</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>C</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>D</mi> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}g'={\frac {pg\pi }{M}}\\\hline {\begin{aligned}g&={\sqrt {-1}}ct+ix+jy+kz\\g'&={\sqrt {-1}}ct'+ix'+jy'+kz'\\p&=(D+{\sqrt {-1}}D')+i(A+{\sqrt {-1}}A')+j(B+{\sqrt {-1}}B')+k(C+{\sqrt {-1}}C')\\\pi &=(D-{\sqrt {-1}}D')-i(A-{\sqrt {-1}}A')-j(B-{\sqrt {-1}}B')-k(C-{\sqrt {-1}}C')\\M&=\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\\&AA'+BB'+CC'+DD'=0\\&A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9006215fc75d2d66b67b1c8b1de581aad6ba7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.171ex; width:75.785ex; height:27.509ex;" alt="{\displaystyle {\begin{matrix}g'={\frac {pg\pi }{M}}\\\hline {\begin{aligned}g&={\sqrt {-1}}ct+ix+jy+kz\\g'&={\sqrt {-1}}ct'+ix'+jy'+kz'\\p&=(D+{\sqrt {-1}}D')+i(A+{\sqrt {-1}}A')+j(B+{\sqrt {-1}}B')+k(C+{\sqrt {-1}}C')\\\pi &=(D-{\sqrt {-1}}D')-i(A-{\sqrt {-1}}A')-j(B-{\sqrt {-1}}B')-k(C-{\sqrt {-1}}C')\\M&=\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\\&AA'+BB'+CC'+DD'=0\\&A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Conway_(1911),_Silberstein_(1911)_–_Quaternions"> Conway (1911), Silberstein (1911) – Quaternions</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=22" title="Edit section: Conway (1911), Silberstein (1911) – Quaternions">edit</a>]</div> <a href="/wiki/Arthur_W._Conway" title="Arthur W. Conway">Arthur W. Conway</a> in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:<a href="#cite_note-86">[R 59]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{aligned}{\mathtt {D}}&=\mathbf {a} ^{-1}{\mathtt {D}}'\mathbf {a} ^{-1}\\{\mathtt {\sigma }}&=\mathbf {a} {\mathtt {\sigma }}'\mathbf {a} ^{-1}\end{aligned}}\\e=\mathbf {a} ^{-1}e'\mathbf {a} ^{-1}\\\hline a=\left(1-hc^{-1}\lambda \right)^{\frac {1}{2}}\left(1+c^{-2}\lambda ^{2}\right)^{-{\frac {1}{4}}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="monospace">D</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="monospace">D</mi> </mrow> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>e</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>h</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{aligned}{\mathtt {D}}&=\mathbf {a} ^{-1}{\mathtt {D}}'\mathbf {a} ^{-1}\\{\mathtt {\sigma }}&=\mathbf {a} {\mathtt {\sigma }}'\mathbf {a} ^{-1}\end{aligned}}\\e=\mathbf {a} ^{-1}e'\mathbf {a} ^{-1}\\\hline a=\left(1-hc^{-1}\lambda \right)^{\frac {1}{2}}\left(1+c^{-2}\lambda ^{2}\right)^{-{\frac {1}{4}}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557f926f4c05c62527600e01198596fb71a9f38f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:35.736ex; height:15.509ex;" alt="{\displaystyle {\begin{matrix}{\begin{aligned}{\mathtt {D}}&=\mathbf {a} ^{-1}{\mathtt {D}}'\mathbf {a} ^{-1}\\{\mathtt {\sigma }}&=\mathbf {a} {\mathtt {\sigma }}'\mathbf {a} ^{-1}\end{aligned}}\\e=\mathbf {a} ^{-1}e'\mathbf {a} ^{-1}\\\hline a=\left(1-hc^{-1}\lambda \right)^{\frac {1}{2}}\left(1+c^{-2}\lambda ^{2}\right)^{-{\frac {1}{4}}}\end{matrix}}}"></dd></dl> Also <a href="/wiki/Ludwik_Silberstein" title="Ludwik Silberstein">Ludwik Silberstein</a> in November 1911<a href="#cite_note-87">[R 60]</a> as well as in 1914,<a href="#cite_note-88">[28]</a> formulated the Lorentz transformation in terms of velocity v: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}q'=QqQ\\\hline {\begin{aligned}q&=\mathbf {r} +l=xi+yj+zk+\iota ct\\q&'=\mathbf {r} '+l'=x'i+y'j+z'k+\iota ct'\\Q&={\frac {1}{\sqrt {2}}}\left({\sqrt {1+\gamma }}+\mathrm {u} {\sqrt {1-\gamma }}\right)\\&=\cos \alpha +\mathrm {u} \sin \alpha =e^{\alpha \mathrm {u} }\\&\left\{\gamma =\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha =\operatorname {arctg} \ \left(\iota {\frac {v}{c}}\right)\right\}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>Q</mi> <mi>q</mi> <mi>Q</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mi>l</mi> <mo>=</mo> <mi>x</mi> <mi>i</mi> <mo>+</mo> <mi>y</mi> <mi>j</mi> <mo>+</mo> <mi>z</mi> <mi>k</mi> <mo>+</mo> <mi>ι</mi> <mi>c</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <msup> <mi></mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>i</mi> <mo>+</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mi>j</mi> <mo>+</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mi>k</mi> <mo>+</mo> <mi>ι</mi> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>{</mo> <mrow> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mi>α</mi> <mo>=</mo> <mi>arctg</mi> <mo>⁡</mo> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>ι</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}q'=QqQ\\\hline {\begin{aligned}q&=\mathbf {r} +l=xi+yj+zk+\iota ct\\q&'=\mathbf {r} '+l'=x'i+y'j+z'k+\iota ct'\\Q&={\frac {1}{\sqrt {2}}}\left({\sqrt {1+\gamma }}+\mathrm {u} {\sqrt {1-\gamma }}\right)\\&=\cos \alpha +\mathrm {u} \sin \alpha =e^{\alpha \mathrm {u} }\\&\left\{\gamma =\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha =\operatorname {arctg} \ \left(\iota {\frac {v}{c}}\right)\right\}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca64b0c599e38e7c493601846b337a2cf5366be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:47.63ex; height:24.509ex;" alt="{\displaystyle {\begin{matrix}q'=QqQ\\\hline {\begin{aligned}q&=\mathbf {r} +l=xi+yj+zk+\iota ct\\q&'=\mathbf {r} '+l'=x'i+y'j+z'k+\iota ct'\\Q&={\frac {1}{\sqrt {2}}}\left({\sqrt {1+\gamma }}+\mathrm {u} {\sqrt {1-\gamma }}\right)\\&=\cos \alpha +\mathrm {u} \sin \alpha =e^{\alpha \mathrm {u} }\\&\left\{\gamma =\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha =\operatorname {arctg} \ \left(\iota {\frac {v}{c}}\right)\right\}\end{aligned}}\end{matrix}}}"></dd></dl> Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book. <div class="mw-heading mw-heading3"><h3 id="Ignatowski_(1910/11),_Herglotz_(1911),_and_others_–_Vector_transformation"> Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=23" title="Edit section: Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Lorentz_transformation#Vector_transformations" title="Lorentz transformation">Lorentz transformation § Vector transformations</a></div> <a href="/wiki/Vladimir_Ignatowski" title="Vladimir Ignatowski">Vladimir Ignatowski</a> (1910, published 1911) showed how to reformulate the Lorentz transformation in order to allow for arbitrary velocities and coordinates:<a href="#cite_note-89">[R 61]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{matrix}{\mathfrak {v}}={\frac {{\mathfrak {v}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {v}}'+pq{\mathfrak {c}}_{0}}{p\left(1+nq{\mathfrak {c}}_{0}{\mathfrak {v}}'\right)}}&\left|{\begin{aligned}{\mathfrak {A}}'&={\mathfrak {A}}+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}-pqb{\mathfrak {c}}_{0}\\b'&=pb-pqn{\mathfrak {A}}{\mathfrak {c}}_{0}\\\\{\mathfrak {A}}&={\mathfrak {A}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}'+pqb'{\mathfrak {c}}_{0}\\b&=pb'+pqn{\mathfrak {A}}'{\mathfrak {c}}_{0}\end{aligned}}\right.\end{matrix}}\\\left[{\mathfrak {v}}=\mathbf {u} ,\ {\mathfrak {A}}=\mathbf {x} ,\ b=t,\ {\mathfrak {c}}_{0}={\frac {\mathbf {v} }{v}},\ p=\gamma ,\ n={\frac {1}{c^{2}}}\right]\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">v</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">v</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">v</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mi>p</mi> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">v</mi> </mrow> </mrow> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mi>b</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mi>b</mi> <mo>−</mo> <mi>p</mi> <mi>q</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mi>p</mi> <mi>q</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>p</mi> <mi>q</mi> <mi>n</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">v</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>p</mi> <mo>=</mo> <mi>γ</mi> <mo>,</mo> <mtext> </mtext> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{matrix}{\mathfrak {v}}={\frac {{\mathfrak {v}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {v}}'+pq{\mathfrak {c}}_{0}}{p\left(1+nq{\mathfrak {c}}_{0}{\mathfrak {v}}'\right)}}&\left|{\begin{aligned}{\mathfrak {A}}'&={\mathfrak {A}}+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}-pqb{\mathfrak {c}}_{0}\\b'&=pb-pqn{\mathfrak {A}}{\mathfrak {c}}_{0}\\\\{\mathfrak {A}}&={\mathfrak {A}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}'+pqb'{\mathfrak {c}}_{0}\\b&=pb'+pqn{\mathfrak {A}}'{\mathfrak {c}}_{0}\end{aligned}}\right.\end{matrix}}\\\left[{\mathfrak {v}}=\mathbf {u} ,\ {\mathfrak {A}}=\mathbf {x} ,\ b=t,\ {\mathfrak {c}}_{0}={\frac {\mathbf {v} }{v}},\ p=\gamma ,\ n={\frac {1}{c^{2}}}\right]\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bff07b58a3ef41af83b9007edbca88b912f2d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:60.612ex; height:20.509ex;" alt="{\displaystyle {\begin{matrix}{\begin{matrix}{\mathfrak {v}}={\frac {{\mathfrak {v}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {v}}'+pq{\mathfrak {c}}_{0}}{p\left(1+nq{\mathfrak {c}}_{0}{\mathfrak {v}}'\right)}}&\left|{\begin{aligned}{\mathfrak {A}}'&={\mathfrak {A}}+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}-pqb{\mathfrak {c}}_{0}\\b'&=pb-pqn{\mathfrak {A}}{\mathfrak {c}}_{0}\\\\{\mathfrak {A}}&={\mathfrak {A}}'+(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {A}}'+pqb'{\mathfrak {c}}_{0}\\b&=pb'+pqn{\mathfrak {A}}'{\mathfrak {c}}_{0}\end{aligned}}\right.\end{matrix}}\\\left[{\mathfrak {v}}=\mathbf {u} ,\ {\mathfrak {A}}=\mathbf {x} ,\ b=t,\ {\mathfrak {c}}_{0}={\frac {\mathbf {v} }{v}},\ p=\gamma ,\ n={\frac {1}{c^{2}}}\right]\end{matrix}}}"></dd></dl> <a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Gustav Herglotz</a> (1911)<a href="#cite_note-90">[R 62]</a> also showed how to formulate the transformation in order to allow for arbitrary velocities and coordinates v=(vx, vy, vz) and r=(x, y, z): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{0}&=x+\alpha u(ux+vy+wz)-\beta ut\\y^{0}&=y+\alpha v(ux+vy+wz)-\beta vt\\z^{0}&=z+\alpha w(ux+vy+wz)-\beta wt\\t^{0}&=-\beta (ux+vy+wz)+\beta t\\&\alpha ={\frac {1}{{\sqrt {1-s^{2}}}\left(1+{\sqrt {1-s^{2}}}\right)}},\ \beta ={\frac {1}{\sqrt {1-s^{2}}}}\end{aligned}}\right|&{\begin{aligned}x'&=x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\y'&=y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\z'&=z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\t'&=-\gamma \left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\&\alpha ={\frac {\gamma ^{2}}{\gamma +1}},\ \gamma ={\frac {1}{\sqrt {1-v^{2}}}}\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>original</mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>modern</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>α</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>β</mi> <mi>u</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>α</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>β</mi> <mi>v</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mi>α</mi> <mi>w</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>β</mi> <mi>w</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>α</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>γ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>α</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>γ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mi>α</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>γ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>γ</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>γ</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{0}&=x+\alpha u(ux+vy+wz)-\beta ut\\y^{0}&=y+\alpha v(ux+vy+wz)-\beta vt\\z^{0}&=z+\alpha w(ux+vy+wz)-\beta wt\\t^{0}&=-\beta (ux+vy+wz)+\beta t\\&\alpha ={\frac {1}{{\sqrt {1-s^{2}}}\left(1+{\sqrt {1-s^{2}}}\right)}},\ \beta ={\frac {1}{\sqrt {1-s^{2}}}}\end{aligned}}\right|&{\begin{aligned}x'&=x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\y'&=y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\z'&=z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\t'&=-\gamma \left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\&\alpha ={\frac {\gamma ^{2}}{\gamma +1}},\ \gamma ={\frac {1}{\sqrt {1-v^{2}}}}\end{aligned}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67229a3bbd3e7e5748fed75b552380205fedd21e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:92.719ex; height:25.509ex;" alt="{\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{0}&=x+\alpha u(ux+vy+wz)-\beta ut\\y^{0}&=y+\alpha v(ux+vy+wz)-\beta vt\\z^{0}&=z+\alpha w(ux+vy+wz)-\beta wt\\t^{0}&=-\beta (ux+vy+wz)+\beta t\\&\alpha ={\frac {1}{{\sqrt {1-s^{2}}}\left(1+{\sqrt {1-s^{2}}}\right)}},\ \beta ={\frac {1}{\sqrt {1-s^{2}}}}\end{aligned}}\right|&{\begin{aligned}x'&=x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\y'&=y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\z'&=z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\t'&=-\gamma \left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\&\alpha ={\frac {\gamma ^{2}}{\gamma +1}},\ \gamma ={\frac {1}{\sqrt {1-v^{2}}}}\end{aligned}}\end{matrix}}}"></dd></dl> This was simplified using vector notation by <a href="/wiki/Ludwik_Silberstein" title="Ludwik Silberstein">Ludwik Silberstein</a> (1911 on the left, 1914 on the right):<a href="#cite_note-91">[R 63]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {ru} )\mathbf {u} +i\beta \gamma lu\\l'&=\gamma \left[l-i\beta (\mathbf {ru} )\right]\end{aligned}}&{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\left[{\frac {\gamma -1}{v^{2}}}(\mathbf {vr} )-\gamma t\right]\mathbf {v} \\t'&=\gamma \left[t-{\frac {1}{c^{2}}}(\mathbf {vr} )\right]\end{aligned}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em" columnlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mi>i</mi> <mi>β</mi> <mi>γ</mi> <mi>l</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>l</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>[</mo> <mrow> <mi>l</mi> <mo>−</mo> <mi>i</mi> <mi>β</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>γ</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>[</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {ru} )\mathbf {u} +i\beta \gamma lu\\l'&=\gamma \left[l-i\beta (\mathbf {ru} )\right]\end{aligned}}&{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\left[{\frac {\gamma -1}{v^{2}}}(\mathbf {vr} )-\gamma t\right]\mathbf {v} \\t'&=\gamma \left[t-{\frac {1}{c^{2}}}(\mathbf {vr} )\right]\end{aligned}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490144e75ad5c53bbf948b49a3d834c3a8b62f0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:64.926ex; height:13.509ex;" alt="{\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {ru} )\mathbf {u} +i\beta \gamma lu\\l'&=\gamma \left[l-i\beta (\mathbf {ru} )\right]\end{aligned}}&{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\left[{\frac {\gamma -1}{v^{2}}}(\mathbf {vr} )-\gamma t\right]\mathbf {v} \\t'&=\gamma \left[t-{\frac {1}{c^{2}}}(\mathbf {vr} )\right]\end{aligned}}\end{array}}}"></dd></dl> Equivalent formulas were also given by <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> (1921),<a href="#cite_note-92">[29]</a> with <a href="/wiki/Erwin_Madelung" title="Erwin Madelung">Erwin Madelung</a> (1922) providing the matrix form<a href="#cite_note-93">[30]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c|c|c|c|c}&x&y&z&t\\\hline x'&1-{\frac {v_{x}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{x}}{\sqrt {1-\beta ^{2}}}}\\y'&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{y}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{y}}{\sqrt {1-\beta ^{2}}}}\\z'&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{z}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{z}}{\sqrt {1-\beta ^{2}}}}\\t'&{\frac {-v_{x}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{y}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{z}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid solid solid"> <mtr> <mtd /> <mtd> <mi>x</mi> </mtd> <mtd> <mi>y</mi> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|c|c|c|c}&x&y&z&t\\\hline x'&1-{\frac {v_{x}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{x}}{\sqrt {1-\beta ^{2}}}}\\y'&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{y}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{y}}{\sqrt {1-\beta ^{2}}}}\\z'&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{z}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{z}}{\sqrt {1-\beta ^{2}}}}\\t'&{\frac {-v_{x}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{y}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{z}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/862b9c40bc19e77249623828536d128eb676f465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.338ex; width:83.055ex; height:33.843ex;" alt="{\displaystyle {\begin{array}{c|c|c|c|c}&x&y&z&t\\\hline x'&1-{\frac {v_{x}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{x}}{\sqrt {1-\beta ^{2}}}}\\y'&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{y}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{y}}{\sqrt {1-\beta ^{2}}}}\\z'&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{z}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{z}}{\sqrt {1-\beta ^{2}}}}\\t'&{\frac {-v_{x}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{y}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{z}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}\end{array}}}"></dd></dl> These formulas were called "general Lorentz transformation without rotation" by <a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Christian Møller</a> (1952),<a href="#cite_note-94">[31]</a> who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a <a href="/wiki/Three-dimensional_rotation_operator" class="mw-redirect" title="Three-dimensional rotation operator">rotation operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c2461a0bd159fa416eeb2bd7a4ac0fed0262ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.934ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {D}}}">. In this case, v′=(v′x, v′y, v′z) is not equal to -v=(-vx, -vy, -vz), but the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} '=-{\mathfrak {D}}\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} '=-{\mathfrak {D}}\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb79e89dc87b9a1e59cfceb103d1e402a25ac08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.347ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} '=-{\mathfrak {D}}\mathbf {v} }"> holds instead, with the result <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}{\begin{aligned}\mathbf {x} '&={\mathfrak {D}}^{-1}\mathbf {x} -\mathbf {v} '\left\{\left(\gamma -1\right)(\mathbf {x\cdot v} )/v^{2}-\gamma t\right\}\\t'&=\gamma \left(t-(\mathbf {v} \cdot \mathbf {x} )/c^{2}\right)\end{aligned}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo>⋅</mo> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>γ</mi> <mi>t</mi> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{\begin{aligned}\mathbf {x} '&={\mathfrak {D}}^{-1}\mathbf {x} -\mathbf {v} '\left\{\left(\gamma -1\right)(\mathbf {x\cdot v} )/v^{2}-\gamma t\right\}\\t'&=\gamma \left(t-(\mathbf {v} \cdot \mathbf {x} )/c^{2}\right)\end{aligned}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06235cd47920f955b21cc0eb7c578b1de6c55caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.467ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{c}{\begin{aligned}\mathbf {x} '&={\mathfrak {D}}^{-1}\mathbf {x} -\mathbf {v} '\left\{\left(\gamma -1\right)(\mathbf {x\cdot v} )/v^{2}-\gamma t\right\}\\t'&=\gamma \left(t-(\mathbf {v} \cdot \mathbf {x} )/c^{2}\right)\end{aligned}}\end{array}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Borel_(1913–14)_–_Cayley–Hermite_parameter"> Borel (1913–14) – Cayley–Hermite parameter</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=24" title="Edit section: Borel (1913–14) – Cayley–Hermite parameter">edit</a>]</div> <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a> (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:<a href="#cite_note-95">[R 64]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\\hline {\scriptstyle {\begin{aligned}\delta a&=\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta b&=2(\lambda \mu +\nu \rho ),&\delta c&=-2(\lambda \nu +\mu \rho ),\\\delta a'&=2(\lambda \mu -\nu \rho ),&\delta b'&=-\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta c'&=2(\lambda \rho -\mu \nu ),\\\delta a''&=2(\lambda \nu -\mu \rho ),&\delta b''&=2(\lambda \rho +\mu \nu ),&\delta c''&=-\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}+\rho ^{2}\right),\end{aligned}}}\\\left(\delta =\lambda ^{2}+\mu ^{2}-\rho ^{2}-\nu ^{2}\right)\\\lambda =\nu =0\rightarrow {\text{Hyperbolic rotation}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid none"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>δ</mi> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>μ</mi> <mo>+</mo> <mi>ν</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <mi>c</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>ν</mi> <mo>+</mo> <mi>μ</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>δ</mi> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>μ</mi> <mo>−</mo> <mi>ν</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <msup> <mi>c</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>ρ</mi> <mo>−</mo> <mi>μ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>δ</mi> <msup> <mi>a</mi> <mo>″</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>ν</mi> <mo>−</mo> <mi>μ</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <msup> <mi>b</mi> <mo>″</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>ρ</mi> <mo>+</mo> <mi>μ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>δ</mi> <msup> <mi>c</mi> <mo>″</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>λ</mi> <mo>=</mo> <mi>ν</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hyperbolic rotation</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\\hline {\scriptstyle {\begin{aligned}\delta a&=\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta b&=2(\lambda \mu +\nu \rho ),&\delta c&=-2(\lambda \nu +\mu \rho ),\\\delta a'&=2(\lambda \mu -\nu \rho ),&\delta b'&=-\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta c'&=2(\lambda \rho -\mu \nu ),\\\delta a''&=2(\lambda \nu -\mu \rho ),&\delta b''&=2(\lambda \rho +\mu \nu ),&\delta c''&=-\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}+\rho ^{2}\right),\end{aligned}}}\\\left(\delta =\lambda ^{2}+\mu ^{2}-\rho ^{2}-\nu ^{2}\right)\\\lambda =\nu =0\rightarrow {\text{Hyperbolic rotation}}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a8bc79f1f94a322c95d5e571461aa304f1665d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:62.138ex; height:20.176ex;" alt="{\displaystyle {\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\\hline {\scriptstyle {\begin{aligned}\delta a&=\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta b&=2(\lambda \mu +\nu \rho ),&\delta c&=-2(\lambda \nu +\mu \rho ),\\\delta a'&=2(\lambda \mu -\nu \rho ),&\delta b'&=-\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta c'&=2(\lambda \rho -\mu \nu ),\\\delta a''&=2(\lambda \nu -\mu \rho ),&\delta b''&=2(\lambda \rho +\mu \nu ),&\delta c''&=-\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}+\rho ^{2}\right),\end{aligned}}}\\\left(\delta =\lambda ^{2}+\mu ^{2}-\rho ^{2}-\nu ^{2}\right)\\\lambda =\nu =0\rightarrow {\text{Hyperbolic rotation}}\end{matrix}}}"></dd></dl> In four dimensions:<a href="#cite_note-96">[R 65]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\\hline {\scriptstyle {\begin{aligned}&\left(\mu ^{2}+\nu ^{2}-\alpha ^{2}\right)\cos \varphi +\left(\lambda ^{2}-\beta ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }&&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi -\gamma \operatorname {sh} {\theta }\\&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi +\gamma \operatorname {sh} {\theta }&&\left(\mu ^{2}+\nu ^{2}-\beta ^{2}\right)\cos \varphi +\left(\mu ^{2}-\alpha ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }\\&-(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi -\beta \operatorname {sh} {\theta }&&-(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\lambda \sin \varphi +\alpha \operatorname {sh} {\theta }\\&(\gamma \mu -\beta \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }&&-(\alpha \nu -\lambda \gamma )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\\\&\quad -(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi +\beta \operatorname {sh} {\theta }&&\quad (\beta \nu -\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }\\&\quad -(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })-\lambda \sin \varphi -\alpha \operatorname {sh} {\theta }&&\quad (\lambda \gamma -\alpha \nu )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\&\quad \left(\lambda ^{2}+\mu ^{2}-\gamma ^{2}\right)\cos \varphi +\left(\nu ^{2}-\alpha ^{2}-\beta ^{2}\right)\operatorname {ch} {\theta }&&\quad (\alpha \mu -\beta \lambda )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }\\&\quad (\beta \gamma -\alpha \mu )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }&&\quad -\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\cos \varphi +\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}\right)\operatorname {ch} {\theta }\end{aligned}}}\\\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}-\lambda ^{2}-\mu ^{2}-\nu ^{2}=-1\right)\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid none"> <mtr> <mtd> <mi>F</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>β</mi> <mo>+</mo> <mi>λ</mi> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>γ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>β</mi> <mo>+</mo> <mi>λ</mi> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>γ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>γ</mi> <mo>+</mo> <mi>λ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>μ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>β</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>μ</mi> <mo>+</mo> <mi>μ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>α</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>λ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>ν</mi> <mo>−</mo> <mi>λ</mi> <mi>γ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>μ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>γ</mi> <mo>+</mo> <mi>λ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>μ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>β</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>β</mi> <mi>ν</mi> <mo>−</mo> <mi>μ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>λ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>−</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>μ</mi> <mo>+</mo> <mi>μ</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>λ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>α</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>γ</mi> <mo>−</mo> <mi>α</mi> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>μ</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>α</mi> <mi>μ</mi> <mo>−</mo> <mi>β</mi> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ν</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>β</mi> <mi>γ</mi> <mo>−</mo> <mi>α</mi> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>ν</mi> <mi>sh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ch</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\\hline {\scriptstyle {\begin{aligned}&\left(\mu ^{2}+\nu ^{2}-\alpha ^{2}\right)\cos \varphi +\left(\lambda ^{2}-\beta ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }&&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi -\gamma \operatorname {sh} {\theta }\\&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi +\gamma \operatorname {sh} {\theta }&&\left(\mu ^{2}+\nu ^{2}-\beta ^{2}\right)\cos \varphi +\left(\mu ^{2}-\alpha ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }\\&-(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi -\beta \operatorname {sh} {\theta }&&-(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\lambda \sin \varphi +\alpha \operatorname {sh} {\theta }\\&(\gamma \mu -\beta \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }&&-(\alpha \nu -\lambda \gamma )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\\\&\quad -(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi +\beta \operatorname {sh} {\theta }&&\quad (\beta \nu -\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }\\&\quad -(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })-\lambda \sin \varphi -\alpha \operatorname {sh} {\theta }&&\quad (\lambda \gamma -\alpha \nu )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\&\quad \left(\lambda ^{2}+\mu ^{2}-\gamma ^{2}\right)\cos \varphi +\left(\nu ^{2}-\alpha ^{2}-\beta ^{2}\right)\operatorname {ch} {\theta }&&\quad (\alpha \mu -\beta \lambda )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }\\&\quad (\beta \gamma -\alpha \mu )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }&&\quad -\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\cos \varphi +\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}\right)\operatorname {ch} {\theta }\end{aligned}}}\\\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}-\lambda ^{2}-\mu ^{2}-\nu ^{2}=-1\right)\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150b142b01117cc72b540148d8405408b3b59d40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.838ex; width:70.503ex; height:34.843ex;" alt="{\displaystyle {\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\\hline {\scriptstyle {\begin{aligned}&\left(\mu ^{2}+\nu ^{2}-\alpha ^{2}\right)\cos \varphi +\left(\lambda ^{2}-\beta ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }&&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi -\gamma \operatorname {sh} {\theta }\\&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi +\gamma \operatorname {sh} {\theta }&&\left(\mu ^{2}+\nu ^{2}-\beta ^{2}\right)\cos \varphi +\left(\mu ^{2}-\alpha ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }\\&-(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi -\beta \operatorname {sh} {\theta }&&-(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\lambda \sin \varphi +\alpha \operatorname {sh} {\theta }\\&(\gamma \mu -\beta \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }&&-(\alpha \nu -\lambda \gamma )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\\\&\quad -(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi +\beta \operatorname {sh} {\theta }&&\quad (\beta \nu -\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }\\&\quad -(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })-\lambda \sin \varphi -\alpha \operatorname {sh} {\theta }&&\quad (\lambda \gamma -\alpha \nu )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\&\quad \left(\lambda ^{2}+\mu ^{2}-\gamma ^{2}\right)\cos \varphi +\left(\nu ^{2}-\alpha ^{2}-\beta ^{2}\right)\operatorname {ch} {\theta }&&\quad (\alpha \mu -\beta \lambda )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }\\&\quad (\beta \gamma -\alpha \mu )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }&&\quad -\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\cos \varphi +\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}\right)\operatorname {ch} {\theta }\end{aligned}}}\\\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}-\lambda ^{2}-\mu ^{2}-\nu ^{2}=-1\right)\end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gruner_(1921)_–_Trigonometric_Lorentz_boosts"> Gruner (1921) – Trigonometric Lorentz boosts</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=25" title="Edit section: Gruner (1921) – Trigonometric Lorentz boosts">edit</a>]</div> In order to simplify the graphical representation of Minkowski space, <a href="/wiki/Paul_Gruner" title="Paul Gruner">Paul Gruner</a> (1921) (with the aid of Josef Sauter) developed what is now called <a href="/wiki/Loedel_diagram" class="mw-redirect" title="Loedel diagram">Loedel diagrams</a>, using the following relations:<a href="#cite_note-97">[R 66]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}v=\alpha \cdot c;\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}\\\sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi \\\hline x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ,\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mi>v</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mi>c</mi> <mo>;</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>α</mi> <mo>;</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>α</mi> <mi>β</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>t</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}v=\alpha \cdot c;\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}\\\sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi \\\hline x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ,\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi \end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889079c45643ce30c8deebc9e082d930b6cb8e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:46.44ex; height:13.843ex;" alt="{\displaystyle {\begin{matrix}v=\alpha \cdot c;\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}\\\sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi \\\hline x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ,\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi \end{matrix}}}"></dd></dl> In another paper Gruner used the alternative relations:<a href="#cite_note-98">[R 67]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\alpha ={\frac {v}{c}};\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}};\\\cos \theta =\alpha ={\frac {v}{c}};\ \sin \theta ={\frac {1}{\beta }};\ \cot \theta =\alpha \cdot \beta \\\hline x'={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\quad t'={\frac {t}{\sin \theta }}-x\cdot \cot \theta \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>;</mo> <mtext> </mtext> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>;</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> <mo>;</mo> <mtext> </mtext> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>t</mi> <mo>⋅</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mspace width="1em" /> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> <mo>⋅</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\alpha ={\frac {v}{c}};\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}};\\\cos \theta =\alpha ={\frac {v}{c}};\ \sin \theta ={\frac {1}{\beta }};\ \cot \theta =\alpha \cdot \beta \\\hline x'={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\quad t'={\frac {t}{\sin \theta }}-x\cdot \cot \theta \end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab6b21b7631cfc21d9bd195e3c20724c68450fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:44.091ex; height:14.176ex;" alt="{\displaystyle {\begin{matrix}\alpha ={\frac {v}{c}};\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}};\\\cos \theta =\alpha ={\frac {v}{c}};\ \sin \theta ={\frac {1}{\beta }};\ \cot \theta =\alpha \cdot \beta \\\hline x'={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\quad t'={\frac {t}{\sin \theta }}-x\cdot \cot \theta \end{matrix}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=26" title="Edit section: See also">edit</a>]</div> <ul><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/History_of_special_relativity" title="History of special relativity">History of special relativity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=27" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Historical_mathematical_sources">Historical mathematical sources</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=28" title="Edit section: Historical mathematical sources">edit</a>]</div> <a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> Learning materials related to <a href="https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/mathsource" class="extiw" title="v:History of Topics in Special Relativity/mathsource">History of Topics in Special Relativity/mathsource</a> at Wikiversity <div class="mw-heading mw-heading3"><h3 id="Historical_relativity_sources">Historical relativity sources</h3>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=29" title="Edit section: Historical relativity sources">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 reflist-columns-3"> <ol class="references"> <li id="cite_note-5"><a href="#cite_ref-5">^</a> Voigt (1887), p. 45 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Lorentz (1915/16), p. 197 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Lorentz (1915/16), p. 198 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Bucherer (1908), p. 762 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Heaviside (1888), p. 324 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Thomson (1889), p. 12 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Searle (1886), p. 333 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Lorentz (1892a), p. 141 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Lorentz (1892b), p. 141 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> Lorentz (1895), p. 37 </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Larmor (1897), p. 229 </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> Larmor (1897/1929), p. 39 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> Larmor (1900), p. 168 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Larmor (1900), p. 174 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> Larmor (1904a), p. 583, 585 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Larmor (1904b), p. 622 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> Lorentz (1899), p. 429 </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Lorentz (1899), p. 439 </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Lorentz (1899), p. 442 </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> Lorentz (1904), p. 812 </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> Lorentz (1904), p. 826 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> Bucherer, p. 129; Definition of s on p. 32 </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> Wien (1904), p. 394 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Cohn (1904a), pp. 1296-1297 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Gans (1905), p. 169 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Poincaré (1900), pp. 272–273 </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> Cohn (1904b), p. 1408 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Abraham (1905), § 42 </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Poincaré (1905), p. 1505 </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Poincaré (1905/06), pp. 129ff </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> Poincaré (1905/06), p. 144 </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> Einstein (1905), p. 902 </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> Einstein (1905), § 5 and § 9 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Einstein (1905), § 7 </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Minkowski (1907/15), pp. 927ff </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> Minkowski (1907/08), pp. 53ff </li> <li id="cite_note-mink1-62">^ <a href="#cite_ref-mink1_62-0">a</a> <a href="#cite_ref-mink1_62-1">b</a> Minkowski (1907/08), p. 59 </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> Minkowski (1907/08), pp. 65–66, 81–82 </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> Minkowski (1908/09), p. 77 </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> Sommerfeld (1909), p. 826ff. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> Frank (1909), pp. 423-425 </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> Bateman (1909/10), pp. 223ff </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> Cunningham (1909/10), pp. 77ff </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> Klein (1910) </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> Cartan (1912), p. 23 </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> Poincaré (1912/21), p. 145 </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> Herglotz (1909/10), pp. 404-408 </li> <li id="cite_note-var1-74">^ <a href="#cite_ref-var1_74-0">a</a> <a href="#cite_ref-var1_74-1">b</a> Varićak (1910), p. 93 </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> Varićak (1910), p. 94 </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> Plummer (1910), p. 256 </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> Ignatowski (1910), pp. 973–974 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> Ignatowski (1910/11), p. 13 </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> Frank & Rothe (1911), pp. 825ff; (1912), p. 750ff. </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> Klein (1908), p. 165 </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> Noether (1910), pp. 939–943 </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> Klein (1910), p. 300 </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> Klein (1911), pp. 602ff. </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> Conway (1911), p. 8 </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> Silberstein (1911/12), p. 793 </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> Ignatowski (1910/11a), p. 23; (1910/11b), p. 22 </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> Herglotz (1911), p. 497 </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> Silberstein (1911/12), p. 792; (1914), p. 123 </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Borel (1913/14), p. 39 </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> Borel (1913/14), p. 41 </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> Gruner (1921a), </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> Gruner (1921b) </li> </ol></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAbraham,_M.1905" class="citation book cs1">Abraham, M. (1905). <a class="external text" href="https://en.wikisource.org/wiki/de:Elektromagnetische_Theorie_der_Strahlung_(1905)">"§ 42. Die Lichtzeit in einem gleichförmig bewegten System" </a>. Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung. Leipzig: Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A7+42.+Die+Lichtzeit+in+einem+gleichf%C3%B6rmig+bewegten+System&rft.btitle=Theorie+der+Elektrizit%C3%A4t%3A+Elektromagnetische+Theorie+der+Strahlung&rft.place=Leipzig&rft.pub=Teubner&rft.date=1905&rft.au=Abraham%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBateman,_Harry1910" class="citation journal cs1">Bateman, Harry (1910) [1909]. <a class="external text" href="https://en.wikisource.org/wiki/en:The_Transformation_of_the_Electrodynamical_Equations">"The Transformation of the Electrodynamical Equations" </a>. 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Archives des sciences physiques et naturelles. 5. 3: 295–296.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archives+des+sciences+physiques+et+naturelles&rft.atitle=Repr%C3%A9sentation+g%C3%A9om%C3%A9trique+%C3%A9l%C3%A9mentaire+des+formules+de+la+th%C3%A9orie+de+la+relativit%C3%A9&rft.volume=3&rft.pages=295-296&rft.date=1921&rft.au=Gruner%2C+Paul&rft.au=Sauter%2C+Josef&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k2991536%2Ff295.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner,_Paul1921b" class="citation journal cs1">Gruner, Paul (1921b). 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Physikalische Zeitschrift. 22: 384–385.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Eine+elementare+geometrische+Darstellung+der+Transformationsformeln+der+speziellen+Relativit%C3%A4tstheorie&rft.volume=22&rft.pages=384-385&rft.date=1921&rft.au=Gruner%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeaviside,_Oliver1889" class="citation cs2">Heaviside, Oliver (1889), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431195">"On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric"</a>, Philosophical Magazine, 5, 27 (167): 324–339, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786448908628362">10.1080/14786448908628362</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+Electromagnetic+Effects+due+to+the+Motion+of+Electrification+through+a+Dielectric&rft.volume=27&rft.issue=167&rft.pages=324-339&rft.date=1889&rft_id=info%3Adoi%2F10.1080%2F14786448908628362&rft.au=Heaviside%2C+Oliver&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431195&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerglotz,_Gustav1910" class="citation cs2">Herglotz, Gustav (1910) [1909], <a rel="nofollow" class="external text" href="https://zenodo.org/record/1424161">"Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper"</a> [Wikisource translation: <a href="https://en.wikisource.org/wiki/Translation:On_bodies_that_are_to_be_designated_as_%22rigid%22" class="extiw" title="s:Translation:On bodies that are to be designated as "rigid"">On bodies that are to be designated as "rigid" from the standpoint of the relativity principle</a>], Annalen der Physik, 336 (2): 393–415, <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/1910AnP...336..393H">1910AnP...336..393H</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.19103360208">10.1002/andp.19103360208</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=%C3%9Cber+den+vom+Standpunkt+des+Relativit%C3%A4tsprinzips+aus+als+starr+zu+bezeichnenden+K%C3%B6rper&rft.volume=336&rft.issue=2&rft.pages=393-415&rft.date=1910&rft_id=info%3Adoi%2F10.1002%2Fandp.19103360208&rft_id=info%3Abibcode%2F1910AnP...336..393H&rft.au=Herglotz%2C+Gustav&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1424161&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerglotz,_G.1911" class="citation journal cs1">Herglotz, G. (1911). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k153397.image.f509">"Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie"</a>. Annalen der Physik. 341 (13): 493–533. <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/1911AnP...341..493H">1911AnP...341..493H</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.19113411303">10.1002/andp.19113411303</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=%C3%9Cber+die+Mechanik+des+deformierbaren+K%C3%B6rpers+vom+Standpunkte+der+Relativit%C3%A4tstheorie&rft.volume=341&rft.issue=13&rft.pages=493-533&rft.date=1911&rft_id=info%3Adoi%2F10.1002%2Fandp.19113411303&rft_id=info%3Abibcode%2F1911AnP...341..493H&rft.au=Herglotz%2C+G.&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k153397.image.f509&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988">; English translation by David Delphenich: <a rel="nofollow" class="external text" href="http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/herglotz_-_rel._cont._mech..pdf">On the mechanics of deformable bodies from the standpoint of relativity theory</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIgnatowsky,_W._v.1910" class="citation journal cs1">Ignatowsky, W. v. (1910). <a class="external text" href="https://en.wikisource.org/wiki/de:Einige_allgemeine_Bemerkungen_%C3%BCber_das_Relativit%C3%A4tsprinzip">"Einige allgemeine Bemerkungen über das Relativitätsprinzip" </a>. Physikalische Zeitschrift. 11: 972–976.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Einige+allgemeine+Bemerkungen+%C3%BCber+das+Relativit%C3%A4tsprinzip&rft.volume=11&rft.pages=972-976&rft.date=1910&rft.au=Ignatowsky%2C+W.+v.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIgnatowski,_W._v.1911" class="citation journal cs1">Ignatowski, W. v. (1911) [1910]. <a class="external text" href="https://en.wikisource.org/wiki/de:Das_Relativit%C3%A4tsprinzip_(Ignatowski)">"Das Relativitätsprinzip" </a>. Archiv der Mathematik und Physik. 18: 17–40.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+der+Mathematik+und+Physik&rft.atitle=Das+Relativit%C3%A4tsprinzip&rft.volume=18&rft.pages=17-40&rft.date=1911&rft.au=Ignatowski%2C+W.+v.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIgnatowski,_W._v.1911" class="citation journal cs1">Ignatowski, W. v. (1911). <a class="external text" href="https://en.wikisource.org/wiki/de:Eine_Bemerkung_zu_meiner_Arbeit:_%22Einige_allgemeine_Bemerkungen_zum_Relativit%C3%A4tsprinzip%22">"Eine Bemerkung zu meiner Arbeit: "Einige allgemeine Bemerkungen zum Relativitätsprinzip"" </a>. Physikalische Zeitschrift. 12: 779.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Eine+Bemerkung+zu+meiner+Arbeit%3A+%22Einige+allgemeine+Bemerkungen+zum+Relativit%C3%A4tsprinzip%22&rft.volume=12&rft.pages=779&rft.date=1911&rft.au=Ignatowski%2C+W.+v.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein,_F.1908" class="citation book cs1">Klein, F. (1908). Hellinger, E. (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/elementarmathem00kleigoog">Elementarmethematik vom höheren Standpunkte aus. Teil I. Vorlesung gehalten während des Wintersemesters 1907-08</a>. Leipzig: Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementarmethematik+vom+h%C3%B6heren+Standpunkte+aus.+Teil+I.+Vorlesung+gehalten+w%C3%A4hrend+des+Wintersemesters+1907-08&rft.place=Leipzig&rft.pub=Teubner&rft.date=1908&rft.au=Klein%2C+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarmathem00kleigoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein,_Felix1921" class="citation book cs1">Klein, Felix (1921) [1910]. "Über die geometrischen Grundlagen der Lorentzgruppe". <a class="external text" href="https://en.wikisource.org/wiki/de:%C3%9Cber_die_geometrischen_Grundlagen_der_Lorentzgruppe">Gesammelte Mathematische Abhandlungen </a>. Vol. 1. pp. 533–552. <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%2F978-3-642-51960-4_31">10.1007/978-3-642-51960-4_31</a> (inactive 1 November 2024). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-51898-0" title="Special:BookSources/978-3-642-51898-0"><bdi>978-3-642-51898-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C3%9Cber+die+geometrischen+Grundlagen+der+Lorentzgruppe&rft.btitle=Gesammelte+Mathematische+Abhandlungen&rft.pages=533-552&rft.date=1921&rft_id=info%3Adoi%2F10.1007%2F978-3-642-51960-4_31&rft.isbn=978-3-642-51898-0&rft.au=Klein%2C+Felix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein,_F.Sommerfeld_A.1910" class="citation book cs1">Klein, F.; Sommerfeld A. (1910). Noether, Fr. (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/fkleinundasommer019696mbp">Über die Theorie des Kreisels. Heft IV</a>. Leipzig: Teuber.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%9Cber+die+Theorie+des+Kreisels.+Heft+IV&rft.place=Leipzig&rft.pub=Teuber&rft.date=1910&rft.au=Klein%2C+F.&rft.au=Sommerfeld+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffkleinundasommer019696mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein,_F.1911" class="citation book cs1">Klein, F. (1911). Hellinger, E. (ed.). Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08. 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Part 3: Relations with material media", Mathematical and Physical Papers: Volume II, Cambridge University Press, pp. 2–132, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-53640-1" title="Special:BookSources/978-1-107-53640-1"><bdi>978-1-107-53640-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+a+Dynamical+Theory+of+the+Electric+and+Luminiferous+Medium.+Part+3%3A+Relations+with+material+media&rft.btitle=Mathematical+and+Physical+Papers%3A+Volume+II&rft.pages=2-132&rft.pub=Cambridge+University+Press&rft.date=1929&rft.isbn=978-1-107-53640-1&rft.au=Larmor%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"> (Reprint of Larmor (1897) with new annotations by Larmor.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarmor,_Joseph1900" class="citation cs2">Larmor, Joseph (1900), <a class="external text" href="https://en.wikisource.org/wiki/Aether_and_Matter">Aether and Matter </a>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aether+and+Matter&rft.pub=Cambridge+University+Press&rft.date=1900&rft.au=Larmor%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarmor,_Joseph1904a" class="citation journal cs1">Larmor, Joseph (1904a). <a rel="nofollow" class="external text" href="https://archive.org/details/londonedinburgh671904lond">"On the intensity of the natural radiation from moving bodies and its mechanical reaction"</a>. Philosophical Magazine. 7 (41): <a rel="nofollow" class="external text" href="https://archive.org/details/londonedinburgh671904lond/page/578">578</a>–586. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440409463149">10.1080/14786440409463149</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+intensity+of+the+natural+radiation+from+moving+bodies+and+its+mechanical+reaction&rft.volume=7&rft.issue=41&rft.pages=578-586&rft.date=1904&rft_id=info%3Adoi%2F10.1080%2F14786440409463149&rft.au=Larmor%2C+Joseph&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flondonedinburgh671904lond&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarmor,_Joseph1904b" class="citation journal cs1">Larmor, Joseph (1904b). <a class="external text" href="https://en.wikisource.org/wiki/en:Absence_of_Effects_of_Motion_through_the_Aether">"On the ascertained Absence of Effects of Motion through the Aether, in relation to the Constitution of Matter, and on the FitzGerald-Lorentz Hypothesis" </a>. 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See also the <a rel="nofollow" class="external text" href="http://www.physicsinsights.org/poincare-1900.pdf">English translation</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré,_Henri1906" class="citation cs2">Poincaré, Henri (1906) [1904], <a class="external text" href="https://en.wikisource.org/wiki/The_Principles_of_Mathematical_Physics">"The Principles of Mathematical Physics" </a>, Congress of arts and science, universal exposition, St. Louis, 1904, vol. 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Principles+of+Mathematical+Physics&rft.btitle=Congress+of+arts+and+science%2C+universal+exposition%2C+St.+Louis%2C+1904&rft.place=Boston+and+New+York&rft.pages=604-622&rft.pub=Houghton%2C+Mifflin+and+Company&rft.date=1906&rft.au=Poincar%C3%A9%2C+Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré,_Henri1905" class="citation cs2">Poincaré, Henri (1905), <a class="external text" href="https://en.wikisource.org/wiki/fr:Sur_la_dynamique_de_l%27%C3%A9lectron_(juin)">"Sur la dynamique de l'électron" </a> [<a href="https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(June)" class="extiw" title="s:Translation:On the Dynamics of the Electron (June)">On the Dynamics of the Electron</a>], Comptes Rendus, 140: 1504–1508</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus&rft.atitle=Sur+la+dynamique+de+l%27%C3%A9lectron&rft.volume=140&rft.pages=1504-1508&rft.date=1905&rft.au=Poincar%C3%A9%2C+Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré,_Henri1906" class="citation cs2">Poincaré, Henri (1906) [1905], <a class="external text" href="https://en.wikisource.org/wiki/fr:Sur_la_dynamique_de_l%27%C3%A9lectron_(juillet)">"Sur la dynamique de l'électron" </a> [<a href="https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)" class="extiw" title="s:Translation:On the Dynamics of the Electron (July)">On the Dynamics of the Electron</a>], Rendiconti del Circolo Matematico di Palermo, 21: 129–176, <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/1906RCMP...21..129P">1906RCMP...21..129P</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%2FBF03013466">10.1007/BF03013466</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/2027%2Fuiug.30112063899089">2027/uiug.30112063899089</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:120211823">120211823</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti+del+Circolo+Matematico+di+Palermo&rft.atitle=Sur+la+dynamique+de+l%27%C3%A9lectron&rft.volume=21&rft.pages=129-176&rft.date=1906&rft_id=info%3Ahdl%2F2027%2Fuiug.30112063899089&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120211823%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF03013466&rft_id=info%3Abibcode%2F1906RCMP...21..129P&rft.au=Poincar%C3%A9%2C+Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré,_Henri1921" class="citation journal cs1">Poincaré, Henri (1921) [1912]. <a rel="nofollow" class="external text" href="https://archive.org/stream/actamathematica38upps#page/n153/mode/2up">"Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)"</a>. 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(1912) [1911], <a rel="nofollow" class="external text" href="https://archive.org/details/londonedinburg6231912lond">"Quaternionic form of relativity"</a>, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440508637276">10.1080/14786440508637276</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+London%2C+Edinburgh%2C+and+Dublin+Philosophical+Magazine+and+Journal+of+Science&rft.atitle=Quaternionic+form+of+relativity&rft.volume=23&rft.issue=137&rft.pages=790-809&rft.date=1912&rft_id=info%3Adoi%2F10.1080%2F14786440508637276&rft.au=Silberstein%2C+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flondonedinburg6231912lond&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSommerfeld,_A.1909" class="citation cs2">Sommerfeld, A. 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New York: Birkhäuser. pp. 3–50. <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%2F978-1-4939-7708-6_1">10.1007/978-1-4939-7708-6_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-7708-6" title="Special:BookSources/978-1-4939-7708-6"><bdi>978-1-4939-7708-6</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:31840179">31840179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Figures+of+Light+in+the+Early+History+of+Relativity+%281905%E2%80%931914%29&rft.btitle=Beyond+Einstein&rft.place=New+York&rft.series=Einstein+Studies&rft.pages=3-50&rft.pub=Birkh%C3%A4user&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A31840179%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-1-4939-7708-6_1&rft.isbn=978-1-4939-7708-6&rft.au=Walter%2C+Scott+A.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FWALFOL-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+Lorentz+transformations" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=History_of_Lorentz_transformations&action=edit&section=31" title="Edit section: External links">edit</a>]</div> <ul><li>Mathpages: <a rel="nofollow" class="external text" href="http://mathpages.com/rr/s1-04/1-04.htm">1.4 The Relativity of 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href="/wiki/History_of_Maxwell%27s_equations" title="History of Maxwell's equations">Maxwell's equations</a></li></ul></li> <li><a href="/wiki/History_of_fluid_mechanics" title="History of fluid mechanics">Fluid mechanics</a> <ul><li><a href="/wiki/Timeline_of_fluid_and_continuum_mechanics" title="Timeline of fluid and continuum mechanics">timeline</a></li> <li><a href="/wiki/History_of_aerodynamics" title="History of aerodynamics">Aerodynamics</a></li></ul></li> <li><a href="/wiki/History_of_classical_field_theory" title="History of classical field theory">Field theory</a></li> <li><a href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">Gravitational theory</a> <ul><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">timeline</a></li></ul></li> <li><a href="/wiki/History_of_materials_science" title="History of materials science">Material science</a> <ul><li><a href="/wiki/Timeline_of_materials_technology" title="Timeline of materials technology">timeline</a></li> <li><a href="/wiki/History_of_metamaterials" title="History of metamaterials">Metamaterials</a></li></ul></li> <li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">Mechanics</a> <ul><li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">timeline</a></li> <li><a href="/wiki/History_of_variational_principles_in_physics" title="History of variational principles in physics">Variational principles</a></li></ul></li> <li><a href="/wiki/History_of_optics" title="History of optics">Optics</a> <ul><li><a href="/wiki/History_of_spectroscopy" title="History of spectroscopy">Spectroscopy</a></li></ul></li> <li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">timeline</a></li> <li><a href="/wiki/History_of_energy" title="History of energy">Energy</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">Perpetual motion</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 physics</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>Computational physics <ul><li><a href="/wiki/Timeline_of_computational_physics" title="Timeline of computational physics">timeline</a></li></ul></li> <li>Condensed matter <ul><li><a href="/wiki/Timeline_of_condensed_matter_physics" title="Timeline of condensed matter physics">timeline</a></li> <li><a href="/wiki/History_of_superconductivity" title="History of superconductivity">Superconductivity</a></li></ul></li> <li>Cosmology <ul><li><a href="/wiki/Timeline_of_cosmological_theories" title="Timeline of cosmological theories">timeline</a></li> <li><a href="/wiki/History_of_the_Big_Bang_theory" title="History of the Big Bang theory">Big Bang theory</a></li></ul></li> <li><a href="/wiki/History_of_general_relativity" title="History of general relativity">General relativity</a> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">tests</a></li></ul></li> <li><a href="/wiki/History_of_geophysics" title="History of geophysics">Geophysics</a></li> <li>Nuclear physics <ul><li><a href="/wiki/Discovery_of_nuclear_fission" title="Discovery of nuclear fission">Fission</a></li> <li><a href="/wiki/History_of_nuclear_fusion" title="History of nuclear fusion">Fusion</a></li> <li><a href="/wiki/History_of_nuclear_power" title="History of nuclear power">Power</a></li> <li><a href="/wiki/History_of_nuclear_weapons" title="History of nuclear weapons">Weapons</a></li></ul></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">Quantum mechanics</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">timeline</a></li> <li><a href="/wiki/History_of_atomic_theory" title="History of atomic theory">Atoms</a></li> <li><a href="/wiki/History_of_molecular_theory" title="History of molecular theory">Molecules</a></li> <li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">Quantum field theory</a></li></ul></li> <li><a href="/wiki/History_of_subatomic_physics" title="History of subatomic physics">Subatomic physics</a> <ul><li><a href="/wiki/Timeline_of_atomic_and_subatomic_physics" title="Timeline of atomic and subatomic physics">timeline</a></li></ul></li> <li><a href="/wiki/History_of_special_relativity" title="History of special relativity">Special relativity</a> <ul><li><a href="/wiki/Timeline_of_special_relativity_and_the_speed_of_light" title="Timeline of special relativity and the speed of light">timeline</a></li> <li><a class="mw-selflink selflink">Lorentz transformations</a></li> <li><a href="/wiki/Tests_of_special_relativity" title="Tests of special relativity">tests</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Recent developments</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Quantum information <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/History_of_loop_quantum_gravity" title="History of loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/History_of_nanotechnology" title="History of nanotechnology">Nanotechnology</a></li> <li><a href="/wiki/History_of_string_theory" title="History of string theory">String theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">On specific discoveries</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/Discovery_of_cosmic_microwave_background_radiation" title="Discovery of cosmic microwave background radiation">Cosmic microwave background</a></li> <li><a href="/wiki/Discovery_of_graphene" title="Discovery of graphene">Graphene</a></li> <li><a href="/wiki/First_observation_of_gravitational_waves" title="First observation of gravitational waves">Gravitational waves</a></li> <li>Subatomic particles <ul><li><a href="/wiki/Timeline_of_particle_discoveries" title="Timeline of particle discoveries">timeline</a></li> <li><a href="/wiki/Search_for_the_Higgs_boson" title="Search for the Higgs boson">Higgs boson</a></li> <li><a href="/wiki/Discovery_of_the_neutron" title="Discovery of the neutron">Neutron</a></li></ul></li> <li><a href="/wiki/R%C3%B8mer%27s_determination_of_the_speed_of_light" title="Rømer's determination of the speed of light">Speed of light</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By periods</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/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li> <li><a href="/wiki/Golden_age_of_physics" title="Golden age of physics">Golden age of physics</a></li> <li><a href="/wiki/Golden_age_of_cosmology" title="Golden age of cosmology">Golden age of cosmology</a></li> <li><a href="/wiki/Physics_in_the_medieval_Islamic_world" title="Physics in the medieval Islamic world">Medieval Islamic world</a> <ul><li><a href="/wiki/Astronomy_in_the_medieval_Islamic_world" title="Astronomy in the medieval Islamic world">Astronomy</a></li></ul></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">Noisy intermediate-scale quantum era</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By groups</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/Harvard_Computers" title="Harvard Computers">Harvard Computers</a></li> <li><a href="/wiki/The_Martians_(scientists)" title="The Martians (scientists)">The Martians</a></li> <li><a href="/wiki/Oxford_Calculators" title="Oxford Calculators">Oxford Calculators</a></li> <li><a href="/wiki/Via_Panisperna_boys" title="Via Panisperna boys">Via Panisperna boys</a></li> <li><a href="/wiki/Women_in_physics" title="Women in physics">Women in physics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scientific disputes</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/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein</a></li> <li><a href="/wiki/Chandrasekhar%E2%80%93Eddington_dispute" title="Chandrasekhar–Eddington dispute">Chandrasekhar–Eddington</a></li> <li><a href="/wiki/Galileo_affair" title="Galileo affair">Galileo affair</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton</a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Joule–von Mayer</a></li> <li><a href="/wiki/Great_Debate_(astronomy)" title="Great Debate (astronomy)">Shapley–Curtis</a></li> <li>Relativity priority <ul><li><a href="/wiki/Relativity_priority_dispute" title="Relativity priority dispute">Special relativity</a></li> <li><a href="/wiki/General_relativity_priority_dispute" title="General relativity priority dispute">General relativity</a></li></ul></li> <li><a href="/wiki/Transfermium_Wars" 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History of Lorentz transformations - Wikipedia