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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-relativistic_"spacetime_diagrams""> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Non-relativistic "spacetime diagrams"</span> </div> </a> <ul id="toc-Non-relativistic_"spacetime_diagrams"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Minkowski_diagrams" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Minkowski_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Minkowski diagrams</span> </div> </a> <button aria-controls="toc-Minkowski_diagrams-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Minkowski diagrams subsection</span> </button> <ul id="toc-Minkowski_diagrams-sublist" class="vector-toc-list"> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Overview</span> </div> </a> <ul id="toc-Overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_details" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_details"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Mathematical details</span> </div> </a> <ul id="toc-Mathematical_details-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Loedel_diagrams" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Loedel_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Loedel diagrams</span> </div> </a> <button aria-controls="toc-Loedel_diagrams-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Loedel diagrams subsection</span> </button> <ul id="toc-Loedel_diagrams-sublist" class="vector-toc-list"> <li id="toc-Formulation_via_median_frame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formulation_via_median_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Formulation via median frame</span> </div> </a> <ul id="toc-Formulation_via_median_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>History</span> </div> </a> <ul id="toc-History_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relativistic_phenomena_in_diagrams" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_phenomena_in_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relativistic phenomena in diagrams</span> </div> </a> <button aria-controls="toc-Relativistic_phenomena_in_diagrams-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relativistic phenomena in diagrams subsection</span> </button> <ul id="toc-Relativistic_phenomena_in_diagrams-sublist" class="vector-toc-list"> <li id="toc-Time_dilation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_dilation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Time dilation</span> </div> </a> <ul id="toc-Time_dilation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Length_contraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Length_contraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Length contraction</span> </div> </a> <ul id="toc-Length_contraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constancy_of_the_speed_of_light" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constancy_of_the_speed_of_light"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Constancy of the speed of light</span> </div> </a> <ul id="toc-Constancy_of_the_speed_of_light-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speed_of_light_and_causality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Speed_of_light_and_causality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Speed of light and causality</span> </div> </a> <ul id="toc-Speed_of_light_and_causality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_speed_of_light_as_a_limit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_speed_of_light_as_a_limit"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>The speed of light as a limit</span> </div> </a> <ul id="toc-The_speed_of_light_as_a_limit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Accelerating_observers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Accelerating_observers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Accelerating observers</span> </div> </a> <ul id="toc-Accelerating_observers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Case_of_non-inertial_reference_frames" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Case_of_non-inertial_reference_frames"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Case of non-inertial reference frames</span> </div> </a> <ul id="toc-Case_of_non-inertial_reference_frames-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Spacetime diagram</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" 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Available in 24 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-24" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Minkowski-Diagramm" title="Minkowski-Diagramm – Alemannic" lang="gsw" hreflang="gsw" data-title="Minkowski-Diagramm" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%B3%D9%85_%D9%85%D9%8A%D9%86%D9%83%D9%88%D9%81%D8%B3%D9%83%D9%8A_%D8%A7%D9%84%D8%A8%D9%8A%D8%A7%D9%86%D9%8A" title="رسم مينكوفسكي البياني – Arabic" lang="ar" hreflang="ar" data-title="رسم مينكوفسكي البياني" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diagrama_espai-temps" title="Diagrama espai-temps – Catalan" lang="ca" hreflang="ca" data-title="Diagrama espai-temps" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prostoro%C4%8Dasov%C3%BD_diagram" title="Prostoročasový diagram – Czech" lang="cs" hreflang="cs" data-title="Prostoročasový diagram" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Minkowski-Diagramm" title="Minkowski-Diagramm – German" lang="de" hreflang="de" data-title="Minkowski-Diagramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagrama_de_Minkowski" title="Diagrama de Minkowski – Spanish" lang="es" hreflang="es" data-title="Diagrama de Minkowski" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Minkowskiren_diagrama" title="Minkowskiren diagrama – Basque" lang="eu" hreflang="eu" data-title="Minkowskiren diagrama" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%AF%D8%A7%D8%B1_%D9%85%DB%8C%D9%86%DA%A9%D9%88%D9%81%D8%B3%DA%A9%DB%8C" title="نمودار مینکوفسکی – Persian" lang="fa" hreflang="fa" data-title="نمودار مینکوفسکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diagramme_de_Minkowski" title="Diagramme de Minkowski – French" lang="fr" hreflang="fr" data-title="Diagramme de Minkowski" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%B6%D5%AF%D5%B8%D5%BE%D5%BD%D5%AF%D5%B8%D6%82_%D5%A4%D5%AB%D5%A1%D5%A3%D6%80%D5%A1%D5%B4" title="Մինկովսկու դիագրամ – Armenian" lang="hy" hreflang="hy" data-title="Մինկովսկու դիագրամ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diagram_Minkowski" title="Diagram Minkowski – Indonesian" lang="id" hreflang="id" data-title="Diagram Minkowski" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A8%D7%A9%D7%99%D7%9D_%D7%9E%D7%99%D7%A0%D7%A7%D7%95%D7%91%D7%A1%D7%A7%D7%99" title="תרשים מינקובסקי – Hebrew" lang="he" hreflang="he" data-title="תרשים מינקובסקי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B8%D0%BD%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B5%D0%B2_%D0%B4%D0%B8%D1%98%D0%B0%D0%B3%D1%80%D0%B0%D0%BC" title="Минковскиев дијаграм – Macedonian" lang="mk" hreflang="mk" data-title="Минковскиев дијаграм" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Minkowski-diagram" title="Minkowski-diagram – Dutch" lang="nl" hreflang="nl" data-title="Minkowski-diagram" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%99%82%E7%A9%BA%E5%9B%B3" title="時空図 – Japanese" lang="ja" hreflang="ja" data-title="時空図" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Diagram_czasoprzestrzenny" title="Diagram czasoprzestrzenny – Polish" lang="pl" hreflang="pl" data-title="Diagram czasoprzestrzenny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Diagrama_de_Minkowski" title="Diagrama de Minkowski – Portuguese" lang="pt" hreflang="pt" data-title="Diagrama de Minkowski" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE-%D0%B2%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B4%D0%B8%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B0" title="Пространственно-временная диаграмма – Russian" lang="ru" hreflang="ru" data-title="Пространственно-временная диаграмма" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prostorvremenski_dijagram" title="Prostorvremenski dijagram – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Prostorvremenski dijagram" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Minkowski_diyagram%C4%B1" title="Minkowski diyagramı – Turkish" lang="tr" hreflang="tr" data-title="Minkowski diyagramı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%B0_%D0%9C%D1%96%D0%BD%D0%BA%D0%BE%D0%B2%D1%81%D1%8C%D0%BA%D0%BE%D0%B3%D0%BE" title="Діаграма Мінковського – Ukrainian" lang="uk" hreflang="uk" data-title="Діаграма Мінковського" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%E1%BA%A3n_%C4%91%E1%BB%93_Minkowski" title="Giản đồ Minkowski – Vietnamese" lang="vi" hreflang="vi" data-title="Giản đồ Minkowski" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%99%82%E7%A9%BA%E5%9C%96" title="時空圖 – Cantonese" lang="yue" hreflang="yue" data-title="時空圖" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%97%B5%E5%8F%AF%E5%A4%AB%E6%96%AF%E5%9F%BA%E5%9B%BE" title="闵可夫斯基图 – Chinese" lang="zh" hreflang="zh" data-title="闵可夫斯基图" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177596#sitelinks-wikipedia" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Graph of space and time in special relativity</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_photon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Minkowski_diagram_-_photon.svg/256px-Minkowski_diagram_-_photon.svg.png" decoding="async" width="256" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Minkowski_diagram_-_photon.svg/384px-Minkowski_diagram_-_photon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Minkowski_diagram_-_photon.svg/512px-Minkowski_diagram_-_photon.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>The world line (yellow path) of a <a href="/wiki/Photon" title="Photon">photon</a>, which is at location <i>x</i> = 0 at time <i>ct</i> = 0.</figcaption></figure> <style 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mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">General relativity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction to general relativity</a></li> <li><a href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Classical gravity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Gravity" title="Gravity">Introduction to gravitation</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Relevant mathematics</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Four-vector" title="Four-vector">Four-vector</a></li> <li><a href="/wiki/Derivations_of_the_Lorentz_transformations" title="Derivations of the Lorentz transformations">Derivations of relativity</a></li> <li><a class="mw-selflink selflink">Spacetime diagrams</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Curved_space" title="Curved space">Curved space</a></li> <li><a href="/wiki/Curved_spacetime" title="Curved spacetime">Curved spacetime</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Spacetime_topology" title="Spacetime topology">Spacetime topology</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below"> <div class="hlist"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" 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//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Spacetime" title="Category:Spacetime">Category</a></span></li></ul> </div></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Spacetime" title="Template:Spacetime"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spacetime" title="Template talk:Spacetime"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spacetime" title="Special:EditPage/Template:Spacetime"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>spacetime diagram</b> is a graphical illustration of locations in space at various times, especially in the <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">special theory of relativity</a>. Spacetime diagrams can show the geometry underlying phenomena like <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> and <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a> without mathematical equations. </p><p>The history of an object's location through time traces out a line or curve on a spacetime diagram, referred to as the object's <a href="/wiki/World_line" title="World line">world line</a>. Each point in a spacetime diagram represents a unique position in space and time and is referred to as an <a href="/wiki/Event_(relativity)" title="Event (relativity)">event</a>. </p><p>The most well-known class of spacetime diagrams are known as <b>Minkowski diagrams</b>, developed by <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> in 1908. Minkowski diagrams are two-dimensional graphs that depict events as happening in a <a href="/wiki/Universe" title="Universe">universe</a> consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction_to_kinetic_diagrams">Introduction to kinetic diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=1" title="Edit section: Introduction to kinetic diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Position_versus_time_graphs">Position versus time graphs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=2" title="Edit section: Position versus time graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Distance-time_graph_example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Distance-time_graph_example.svg/220px-Distance-time_graph_example.svg.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Distance-time_graph_example.svg/330px-Distance-time_graph_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Distance-time_graph_example.svg/440px-Distance-time_graph_example.svg.png 2x" data-file-width="524" data-file-height="372" /></a><figcaption>Fig 1-1. Position vs. time graph</figcaption></figure> <p>In the study of 1-dimensional <a href="/wiki/Kinematics" title="Kinematics">kinematics</a>, position vs. time graphs (called x-t graphs for short) provide a useful means to describe motion. Kinematic features besides the object's position are visible by the slope and shape of the lines.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In Fig 1-1, the plotted object moves away from the origin at a positive constant velocity (1.66 m/s) for 6 seconds, halts for 5 seconds, then returns to the origin over a period of 7 seconds at a non-constant speed (but negative velocity). </p><p>At its most basic level, a spacetime diagram is merely a time vs position graph, with the directions of the axes in a usual p-t graph exchanged; that is, the vertical axis refers to temporal and the horizontal axis to spatial coordinate values. Especially when used in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> (SR), the temporal axes of a spacetime diagram are often scaled with the speed of light <span class="texhtml mvar" style="font-style:italic;">c</span>, and thus are often labeled by <span class="texhtml mvar" style="font-style:italic;">ct.</span> This changes the dimension of the addressed physical quantity from <<i>Time</i>> to <<i>Length</i>>, in accordance with the dimension associated with the spatial axis, which is frequently labeled <span class="texhtml mvar" style="font-style:italic;">x.</span> </p> <div class="mw-heading mw-heading3"><h3 id="Standard_configuration_of_reference_frames">Standard configuration of reference frames</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=3" title="Edit section: Standard configuration of reference frames"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Standard_configuration_of_coordinate_systems.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Standard_configuration_of_coordinate_systems.svg/220px-Standard_configuration_of_coordinate_systems.svg.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Standard_configuration_of_coordinate_systems.svg/330px-Standard_configuration_of_coordinate_systems.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Standard_configuration_of_coordinate_systems.svg/440px-Standard_configuration_of_coordinate_systems.svg.png 2x" data-file-width="416" data-file-height="357" /></a><figcaption>Fig 1–2. Galilean diagram of two frames of reference in standard configuration.</figcaption></figure> <p>To ease insight into how spacetime coordinates, measured by observers in different <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">reference frames</a>, compare with each other, it is useful to standardize and simplify the setup. Two <a href="/wiki/Galilean_reference_frame" class="mw-redirect" title="Galilean reference frame">Galilean reference frames</a> (i.e., conventional 3-space frames), S and S′ (pronounced "S prime"), each with observers O and O′ at rest in their respective frames, but measuring the other as moving with speeds ±<i>v</i> are said to be in <i>standard configuration</i>, when: </p> <ul><li>The <i>x</i>, <i>y</i>, <i>z</i> axes of frame S are oriented parallel to the respective primed axes of frame S′.</li> <li>The origins of frames S and S′ coincide at time <i>t</i> = 0 in frame S and also at <i>t</i>′ = 0 in frame S′.<sup id="cite_ref-Collier_2-0" class="reference"><a href="#cite_note-Collier-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 107">: 107 </span></sup></li> <li>Frame S′ moves in the <i>x</i>-direction of frame S with velocity <i>v</i> as measured in frame S.</li></ul> <p>This spatial setting is displayed in the Fig 1-2, in which the temporal coordinates are separately annotated as quantities <i>t</i> and <i>t'</i>. </p><p>In a further step of simplification it is often sufficient to consider just the direction of the observed motion and ignore the other two spatial components, allowing <i>x</i> and <i>ct</i> to be plotted in 2-dimensional spacetime diagrams, as introduced above. </p> <div class="mw-heading mw-heading3"><h3 id="Non-relativistic_"spacetime_diagrams""><span id="Non-relativistic_.22spacetime_diagrams.22"></span>Non-relativistic "spacetime diagrams"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=4" title="Edit section: Non-relativistic "spacetime diagrams""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_Newtonian_physics.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Minkowski_diagram_-_Newtonian_physics.svg/256px-Minkowski_diagram_-_Newtonian_physics.svg.png" decoding="async" width="256" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Minkowski_diagram_-_Newtonian_physics.svg/384px-Minkowski_diagram_-_Newtonian_physics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Minkowski_diagram_-_Newtonian_physics.svg/512px-Minkowski_diagram_-_Newtonian_physics.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 1–3. In Newtonian physics for both observers the event at A is assigned to the same point in time.</figcaption></figure> <p>The black axes labelled <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>ct</i></span> on Fig 1-3 are the coordinate system of an observer, referred to as <i>at rest</i>, and who is positioned at <span class="nowrap"><i>x</i> = 0</span>. This observer's world line is identical with the <span class="texhtml"><i>ct</i></span> time axis. Each parallel line to this axis would correspond also to an object at rest but at another position. The blue line describes an object moving with constant speed <span class="texhtml"><i>v</i></span> to the right, such as a moving observer. </p><p>This blue line labelled <span class="texhtml"><i>ct</i>′</span> may be interpreted as the time axis for the second observer. Together with the <span class="texhtml"><i>x</i></span> axis, which is identical for both observers, it represents their coordinate system. Since the reference frames are in standard configuration, both observers agree on the location of the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of their coordinate systems. The axes for the moving observer are not <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to each other and the scale on their time axis is stretched. To determine the coordinates of a certain event, two lines, each parallel to one of the two axes, must be constructed passing through the event, and their intersections with the axes read off. </p><p>Determining position and time of the event A as an example in the diagram leads to the same time for both observers, as expected. Only for the position different values result, because the moving observer has approached the position of the event A since <span class="texhtml"><i>t</i> = 0</span>. Generally stated, all events on a line parallel to the <span class="texhtml"><i>x</i></span> axis happen simultaneously for both observers. There is only one universal time <span class="nowrap"><i>t</i> = <i>t</i>′</span>, modelling the existence of one common position axis. On the other hand, due to two different time axes the observers usually measure different coordinates for the same event. This graphical translation from <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>t</i></span> to <span class="texhtml"><i>x</i>′</span> and <span class="texhtml"><i>t</i>′</span> and vice versa is described mathematically by the so-called <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Minkowski_diagrams">Minkowski diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=5" title="Edit section: Minkowski diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Overview">Overview</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=6" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_asymmetric.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Minkowski_diagram_-_asymmetric.svg/256px-Minkowski_diagram_-_asymmetric.svg.png" decoding="async" width="256" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Minkowski_diagram_-_asymmetric.svg/384px-Minkowski_diagram_-_asymmetric.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Minkowski_diagram_-_asymmetric.svg/512px-Minkowski_diagram_-_asymmetric.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 2-1 In the theory of relativity each observer assigns the event at A to a different time and location.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MinkBoost2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/MinkBoost2.gif/256px-MinkBoost2.gif" decoding="async" width="256" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/0f/MinkBoost2.gif 1.5x" data-file-width="360" data-file-height="400" /></a><figcaption>Fig 2-2 Minkowski diagram for various speeds of the primed frame, which is moving relative to the unprimed frame. The dashed lines represent the <a href="/wiki/Light_cone" title="Light cone">light cone</a> of a flash of light at the origin.</figcaption></figure> <p>The term Minkowski diagram refers to a specific form of spacetime diagram frequently used in special relativity. A Minkowski diagram is a two-dimensional graphical depiction of a portion of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, usually where space has been curtailed to a single dimension. The units of measurement in these diagrams are taken such that the light cone at an event consists of the lines of <a href="/wiki/Slope" title="Slope">slope</a> plus or minus one through that event.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The horizontal lines correspond to the usual notion of <i>simultaneous events</i> for a stationary observer at the origin. </p><p>A particular Minkowski diagram illustrates the result of a <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>. The Lorentz transformation relates two <a href="/wiki/Inertial_frames_of_reference" class="mw-redirect" title="Inertial frames of reference">inertial frames of reference</a>, where an <a href="/wiki/Observer_(special_relativity)" title="Observer (special relativity)">observer</a> stationary at the event <span class="nowrap">(0, 0)</span> makes a change of <a href="/wiki/Velocity" title="Velocity">velocity</a> along the <span class="texhtml"><i>x</i></span>-axis. As shown in Fig 2-1, the new time axis of the observer forms an angle <span class="texhtml"><i>α</i></span> with the previous time axis, with <span class="texhtml"><i>α</i> < <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></span>. In the new frame of reference the simultaneous events lie parallel to a line inclined by <span class="texhtml"><i>α</i></span> to the previous lines of simultaneity. This is the new <span class="texhtml"><i>x</i></span>-axis. Both the original set of axes and the primed set of axes have the property that they are orthogonal with respect to the <a href="/wiki/Minkowski_inner_product" class="mw-redirect" title="Minkowski inner product">Minkowski inner product</a> or <i>relativistic <a href="/wiki/Dot_product" title="Dot product">dot product</a></i>. The original position on your time line (ct) is perpendicular to position A, the original position on your mutual timeline (x) where (t) is zero. This timeline where timelines come together are positioned then on the same timeline even when there are 2 different positions. The 2 positions are on the 45 degree Event line on the original position of A. Hence position A and position A’ on the Event line and (t)=0, relocate A’ back to position A. </p><p>Whatever the magnitude of <span class="texhtml mvar" style="font-style:italic;">α</span>, the line <span class="nowrap"><i>ct</i> = <i>x</i></span> forms the universal<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bisection" title="Bisection">bisector</a>, as shown in Fig 2-2. </p><p>One frequently encounters Minkowski diagrams where the time <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> are scaled by a factor of <span class="texhtml"><i>c</i></span> such that one unit of <span class="texhtml"><i>x</i></span> equals one unit of <span class="texhtml"><i>t</i></span>. Such a diagram may have units of </p> <ul><li>Approximately 30 centimetres length and <a href="/wiki/Nanosecond" title="Nanosecond">nanoseconds</a></li> <li><a href="/wiki/Astronomical_unit" title="Astronomical unit">Astronomical units</a> and intervals of about 8 minutes and 19 seconds (499 seconds)</li> <li><a href="/wiki/Light_year" class="mw-redirect" title="Light year">Light years</a> and <a href="/wiki/Year" title="Year">years</a></li> <li><a href="/wiki/Light-second" title="Light-second">Light-second</a> and second</li></ul> <p>With that, light paths are represented by lines parallel to the bisector between the axes. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_details">Mathematical details</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=7" title="Edit section: Mathematical details"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MinkScale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/MinkScale.svg/256px-MinkScale.svg.png" decoding="async" width="256" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/MinkScale.svg/384px-MinkScale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/MinkScale.svg/512px-MinkScale.svg.png 2x" data-file-width="1110" data-file-height="1106" /></a><figcaption>Fig 2-3 Different scales on the axes.</figcaption></figure> <p>The angle <span class="texhtml"><i>α</i></span> between the <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>x</i>′</span> axes will be identical with that between the time axes <span class="texhtml"><i>ct</i></span> and <span class="texhtml"><i>ct</i>′</span>. This follows from the second postulate of special relativity, which says that the speed of light is the same for all observers, regardless of their relative motion (see below). The angle <span class="texhtml"><i>α</i></span> is given by<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \alpha ={\frac {v}{c}}=\beta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {v}{c}}=\beta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5944ae23a900cf71d194bb35283f7509ba2bfd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.374ex; height:4.676ex;" alt="{\displaystyle \tan \alpha ={\frac {v}{c}}=\beta .}"></span> </p><p>The corresponding boost from <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>t</i></span> to <span class="texhtml"><i>x</i>′</span> and <span class="texhtml"><i>t</i>′</span> and vice versa is described mathematically by the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>, which can be written </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}ct'&=\gamma (ct-\beta x),\\x'&=\gamma (x-\beta ct)\\\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>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>β</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </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>β</mi> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}ct'&=\gamma (ct-\beta x),\\x'&=\gamma (x-\beta ct)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e9d29f0b456f53791b5416940cef56f046bfe3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.448ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}ct'&=\gamma (ct-\beta x),\\x'&=\gamma (x-\beta ct)\\\end{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</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>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40c4d3b044dcfbea954b7cc2934a40b9393c835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.899ex; height:4.343ex;" alt="{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}"></span> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>. By applying the Lorentz transformation, the spacetime axes obtained for a boosted frame will always correspond to <a href="/wiki/Conjugate_diameters" title="Conjugate diameters">conjugate diameters</a> of a pair of <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolas</a>. </p><p>As illustrated in Fig 2-3, the boosted and unboosted spacetime axes will in general have unequal unit lengths. If <span class="texhtml"><i>U</i></span> is the unit length on the axes of <span class="texhtml"><i>ct</i></span> and <span class="texhtml"><i>x</i></span> respectively, the unit length on the axes of <span class="texhtml"><i>ct</i>′</span> and <span class="texhtml"><i>x</i>′</span> is:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U'=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U'=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f7d35a63d076fb48ea14cf1a40517914a023b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.995ex; height:7.509ex;" alt="{\displaystyle U'=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}\,.}"></span> </p><p>The <span class="texhtml"><i>ct</i></span>-axis represents the worldline of a clock resting in <span class="texhtml"><i>S</i></span>, with <span class="texhtml"><i>U</i></span> representing the duration between two events happening on this worldline, also called the <a href="/wiki/Proper_time" title="Proper time">proper time</a> between these events. Length <span class="texhtml"><i>U</i></span> upon the <span class="texhtml"><i>x</i></span>-axis represents the rest length or <a href="/wiki/Proper_length" title="Proper length">proper length</a> of a rod resting in <span class="texhtml"><i>S</i></span>. The same interpretation can also be applied to distance <span class="texhtml"><i>U</i>′</span> upon the <span class="texhtml"><i>ct</i>′</span>- and <span class="texhtml"><i>x</i>′</span>-axes for clocks and rods resting in <span class="texhtml"><i>S</i>′</span>. </p> <div class="mw-heading mw-heading3"><h3 id="History">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=8" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Minkowski2.png/220px-Minkowski2.png" decoding="async" width="220" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Minkowski2.png/330px-Minkowski2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Minkowski2.png/440px-Minkowski2.png 2x" data-file-width="1436" data-file-height="500" /></a><figcaption>Light cone and hyperbolas in Minkowski (1908)</figcaption></figure> <p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> announced his theory of special relativity in 1905,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> with <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> providing his graphical representation in 1908.<sup id="cite_ref-minko_8-0" class="reference"><a href="#cite_note-minko-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>In Minkowski's 1908 paper there were three diagrams, first to illustrate the Lorentz transformation, then the partition of the plane by the light-cone, and finally illustration of worldlines.<sup id="cite_ref-minko_8-1" class="reference"><a href="#cite_note-minko-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The first diagram used a branch of the <a href="/wiki/Unit_hyperbola" title="Unit hyperbola">unit hyperbola</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t^{2}-x^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t^{2}-x^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ae6dfe241f930a69854337fd7bc74820a033b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.379ex; height:2.676ex;" alt="{\textstyle t^{2}-x^{2}=1}"></span> to show the locus of a unit of proper time depending on velocity, thus illustrating time dilation. The second diagram showed the conjugate hyperbola to calibrate space, where a similar stretching leaves the impression of <a href="/wiki/FitzGerald_contraction" class="mw-redirect" title="FitzGerald contraction">FitzGerald contraction</a>. In 1914 <a href="/wiki/Ludwik_Silberstein" title="Ludwik Silberstein">Ludwik Silberstein</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> included a diagram of "Minkowski's representation of the Lorentz transformation". This diagram included the unit hyperbola, <a href="/wiki/Conjugate_hyperbola" title="Conjugate hyperbola">its conjugate</a>, and a pair of <a href="/wiki/Conjugate_diameters" title="Conjugate diameters">conjugate diameters</a>. Since the 1960s a version of this more complete configuration has been referred to as The Minkowski Diagram, and used as a standard illustration of the <a href="/wiki/Transformation_geometry" title="Transformation geometry">transformation geometry</a> of special relativity. <a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a> has pointed out that the <a href="/wiki/Principle_of_relativity" title="Principle of relativity">principle of relativity</a> is tantamount to the arbitrariness of what hyperbola radius is selected for <a href="/wiki/Time" title="Time">time</a> in the Minkowski diagram. In 1912 <a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Gilbert N. Lewis</a> and <a href="/wiki/Edwin_B._Wilson" class="mw-redirect" title="Edwin B. Wilson">Edwin B. Wilson</a> applied the methods of <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a> to develop the properties of the <a href="/wiki/Non-Euclidean" class="mw-redirect" title="Non-Euclidean">non-Euclidean</a> plane that has Minkowski diagrams.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability#Self-published_sources" title="Wikipedia:Verifiability"><span title="The material near this tag may rely on a self-published source. (December 2024)">self-published source?</span></a></i>]</sup> </p><p>When Taylor and Wheeler composed <i>Spacetime Physics</i> (1966), they did <i>not</i> use the term <i>Minkowski diagram</i> for their spacetime geometry. Instead they included an acknowledgement of Minkowski's contribution to philosophy by the totality of his innovation of 1908.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Loedel_diagrams">Loedel diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=9" title="Edit section: Loedel diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While a frame at rest in a Minkowski diagram has orthogonal spacetime axes, a frame moving relative to the rest frame in a Minkowski diagram has spacetime axes which form an acute angle. This asymmetry of Minkowski diagrams can be misleading, since <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> postulates that any two <a href="/wiki/Inertial_reference_frames" class="mw-redirect" title="Inertial reference frames">inertial reference frames</a> must be physically equivalent. The Loedel diagram is an alternative spacetime diagram that makes the symmetry of inertial references frames much more manifest. </p> <div class="mw-heading mw-heading3"><h3 id="Formulation_via_median_frame">Formulation via median frame</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=10" title="Edit section: Formulation via median frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:254px;max-width:254px"><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:MinkScale3.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MinkScale3.svg/250px-MinkScale3.svg.png" decoding="async" width="250" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MinkScale3.svg/375px-MinkScale3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MinkScale3.svg/500px-MinkScale3.svg.png 2x" data-file-width="1109" data-file-height="1097" /></a></span></div><div class="thumbcaption">Fig. 3-1: View in the median frame</div></div></div><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:MinkScale2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/MinkScale2.svg/250px-MinkScale2.svg.png" decoding="async" width="250" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/MinkScale2.svg/375px-MinkScale2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/MinkScale2.svg/500px-MinkScale2.svg.png 2x" data-file-width="1109" data-file-height="1097" /></a></span></div><div class="thumbcaption">Fig. 3-2: Symmetric diagram</div></div></div></div></div> <p>Several authors showed that there is a frame of reference between the resting and moving ones where their symmetry would be apparent ("median frame").<sup id="cite_ref-mirimanoff_13-0" class="reference"><a href="#cite_note-mirimanoff-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In this frame, the two other frames are moving in opposite directions with equal speed. Using such coordinates makes the units of length and time the same for both axes. If <span class="texhtml"><i>β</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>v</i></span><span class="sr-only">/</span><span class="den"><i>c</i></span></span>⁠</span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</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>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40c4d3b044dcfbea954b7cc2934a40b9393c835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.899ex; height:4.343ex;" alt="{\textstyle \gamma =\left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}}"></span> are given between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015d672f9826104f9027724cb3e74e100c1fab44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S^{\prime }}"></span>, then these expressions are connected with the values in their median frame <i>S</i><sub>0</sub> as follows:<sup id="cite_ref-mirimanoff_13-1" class="reference"><a href="#cite_note-mirimanoff-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(1)&\beta &={\frac {2\beta _{0}}{1+{\beta _{0}}^{2}}},\\[3pt]&(2)&\beta _{0}&={\frac {\gamma -1}{\beta \gamma }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>β</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(1)&\beta &={\frac {2\beta _{0}}{1+{\beta _{0}}^{2}}},\\[3pt]&(2)&\beta _{0}&={\frac {\gamma -1}{\beta \gamma }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798241c562d277c8c32ee1efae1eb249838adb8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:22.747ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}&(1)&\beta &={\frac {2\beta _{0}}{1+{\beta _{0}}^{2}}},\\[3pt]&(2)&\beta _{0}&={\frac {\gamma -1}{\beta \gamma }}.\end{aligned}}}"></span> </p><p>For instance, if <span class="texhtml"><i>β</i> = 0.5</span> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\textstyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1c88dcfc6dc0db92b8e088dc681ad144c4ff64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.343ex;" alt="{\textstyle S^{\prime }}"></span>, then by (2) they are moving in their median frame <i>S</i><sub>0</sub> with approximately <span class="texhtml">±0.268<i>c</i></span> each in opposite directions. On the other hand, if <span class="texhtml"><i>β</i><sub>0</sub> = 0.5</span> in <i>S</i><sub>0</sub>, then by (1) the relative velocity between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015d672f9826104f9027724cb3e74e100c1fab44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S^{\prime }}"></span> in their own rest frames is <span class="texhtml">0.8<i>c</i></span>. The construction of the axes of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\textstyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1c88dcfc6dc0db92b8e088dc681ad144c4ff64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.343ex;" alt="{\textstyle S^{\prime }}"></span> is done in accordance with the ordinary method using <span class="texhtml">tan <i>α</i> = <i>β</i><sub>0</sub></span> with respect to the orthogonal axes of the median frame (Fig. 3–1). </p><p>However, it turns out that when drawing such a symmetric diagram, it is possible to derive the diagram's relations even without mentioning the median frame and <span class="texhtml"><i>β</i><sub>0</sub></span> at all. Instead, the relative velocity <span class="texhtml"><i>β</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>v</i></span><span class="sr-only">/</span><span class="den"><i>c</i></span></span>⁠</span></span> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\textstyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1c88dcfc6dc0db92b8e088dc681ad144c4ff64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.343ex;" alt="{\textstyle S^{\prime }}"></span> can directly be used in the following construction, providing the same result:<sup id="cite_ref-sartori_15-0" class="reference"><a href="#cite_note-sartori-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml"><i>φ</i></span> is the angle between the axes of <span class="texhtml"><i>ct</i>′</span> and <span class="texhtml"><i>ct</i></span> (or between <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>x</i>′</span>), and <span class="texhtml"><i>θ</i></span> between the axes of <span class="texhtml"><i>x</i>′</span> and <span class="texhtml"><i>ct</i>′</span>, it is given:<sup id="cite_ref-sartori_15-1" class="reference"><a href="#cite_note-sartori-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner1_16-0" class="reference"><a href="#cite_note-gruner1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner2_17-0" class="reference"><a href="#cite_note-gruner2-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-shado_18-0" class="reference"><a href="#cite_note-shado-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \varphi =\cos \theta &=\beta ,\\\cos \varphi =\sin \theta &={\frac {1}{\gamma }},\\\tan \varphi =\cot \theta &=\beta \gamma .\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>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>β</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>β</mi> <mi>γ</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \varphi =\cos \theta &=\beta ,\\\cos \varphi =\sin \theta &={\frac {1}{\gamma }},\\\tan \varphi =\cot \theta &=\beta \gamma .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531bd4f7205bc01e87ea9900aadd4176bb0995b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.092ex; margin-bottom: -0.246ex; width:20.033ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\sin \varphi =\cos \theta &=\beta ,\\\cos \varphi =\sin \theta &={\frac {1}{\gamma }},\\\tan \varphi =\cot \theta &=\beta \gamma .\end{aligned}}}"></span> </p><p>Two methods of construction are obvious from Fig. 3-2: the <span class="texhtml"><i>x</i></span>-axis is drawn perpendicular to the <span class="texhtml"><i>ct</i>′</span>-axis, the <span class="texhtml"><i>x</i>′</span> and <span class="texhtml"><i>ct</i></span>-axes are added at angle <span class="texhtml"><i>φ</i></span>; and the <i>x</i>′-axis is drawn at angle <span class="texhtml"><i>θ</i></span> with respect to the <span class="texhtml"><i>ct</i>′</span>-axis, the <span class="texhtml"><i>x</i></span>-axis is added perpendicular to the <span class="texhtml"><i>ct</i>′</span>-axis and the <span class="texhtml"><i>ct</i></span>-axis perpendicular to the <span class="texhtml"><i>x</i>′</span>-axis. </p><p>In a Minkowski diagram, lengths on the page cannot be directly compared to each other, due to warping factor between the axes' unit lengths in a Minkowski diagram. In particular, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087a7d8a39fe35012cbf2f561879f9e975cb4555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\textstyle U}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle U^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle U^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2883cbea553342ae488f3e4753e3d805f4e0f271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.526ex; height:2.343ex;" alt="{\textstyle U^{\prime }}"></span> are the unit lengths of the rest frame axes and moving frame axes, respectively, in a Minkowski diagram, then the two unit lengths are warped relative to each other via the formula: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{\prime }=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{\prime }=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f6fce279efeff5222d1e8d75a3ebd548bd3c1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.961ex; height:7.509ex;" alt="{\displaystyle U^{\prime }=U{\sqrt {\frac {1+\beta ^{2}}{1-\beta ^{2}}}}}"></span> </p><p>By contrast, in a symmetric Loedel diagram, both the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\textstyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1c88dcfc6dc0db92b8e088dc681ad144c4ff64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.343ex;" alt="{\textstyle S^{\prime }}"></span> frame axes are warped by the same factor relative to the median frame and hence have identical unit lengths. This implies that, for a Loedel spacetime diagram, we can directly compare spacetime lengths between different frames as they appear on the page; no unit length scaling/conversion between frames is necessary due to the symmetric nature of the Loedel diagram. </p> <div class="mw-heading mw-heading3"><h3 id="History_2">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=11" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Max_Born" title="Max Born">Max Born</a> (1920) drew Minkowski diagrams by placing the <span class="texhtml"><i>ct</i>′</span>-axis almost perpendicular to the <span class="texhtml"><i>x</i></span>-axis, as well as the <span class="texhtml"><i>ct</i></span>-axis to the <span class="texhtml"><i>x</i>′</span>-axis, in order to demonstrate length contraction and time dilation in the symmetric case of two rods and two clocks moving in opposite direction.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Dmitry_Mirimanoff" title="Dmitry Mirimanoff">Dmitry Mirimanoff</a> (1921) showed that there is always a median frame with respect to two relatively moving frames, and derived the relations between them from the Lorentz transformation. However, he did not give a graphical representation in a diagram.<sup id="cite_ref-mirimanoff_13-2" class="reference"><a href="#cite_note-mirimanoff-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li>Symmetric diagrams were systematically developed by <a href="/wiki/Paul_Gruner" title="Paul Gruner">Paul Gruner</a> in collaboration with Josef Sauter in two papers in 1921. Relativistic effects such as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them.<sup id="cite_ref-gruner1_16-1" class="reference"><a href="#cite_note-gruner1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner2_17-1" class="reference"><a href="#cite_note-gruner2-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Gruner extended this method in subsequent papers (1922–1924), and gave credit to Mirimanoff's treatment as well.<sup id="cite_ref-gruner3_20-0" class="reference"><a href="#cite_note-gruner3-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner4_21-0" class="reference"><a href="#cite_note-gruner4-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner5_22-0" class="reference"><a href="#cite_note-gruner5-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner6_23-0" class="reference"><a href="#cite_note-gruner6-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner7_24-0" class="reference"><a href="#cite_note-gruner7-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gruner8_25-0" class="reference"><a href="#cite_note-gruner8-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>The construction of symmetric Minkowski diagrams was later independently rediscovered by several authors. For instance, starting in 1948, <a href="/wiki/Enrique_Loedel_Palumbo" title="Enrique Loedel Palumbo">Enrique Loedel Palumbo</a> published a series of papers in Spanish language, presenting the details of such an approach.<sup id="cite_ref-loed48_26-0" class="reference"><a href="#cite_note-loed48-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> In 1955, <a href="/w/index.php?title=Henri_Amar&action=edit&redlink=1" class="new" title="Henri Amar (page does not exist)">Henri Amar</a> also published a paper presenting such relations, and gave credit to Loedel in a subsequent paper in 1957.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Some authors of <a href="/wiki/Textbook" title="Textbook">textbooks</a> use symmetric Minkowski diagrams, denoting as <i>Loedel diagrams</i>.<sup id="cite_ref-sartori_15-2" class="reference"><a href="#cite_note-sartori-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-shado_18-1" class="reference"><a href="#cite_note-shado-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Relativistic_phenomena_in_diagrams">Relativistic phenomena in diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=12" title="Edit section: Relativistic phenomena in diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Time_dilation">Time dilation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=13" title="Edit section: Time dilation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LoedelTD.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/LoedelTD.png/400px-LoedelTD.png" decoding="async" width="400" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/LoedelTD.png/600px-LoedelTD.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/LoedelTD.png/800px-LoedelTD.png 2x" data-file-width="1894" data-file-height="958" /></a><figcaption>Fig 4–1. Relativistic time dilation, as depicted in two Loedel spacetime diagrams. Both observers consider the clock of the other as running slower.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_time_dilation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Minkowski_diagram_-_time_dilation.svg/240px-Minkowski_diagram_-_time_dilation.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Minkowski_diagram_-_time_dilation.svg/360px-Minkowski_diagram_-_time_dilation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Minkowski_diagram_-_time_dilation.svg/480px-Minkowski_diagram_-_time_dilation.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 4–2. Relativistic time dilation, as depicted in a single Loedel spacetime diagram. Both observers consider the clock of the other as running slower.</figcaption></figure> <p>Relativistic time dilation refers to the fact that a clock (indicating its <a href="/wiki/Proper_time" title="Proper time">proper time</a> in its rest frame) that moves relative to an observer is observed to run slower. The situation is depicted in the symmetric Loedel diagrams of Fig 4-1. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram. </p><p>In Fig 4-2, the observer whose reference frame is given by the black axes is assumed to move from the origin O towards A. The moving clock has the reference frame given by the blue axes and moves from O to B. For the black observer, all events happening simultaneously with the event at A are located on a straight line parallel to its space axis. This line passes through A and B, so A and B are simultaneous from the reference frame of the observer with black axes. However, the clock that is moving relative to the black observer marks off time along the blue time axis. This is represented by the distance from O to B. Therefore, the observer at A with the black axes notices their clock as reading the distance from O to A while they observe the clock moving relative him or her to read the distance from O to B. Due to the distance from O to B being smaller than the distance from O to A, they conclude that the time passed on the clock moving relative to them is smaller than that passed on their own clock. </p><p>A second observer, having moved together with the clock from O to B, will argue that the black axis clock has only reached C and therefore runs slower. The reason for these apparently paradoxical statements is the different determination of the events happening synchronously at different locations. Due to the principle of relativity, the question of who is right has no answer and does not make sense. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Length_contraction">Length contraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=14" title="Edit section: Length contraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LoedelLC2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/LoedelLC2.png/400px-LoedelLC2.png" decoding="async" width="400" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/LoedelLC2.png/600px-LoedelLC2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/LoedelLC2.png/800px-LoedelLC2.png 2x" data-file-width="2290" data-file-height="1204" /></a><figcaption>Fig 4-3 Relativistic length contraction, as depicted in two Loedel spacetime diagrams. Both observers consider objects moving with the other observer as being shorter.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_length_contraction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Minkowski_diagram_-_length_contraction.svg/220px-Minkowski_diagram_-_length_contraction.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Minkowski_diagram_-_length_contraction.svg/330px-Minkowski_diagram_-_length_contraction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Minkowski_diagram_-_length_contraction.svg/440px-Minkowski_diagram_-_length_contraction.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 4-4 Relativistic length contraction, as depicted in a single Loedel spacetime diagram. Both observers consider objects moving with the other observer as being shorter.</figcaption></figure> <p>Relativistic length contraction refers to the fact that a ruler (indicating its <a href="/wiki/Proper_length" title="Proper length">proper length</a> in its rest frame) that moves relative to an observer is observed to contract/shorten. The situation is depicted in symmetric Loedel diagrams in Fig 4-3. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram. </p><p>In Fig 4-4, the observer is assumed again to move along the <span class="texhtml"><i>ct</i></span>-axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the <span class="texhtml"><i>ct</i>′</span>-axis and the parallel line passing through A and B. For this observer the endpoints of the object at <span class="nowrap"><i>t</i> = 0</span> are O and A. For a second observer moving together with the object, so that for him the object is at rest, it has the proper length OB at <span class="nowrap"><i>t</i>′ = 0</span>. Due to <span class="nowrap">OA < OB</span>. the object is contracted for the first observer. </p><p>The second observer will argue that the first observer has evaluated the endpoints of the object at O and A respectively and therefore at different times, leading to a wrong result due to his motion in the meantime. If the second observer investigates the length of another object with endpoints moving along the <span class="texhtml"><i>ct</i></span>-axis and a parallel line passing through C and D he concludes the same way this object to be contracted from OD to OC. Each observer estimates objects moving with the other observer to be contracted. This apparently paradoxical situation is again a consequence of the relativity of simultaneity as demonstrated by the analysis via Minkowski diagram. </p><p>For all these considerations it was assumed, that both observers take into account the speed of light and their distance to all events they see in order to determine the actual times at which these events happen from their point of view. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Constancy_of_the_speed_of_light">Constancy of the speed of light</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=15" title="Edit section: Constancy of the speed of light"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_3_systems.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Minkowski_diagram_-_3_systems.svg/240px-Minkowski_diagram_-_3_systems.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Minkowski_diagram_-_3_systems.svg/360px-Minkowski_diagram_-_3_systems.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Minkowski_diagram_-_3_systems.svg/480px-Minkowski_diagram_-_3_systems.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 4-5 Minkowski diagram for 3 coordinate systems. For the speeds relative to the system in black <span class="texhtml"><i>v</i>′ = 0.4<i>c</i></span> and <span class="texhtml"><i>v</i>″ = 0.8<i>c</i></span> holds.</figcaption></figure><p> Another postulate of special relativity is the constancy of the speed of light. It says that any observer in an inertial reference frame measuring the vacuum speed of light relative to themself obtains the same value regardless of his own motion and that of the light source. This statement seems to be paradoxical, but it follows immediately from the differential equation yielding this, and the Minkowski diagram agrees. It explains also the result of the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> which was considered to be a mystery before the theory of relativity was discovered, when photons were thought to be waves through an undetectable medium. </p><p>For world lines of photons passing the origin in different directions <span class="texhtml"><i>x</i> = <i>ct</i></span> and <span class="texhtml"><i>x</i> = −<i>ct</i></span> holds. That means any position on such a world line corresponds with steps on <span class="texhtml"><i>x</i></span>- and <span class="texhtml"><i>ct</i></span>-axes of equal absolute value. From the rule for reading off coordinates in coordinate system with tilted axes follows that the two world lines are the angle bisectors of the <span class="texhtml"><i>x</i></span>- and <span class="texhtml"><i>ct</i></span>-axes. As shown in Fig 4-5, the Minkowski diagram illustrates them as being angle bisectors of the <span class="texhtml"><i>x′</i></span>- and <span class="texhtml"><i>ct</i>′</span>-axes as well. That means both observers measure the same speed <span class="texhtml"><i>c</i></span> for both photons. </p><p>Further coordinate systems corresponding to observers with arbitrary velocities can be added to this Minkowski diagram. For all these systems both photon world lines represent the angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the corresponding angle bisector. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> axis is always more flat and the time axis more steep than the photon world lines. The scales on both axes are always identical, but usually different from those of the other coordinate systems. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Speed_of_light_and_causality">Speed of light and causality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=16" title="Edit section: Speed of light and causality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_causality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Minkowski_diagram_-_causality.svg/240px-Minkowski_diagram_-_causality.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Minkowski_diagram_-_causality.svg/360px-Minkowski_diagram_-_causality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Minkowski_diagram_-_causality.svg/480px-Minkowski_diagram_-_causality.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 4-6 Past and future relative to the origin. For the grey areas a corresponding temporal classification is not possible.</figcaption></figure> <p>Straight lines passing the origin which are steeper than both photon world lines correspond with objects moving more slowly than the speed of light. If this applies to an object, then it applies from the viewpoint of all observers, because the world lines of these photons are the angle bisectors for any inertial reference frame. Therefore, any point above the origin and between the world lines of both photons can be reached with a speed smaller than that of the light and can have a cause-and-effect relationship with the origin. This area is the absolute future, because any event there happens later compared to the event represented by the origin regardless of the observer, which is obvious graphically from the Minkowski diagram in Fig 4-6. </p><p>Following the same argument the range below the origin and between the photon world lines is the absolute past relative to the origin. Any event there belongs definitely to the past and can be the cause of an effect at the origin. </p><p>The relationship between any such pairs of event is called <i>timelike</i>, because they have a time distance greater than zero for all observers. A straight line connecting these two events is always the time axis of a possible observer for whom they happen at the same place. Two events which can be connected just with the speed of light are called <i>lightlike</i>. </p><p>In principle a further dimension of space can be added to the Minkowski diagram leading to a three-dimensional representation. In this case the ranges of future and past become <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cones</a> with apexes touching each other at the origin. They are called <a href="/wiki/Light_cones" class="mw-redirect" title="Light cones">light cones</a>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="The_speed_of_light_as_a_limit">The speed of light as a limit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=17" title="Edit section: The speed of light as a limit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_diagram_-_time_travel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Minkowski_diagram_-_time_travel.svg/240px-Minkowski_diagram_-_time_travel.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Minkowski_diagram_-_time_travel.svg/360px-Minkowski_diagram_-_time_travel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Minkowski_diagram_-_time_travel.svg/480px-Minkowski_diagram_-_time_travel.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Fig 4-7 Sending a message at superluminal speed from O via A to B into the past. Both observers consider the temporal order of the pairs of events O and A as well as A and B different.</figcaption></figure> <p>Following the same argument, all straight lines passing through the origin and which are more nearly horizontal than the photon world lines, would correspond to objects or signals moving <a href="/wiki/Faster-than-light" title="Faster-than-light">faster than light</a> regardless of the speed of the observer. Therefore, no event outside the light cones can be reached from the origin, even by a light-signal, nor by any object or signal moving with less than the speed of light. Such pairs of events are called <i>spacelike</i> because they have a finite spatial distance different from zero for all observers. On the other hand, a straight line connecting such events is always the space coordinate axis of a possible observer for whom they happen at the same time. By a slight variation of the velocity of this coordinate system in both directions it is always possible to find two inertial reference frames whose observers estimate the chronological order of these events to be different. </p><p>Given an object moving faster than light, say from O to A in Fig 4-7, then for any observer watching the object moving from O to A, another observer can be found (moving at less than the speed of light with respect to the first) for whom the object moves from A to O. The question of which observer is right has no unique answer, and therefore makes no physical sense. Any such moving object or signal would violate the principle of causality. </p><p>Also, any general technical means of sending signals faster than light would permit information to be sent into the originator's own past. In the diagram, an observer at O in the <span class="texhtml"><i>x</i>-<i>ct</i></span> system sends a message moving faster than light to A. At A, it is received by another observer, moving so as to be in the <span class="texhtml"><i>x</i>′-<i>ct</i>′</span> system, who sends it back, again faster than light, arriving at B. But B is in the past relative to O. The absurdity of this process becomes obvious when both observers subsequently confirm that they received no message at all, but all messages were directed towards the other observer as can be seen graphically in the Minkowski diagram. Furthermore, if it were possible to accelerate an observer to the speed of light, their space and time axes would coincide with their angle bisector. The coordinate system would collapse, in concordance with the fact that due to <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a>, time would effectively stop passing for them. </p><p>These considerations show that the speed of light as a limit is a consequence of the properties of spacetime, and not of the properties of objects such as technologically imperfect space ships. The prohibition of faster-than-light motion, therefore, has nothing in particular to do with electromagnetic waves or light, but comes as a consequence of the structure of spacetime. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Accelerating_observers">Accelerating observers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=18" title="Edit section: Accelerating observers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Momentarily_Comoving_Reference_Frame.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/220px-Momentarily_Comoving_Reference_Frame.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/330px-Momentarily_Comoving_Reference_Frame.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Momentarily_Comoving_Reference_Frame.gif/440px-Momentarily_Comoving_Reference_Frame.gif 2x" data-file-width="501" data-file-height="501" /></a><figcaption>Fig 5-1 The momentarily co-moving reference frames of an accelerating particle as observed from a stationary frame</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lorentz_transform_of_world_line.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Lorentz_transform_of_world_line.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Fig 5-2 The momentarily co-moving inertial frames along the world line of a rapidly accelerating observer (origin).</figcaption></figure> <p>It is often, incorrectly, asserted that special relativity cannot handle accelerating particles or accelerating reference frames. In reality, accelerating particles present no difficulty at all in special relativity. On the other hand, accelerating <i>frames</i> do require some special treatment, However, as long as one is dealing with flat, Minkowskian spacetime, special relativity can handle the situation. It is only in the presence of gravitation that general relativity is required.<sup id="cite_ref-Gibbs1996_30-0" class="reference"><a href="#cite_note-Gibbs1996-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity. This is not a difficult situation to handle. Accelerating frames require that one understand the concept of a <i>momentarily comoving reference frame</i> (MCRF), which is to say, a frame traveling at the same instantaneous velocity of a particle at any given instant. </p><p>Consider the animation in Fig 5–1. The curved line represents the world line of a particle that undergoes continuous acceleration, including complete changes of direction in the positive and negative <i>x</i>-directions. The red axes are the axes of the MCRF for each point along the particle's trajectory. The coordinates of events in the unprimed (stationary) frame can be related to their coordinates in any momentarily co-moving primed frame using the Lorentz transformations. </p><p>Fig 5-2 illustrates the changing views of spacetime along the world line of a rapidly accelerating particle. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle ct'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle ct'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd8e34917cbee2b605a05b3953b7d69b432deaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.343ex;" alt="{\textstyle ct'}"></span> axis (not drawn) is vertical, while the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd4bbf7bc3c37c71d0a1f9e4ef6c504c8f9d5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.343ex;" alt="{\textstyle x'}"></span> axis (not drawn) is horizontal. The dashed line is the spacetime trajectory ("world line") of the particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and they intersect at that event. The small dots are other arbitrary events in the spacetime. </p><p>The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. Bends in the world line represent particle acceleration. As the particle accelerates, its view of spacetime changes. These changes in view are governed by the Lorentz transformations. Also note that: </p> <ul><li>the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.</li> <li>events which were simultaneous before an acceleration (horizontally spaced events) are at different times afterwards due to the relativity of simultaneity,</li> <li>events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and</li> <li>the world line always remains within the future and past light cones of the current event.</li></ul> <p>If one imagines each event to be the flashing of a light, then the events that are within the past light cone of the observer are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the velocity relative to the observer. </p> <div class="mw-heading mw-heading2"><h2 id="Case_of_non-inertial_reference_frames">Case of non-inertial reference frames</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=19" title="Edit section: Case of non-inertial reference frames"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:MinkowskiDiagram-Inertial-FreeFallRocket.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/MinkowskiDiagram-Inertial-FreeFallRocket.png/220px-MinkowskiDiagram-Inertial-FreeFallRocket.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/MinkowskiDiagram-Inertial-FreeFallRocket.png/330px-MinkowskiDiagram-Inertial-FreeFallRocket.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/MinkowskiDiagram-Inertial-FreeFallRocket.png/440px-MinkowskiDiagram-Inertial-FreeFallRocket.png 2x" data-file-width="1025" data-file-height="1000" /></a><figcaption>Fig 6-1 Minkowski diagram in an inertial reference frame. On the left, the vertical world line of the falling object. On the right, the hyperbolic world line of the rocket.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MinkowskiDiagram-NonInertial-FreeFallRocket.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinkowskiDiagram-NonInertial-FreeFallRocket.png/220px-MinkowskiDiagram-NonInertial-FreeFallRocket.png" decoding="async" width="220" height="368" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinkowskiDiagram-NonInertial-FreeFallRocket.png/330px-MinkowskiDiagram-NonInertial-FreeFallRocket.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinkowskiDiagram-NonInertial-FreeFallRocket.png/440px-MinkowskiDiagram-NonInertial-FreeFallRocket.png 2x" data-file-width="800" data-file-height="1339" /></a><figcaption>Fig 6-2 Minkowski diagram in a non-inertial reference frame. On the left, the world line of the falling object. On the right, the vertical world line of the rocket.</figcaption></figure> <p>The photon world lines are determined using the metric with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d\tau =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d\tau =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a976c9888708ee39df5df5e1e8d56864d3fe6202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.679ex; height:2.176ex;" alt="{\textstyle d\tau =0}"></span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Light_cone" title="Light cone">light cones</a> are deformed according to the position. In an inertial reference frame a free particle has a straight world line. In a <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frame</a> the world line of a free particle is curved. </p><p>Take the example of the fall of an object dropped without initial velocity from a rocket. The rocket has a uniformly accelerated motion with respect to an inertial reference frame. As can be seen from Fig 6-2 of a Minkowski diagram in a non-inertial reference frame, the object once dropped, gains speed, reaches a maximum, and then sees its speed decrease and asymptotically cancel on the <a href="/wiki/Event_horizon" title="Event horizon">horizon</a> where its proper time freezes at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901d6e7c875a4eadb0d260f7b1afdf7bd2d3a5bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.304ex; height:2.343ex;" alt="{\textstyle t_{\text{H}}}"></span>. The velocity is measured by an observer at rest in the accelerated rocket. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li></ul> <ul><li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/science/physics/one-dimensional-motion/displacement-velocity-time/a/position-vs-time-graphs">"What are position vs. time graphs?"</a>. Khan Academy<span class="reference-accessdate">. Retrieved <span class="nowrap">19 November</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+are+position+vs.+time+graphs%3F&rft.pub=Khan+Academy&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fscience%2Fphysics%2Fone-dimensional-motion%2Fdisplacement-velocity-time%2Fa%2Fposition-vs-time-graphs&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Collier-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Collier_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollier2017" class="citation book cs1">Collier, Peter (2017). <i>A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity</i> (3rd ed.). Incomprehensible Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780957389465" title="Special:BookSources/9780957389465"><bdi>9780957389465</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Most+Incomprehensible+Thing%3A+Notes+Towards+a+Very+Gentle+Introduction+to+the+Mathematics+of+Relativity&rft.edition=3rd&rft.pub=Incomprehensible+Books&rft.date=2017&rft.isbn=9780957389465&rft.aulast=Collier&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Mermin (1968) Chapter 17</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">See <a href="/wiki/Vladimir_Karapetoff" title="Vladimir Karapetoff">Vladimir Karapetoff</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDemtröder2016" class="citation book cs1">Demtröder, Wolfgang (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TWUSDgAAQBAJ"><i>Mechanics and Thermodynamics</i></a> (illustrated ed.). Springer. pp. 92–93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-27877-3" title="Special:BookSources/978-3-319-27877-3"><bdi>978-3-319-27877-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+and+Thermodynamics&rft.pages=92-93&rft.edition=illustrated&rft.pub=Springer&rft.date=2016&rft.isbn=978-3-319-27877-3&rft.aulast=Demtr%C3%B6der&rft.aufirst=Wolfgang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTWUSDgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TWUSDgAAQBAJ&pg=PA93">Extract of page 93</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreund2008" class="citation book cs1">Freund, Jürgen (2008). <i>Special Relativity for Beginners: A Textbook for Undergraduates</i>. World Scientific. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9812771599" title="Special:BookSources/978-9812771599"><bdi>978-9812771599</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+Relativity+for+Beginners%3A+A+Textbook+for+Undergraduates&rft.pages=49&rft.pub=World+Scientific&rft.date=2008&rft.isbn=978-9812771599&rft.aulast=Freund&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1905" class="citation journal cs1">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf">"Zur Elektrodynamik bewegter Körper"</a> [On the electrodynamics of moving bodies] <span class="cs1-format">(PDF)</span>. <i>Annalen der Physik</i>. <b>322</b> (10): 891–921. <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/1905AnP...322..891E">1905AnP...322..891E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053221004">10.1002/andp.19053221004</a></span>.</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=Zur+Elektrodynamik+bewegter+K%C3%B6rper&rft.volume=322&rft.issue=10&rft.pages=891-921&rft.date=1905&rft_id=info%3Adoi%2F10.1002%2Fandp.19053221004&rft_id=info%3Abibcode%2F1905AnP...322..891E&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=http%3A%2F%2Fwww.physik.uni-augsburg.de%2Fannalen%2Fhistory%2Feinstein-papers%2F1905_17_891-921.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span>. See also: <a rel="nofollow" class="external text" href="http://www.fourmilab.ch/etexts/einstein/specrel/">English translation</a>.</span> </li> <li id="cite_note-minko-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-minko_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-minko_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinkowski1909" class="citation journal cs1">Minkowski, Hermann (1909). <span class="cs1-ws-icon" title="s:de:Raum und Zeit (Minkowski)"><a class="external text" href="https://en.wikisource.org/wiki/de:Raum_und_Zeit_(Minkowski)">"Raum und Zeit" </a></span> [Space and time]. <i>Physikalische Zeitschrift</i>. <b>10</b>: 75–88.</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=Raum+und+Zeit&rft.volume=10&rft.pages=75-88&rft.date=1909&rft.aulast=Minkowski&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> Various English translations on Wikisource: <a href="https://en.wikisource.org/wiki/Space_and_Time" class="extiw" title="s:Space and Time">Space and Time</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilberstein1914" class="citation book cs1">Silberstein, Ludwik (1914). <a rel="nofollow" class="external text" href="https://archive.org/details/theoryofrelativi00silbrich"><i>The Theory of Relativity</i></a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryofrelativi00silbrich/page/131">131</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Relativity&rft.pages=131&rft.date=1914&rft.aulast=Silberstein&rft.aufirst=Ludwik&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryofrelativi00silbrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilsonLewis1912" class="citation journal cs1"><a href="/wiki/Edwin_B._Wilson" class="mw-redirect" title="Edwin B. Wilson">Wilson, Edwin B.</a>; <a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis, Gilbert N.</a> (1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics". <i>Proceedings of the American Academy of Arts and Sciences</i>. <b>48</b> (11): 387–507. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F20022840">10.2307/20022840</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20022840">20022840</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=The+Space-time+Manifold+of+Relativity.+The+Non-Euclidean+Geometry+of+Mechanics+and+Electromagnetics&rft.volume=48&rft.issue=11&rft.pages=387-507&rft.date=1912&rft_id=info%3Adoi%2F10.2307%2F20022840&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20022840%23id-name%3DJSTOR&rft.aulast=Wilson&rft.aufirst=Edwin+B.&rft.au=Lewis%2C+Gilbert+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ca.geocities.com/cocklebio/synsptm.html">Synthetic Spacetime</a>, a digest of the axioms used, and theorems proved, by Wilson and Lewis. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091027012400/http://ca.geocities.com/cocklebio/synsptm.html">Archived</a> 2009-10-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorWheeler1966" class="citation book cs1">Taylor, Edwin F.; Wheeler (1966). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/spacetimephysics0000tayl"><i>Spacetime physics</i></a></span>. San Francisco: W. H. Freeman. p. 37. <q>Minkowski's insight is central to the understanding of the physical world. It focuses attention on those quantities, such as interval, which are the same in all frames of reference. It brings out the relative character of quantities, such as velocity, energy, time, distance, which depend on the frame of reference.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+physics&rft.place=San+Francisco&rft.pages=37&rft.pub=W.+H.+Freeman&rft.date=1966&rft.aulast=Taylor&rft.aufirst=Edwin+F.&rft.au=Wheeler&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspacetimephysics0000tayl&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-mirimanoff-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-mirimanoff_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mirimanoff_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mirimanoff_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMirimanoff1921" class="citation journal cs1 cs1-prop-foreign-lang-source">Mirimanoff, Dmitry (1921). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f682.image">"La transformation de Lorentz-Einstein et le temps universel de M. Ed. Guillaume"</a>. <i>Archives des sciences physiques et naturelles (Supplement)</i>. 5 (in French). <b>3</b>: 46–48.</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+%28Supplement%29&rft.atitle=La+transformation+de+Lorentz-Einstein+et+le+temps+universel+de+M.+Ed.+Guillaume&rft.volume=3&rft.pages=46-48&rft.date=1921&rft.aulast=Mirimanoff&rft.aufirst=Dmitry&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k2991536%2Ff682.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> (Translation: <a href="https://en.wikisource.org/wiki/en:Translation:The_Lorentz-Einstein_transformation_and_the_universal_time_of_Ed._Guillaume" class="extiw" title="s:en:Translation:The Lorentz-Einstein transformation and the universal time of Ed. Guillaume">The Lorentz–Einstein transformation and the universal time of Ed. Guillaume</a>)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShadowitz2012" class="citation book cs1">Shadowitz, Albert (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=31hjdifsTeQC&pg=PA460&dq=%22distortion%22"><i>The Electromagnetic Field</i></a> (1975 reprinted ed.). Courier Dover Publications. p. 460. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486132013" title="Special:BookSources/978-0486132013"><bdi>978-0486132013</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Electromagnetic+Field&rft.pages=460&rft.edition=1975+reprinted&rft.pub=Courier+Dover+Publications&rft.date=2012&rft.isbn=978-0486132013&rft.aulast=Shadowitz&rft.aufirst=Albert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D31hjdifsTeQC%26pg%3DPA460%26dq%3D%2522distortion%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-sartori-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-sartori_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sartori_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sartori_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSartori1996" class="citation book cs1">Sartori, Leo (1996). <i>Understanding Relativity: A simplified approach to Einstein's theories</i>. University of California Press. pp. 151ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-520-20029-2" title="Special:BookSources/0-520-20029-2"><bdi>0-520-20029-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Relativity%3A+A+simplified+approach+to+Einstein%27s+theories&rft.pages=151ff&rft.pub=University+of+California+Press&rft.date=1996&rft.isbn=0-520-20029-2&rft.aulast=Sartori&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-gruner1-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-gruner1_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gruner1_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrunerSauter1921" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul; Sauter, Josef (1921). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f295.image">"Représentation géométrique élémentaire des formules de la théorie de la relativité"</a> [Elementary geometric representation of the formulae of the theory of relativity]. <i>Archives des sciences physiques et naturelles</i>. 5 (in French). <b>3</b>: 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.aulast=Gruner&rft.aufirst=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%3ASpacetime+diagram" class="Z3988"></span> (Translation: <a href="https://en.wikisource.org/wiki/en:Translation:Elementary_geometric_representation_of_the_formulas_of_the_special_theory_of_relativity" class="extiw" title="s:en:Translation:Elementary geometric representation of the formulas of the special theory of relativity">Elementary geometric representation of the formulas of the special theory of relativity</a>)</span> </li> <li id="cite_note-gruner2-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-gruner2_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gruner2_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1921" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1921). "Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie" [An elementary geometric representation of the transformation formulae of the special theory of relativity]. <i>Physikalische Zeitschrift</i> (in German). <b>22</b>: 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.aulast=Gruner&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> (translation: <a href="https://en.wikisource.org/wiki/Translation:An_elementary_geometrical_representation_of_the_transformation_formulas_of_the_special_theory_of_relativity" class="extiw" title="s:Translation:An elementary geometrical representation of the transformation formulas of the special theory of relativity">An elementary geometrical representation of the transformation formulas of the special theory of relativity</a>)</span> </li> <li id="cite_note-shado-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-shado_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-shado_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShadowitz1988" class="citation book cs1">Shadowitz, Albert (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/specialrelativit0000shad"><i>Special Relativity</i></a></span> (reprinted 1968 ed.). Courier Dover. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/specialrelativit0000shad/page/20">20–22</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65743-4" title="Special:BookSources/0-486-65743-4"><bdi>0-486-65743-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+Relativity&rft.pages=20-22&rft.edition=reprinted+1968&rft.pub=Courier+Dover&rft.date=1988&rft.isbn=0-486-65743-4&rft.aulast=Shadowitz&rft.aufirst=Albert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspecialrelativit0000shad&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorn1920" class="citation book cs1 cs1-prop-foreign-lang-source">Born, Max (1920). <a rel="nofollow" class="external text" href="http://catalog.hathitrust.org/Record/006663730"><i>Die Relativitätstheorie Einsteins</i></a> [<i>Einstein's Theory of Relativity</i>]. Naturwissenschaftliche monographien und lehrbücher (in German). Vol. 3 (1st ed.). Springer. pp. 177–180.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Relativit%C3%A4tstheorie+Einsteins&rft.series=Naturwissenschaftliche+monographien+und+lehrb%C3%BCcher&rft.pages=177-180&rft.edition=1st&rft.pub=Springer&rft.date=1920&rft.aulast=Born&rft.aufirst=Max&rft_id=http%3A%2F%2Fcatalog.hathitrust.org%2FRecord%2F006663730&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> See also <a rel="nofollow" class="external text" href="https://books.google.com/books?id=de_6AAAAQBAJ&pg=PA187">Reprint (2013) of third edition (1922) at Google books, p. 187</a></span> </li> <li id="cite_note-gruner3-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner3_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1922" class="citation book cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1922). <i>Elemente der Relativitätstheorie</i> [<i>Elements of the theory of relativity</i>] (in German). Bern: P. Haupt.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elemente+der+Relativit%C3%A4tstheorie&rft.place=Bern&rft.pub=P.+Haupt&rft.date=1922&rft.aulast=Gruner&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-gruner4-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner4_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1922" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1922). "Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt I" [Graphical representation of the special theory of relativity in the four-dimensional spacetime world I]. <i>Zeitschrift für Physik</i> (in German). <b>10</b> (1): 22–37. <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/1922ZPhy...10...22G">1922ZPhy...10...22G</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%2FBF01332542">10.1007/BF01332542</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:123131527">123131527</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Graphische+Darstellung+der+speziellen+Relativit%C3%A4tstheorie+in+der+vierdimensionalen+Raum-Zeit-Welt+I&rft.volume=10&rft.issue=1&rft.pages=22-37&rft.date=1922&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123131527%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01332542&rft_id=info%3Abibcode%2F1922ZPhy...10...22G&rft.aulast=Gruner&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-gruner5-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner5_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1922" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1922). "Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt II" [Graphical representation of the special theory of relativity in the four-dimensional spacetime world II]. <i>Zeitschrift für Physik</i> (in German). <b>10</b> (1): 227–235. <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/1922ZPhy...10..227G">1922ZPhy...10..227G</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%2FBF01332563">10.1007/BF01332563</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:186220809">186220809</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Graphische+Darstellung+der+speziellen+Relativit%C3%A4tstheorie+in+der+vierdimensionalen+Raum-Zeit-Welt+II&rft.volume=10&rft.issue=1&rft.pages=227-235&rft.date=1922&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186220809%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01332563&rft_id=info%3Abibcode%2F1922ZPhy...10..227G&rft.aulast=Gruner&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-gruner6-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner6_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1921" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1921). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k299154k/f235.image">"a) Représentation graphique de l'univers espace-temps à quatre dimensions. b) Représentation graphique du temps universel dans la théorie de la relativité"</a> [a) Graphical representation of the four-dimensional spacetime universe. b) Graphical representation of universal time in the theory of relativity]. <i>Archives des sciences physiques et naturelles</i>. 5 (in French). <b>4</b>: 234–236.</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=a%29+Repr%C3%A9sentation+graphique+de+l%27univers+espace-temps+%C3%A0+quatre+dimensions.+b%29+Repr%C3%A9sentation+graphique+du+temps+universel+dans+la+th%C3%A9orie+de+la+relativit%C3%A9&rft.volume=4&rft.pages=234-236&rft.date=1921&rft.aulast=Gruner&rft.aufirst=Paul&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k299154k%2Ff235.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> (translation: <a href="https://en.wikisource.org/wiki/en:Translation:Graphical_representation_of_the_four-dimensional_space-time_universe" class="extiw" title="s:en:Translation:Graphical representation of the four-dimensional space-time universe">Graphical representation of the four-dimensional space-time universe</a>)</span> </li> <li id="cite_note-gruner7-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner7_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1922" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1922). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1711705">"Die Bedeutung 'reduzierter' orthogonaler Koordinatensysteme für die Tensoranalysis und die spezielle Relativitätstheorie"</a> [The importance of "reduced" orthogonal coordinate-systems for tensor analysis and the special theory of relativity]. <i>Zeitschrift für Physik</i> (in German). <b>10</b> (1): 236–242. <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/1922ZPhy...10..236G">1922ZPhy...10..236G</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%2FBF01332564">10.1007/BF01332564</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:120593068">120593068</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Die+Bedeutung+%27reduzierter%27+orthogonaler+Koordinatensysteme+f%C3%BCr+die+Tensoranalysis+und+die+spezielle+Relativit%C3%A4tstheorie&rft.volume=10&rft.issue=1&rft.pages=236-242&rft.date=1922&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120593068%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01332564&rft_id=info%3Abibcode%2F1922ZPhy...10..236G&rft.aulast=Gruner&rft.aufirst=Paul&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1711705&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-gruner8-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-gruner8_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGruner1924" class="citation journal cs1 cs1-prop-foreign-lang-source">Gruner, Paul (1924). "Geometrische Darstellungen der speziellen Relativitätstheorie, insbesondere des elektromagnetischen Feldes bewegter Körper" [Geometrich representations of the special theory of relativity, in particular the electromagnetic field of moving bodies]. <i>Zeitschrift für Physik</i> (in German). <b>21</b> (1): 366–371. <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/1924ZPhy...21..366G">1924ZPhy...21..366G</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%2FBF01328285">10.1007/BF01328285</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:121376032">121376032</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Geometrische+Darstellungen+der+speziellen+Relativit%C3%A4tstheorie%2C+insbesondere+des+elektromagnetischen+Feldes+bewegter+K%C3%B6rper&rft.volume=21&rft.issue=1&rft.pages=366-371&rft.date=1924&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121376032%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01328285&rft_id=info%3Abibcode%2F1924ZPhy...21..366G&rft.aulast=Gruner&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-loed48-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-loed48_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoedel1948" class="citation journal cs1 cs1-prop-foreign-lang-source">Loedel, Enrique (1948). <a rel="nofollow" class="external text" href="https://archive.org/details/Loedel1948AberracionYRelatividad">"Aberración y Relatividad"</a> [Aberration and Relativity]. <i>Anales de la Sociedad Cientifica Argentina</i> (in Spanish). <b>145</b>: <a rel="nofollow" class="external text" href="https://archive.org/details/Loedel1948AberracionYRelatividad/page/n4">3</a>–13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Anales+de+la+Sociedad+Cientifica+Argentina&rft.atitle=Aberraci%C3%B3n+y+Relatividad&rft.volume=145&rft.pages=3-13&rft.date=1948&rft.aulast=Loedel&rft.aufirst=Enrique&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FLoedel1948AberracionYRelatividad&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><i><a href="//archive.org/details/LoedelFisicaRelativista1955" class="extiw" title="iarchive:LoedelFisicaRelativista1955">Fisica relativista</a></i>, Kapelusz Editorial, Buenos Aires, Argentina (1955).</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmar1955" class="citation journal cs1">Amar, Henri (1955). "New Geometric Representation of the Lorentz Transformation". <i>American Journal of Physics</i>. <b>23</b> (8): 487–489. <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/1955AmJPh..23..487A">1955AmJPh..23..487A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1934074">10.1119/1.1934074</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=New+Geometric+Representation+of+the+Lorentz+Transformation&rft.volume=23&rft.issue=8&rft.pages=487-489&rft.date=1955&rft_id=info%3Adoi%2F10.1119%2F1.1934074&rft_id=info%3Abibcode%2F1955AmJPh..23..487A&rft.aulast=Amar&rft.aufirst=Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmarLoedel1957" class="citation journal cs1">Amar, Henri; Loedel, Enrique (1957). "Geometric Representation of the Lorentz Transformation". <i>American Journal of Physics</i>. <b>25</b> (5): 326–327. <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/1957AmJPh..25..326A">1957AmJPh..25..326A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1934453">10.1119/1.1934453</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Geometric+Representation+of+the+Lorentz+Transformation&rft.volume=25&rft.issue=5&rft.pages=326-327&rft.date=1957&rft_id=info%3Adoi%2F10.1119%2F1.1934453&rft_id=info%3Abibcode%2F1957AmJPh..25..326A&rft.aulast=Amar&rft.aufirst=Henri&rft.au=Loedel%2C+Enrique&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Gibbs1996-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gibbs1996_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbs" class="citation web cs1">Gibbs, Philip. <a rel="nofollow" class="external text" href="https://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html">"Can Special Relativity Handle Acceleration?"</a>. <i>The Original Usenet Physics FAQ</i>. University of California, Riverside<span class="reference-accessdate">. Retrieved <span class="nowrap">6 November</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Original+Usenet+Physics+FAQ&rft.atitle=Can+Special+Relativity+Handle+Acceleration%3F&rft.aulast=Gibbs&rft.aufirst=Philip&rft_id=https%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FRelativity%2FSR%2Facceleration.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRouaud2021" class="citation book cs1">Rouaud, Mathieu (2021). <a rel="nofollow" class="external text" href="http://www.voyagepourproxima.fr/SR.pdf"><i>Special relativity, a geometric approach: course with exercises and answers followed by the conference "Interstellar travel and antimatter"</i></a> <span class="cs1-format">(PDF)</span>. Querrien: Mathieu Rouaud. p. 534. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-9549309-3-0" title="Special:BookSources/978-2-9549309-3-0"><bdi>978-2-9549309-3-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+relativity%2C+a+geometric+approach%3A+course+with+exercises+and+answers+followed+by+the+conference+%22Interstellar+travel+and+antimatter%22&rft.place=Querrien&rft.pages=534&rft.pub=Mathieu+Rouaud&rft.date=2021&rft.isbn=978-2-9549309-3-0&rft.aulast=Rouaud&rft.aufirst=Mathieu&rft_id=http%3A%2F%2Fwww.voyagepourproxima.fr%2FSR.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrench1968" class="citation book cs1"><a href="/wiki/Anthony_French" title="Anthony French">French, A. P.</a> (1968). <i>Special relativity</i>. The MIT introductory physics series. New York: Norton. pp. 82–83. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-17-177075-9" title="Special:BookSources/978-0-17-177075-9"><bdi>978-0-17-177075-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+relativity&rft.place=New+York&rft.series=The+MIT+introductory+physics+series&rft.pages=82-83&rft.pub=Norton&rft.date=1968&rft.isbn=978-0-17-177075-9&rft.aulast=French&rft.aufirst=A.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlass1975" class="citation journal cs1">Glass, E. N. (1975-11-01). <a rel="nofollow" class="external text" href="https://pubs.aip.org/ajp/article/43/11/1013/1049864/Lorentz-boosts-and-Minkowski-diagrams">"Lorentz boosts and Minkowski diagrams"</a>. <i>American Journal of Physics</i>. <b>43</b> (11): 1013–1014. <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/1975AmJPh..43.1013G">1975AmJPh..43.1013G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.9953">10.1119/1.9953</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Lorentz+boosts+and+Minkowski+diagrams&rft.volume=43&rft.issue=11&rft.pages=1013-1014&rft.date=1975-11-01&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.9953&rft_id=info%3Abibcode%2F1975AmJPh..43.1013G&rft.aulast=Glass&rft.aufirst=E.+N.&rft_id=https%3A%2F%2Fpubs.aip.org%2Fajp%2Farticle%2F43%2F11%2F1013%2F1049864%2FLorentz-boosts-and-Minkowski-diagrams&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMermin1968" class="citation book cs1">Mermin, N. David (1968). "17: Minkowski diagrams: The Geometry of Spacetime". <a rel="nofollow" class="external text" href="https://archive.org/details/spacetimeinspeci0000merm"><i>Space and time in special relativity</i></a>. New York: McGraw-Hill. pp. 155–199.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=17%3A+Minkowski+diagrams%3A+The+Geometry+of+Spacetime&rft.btitle=Space+and+time+in+special+relativity&rft.place=New+York&rft.pages=155-199&rft.pub=McGraw-Hill&rft.date=1968&rft.aulast=Mermin&rft.aufirst=N.+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspacetimeinspeci0000merm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRindler2001" class="citation book cs1">Rindler, Wolfgang (2001). <i>Relativity: special, general, and cosmological</i> (2nd ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-856732-5" title="Special:BookSources/978-0-19-856732-5"><bdi>978-0-19-856732-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%3A+special%2C+general%2C+and+cosmological&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=978-0-19-856732-5&rft.aulast=Rindler&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosser1971" class="citation book cs1">Rosser, William G. V. (1971). <i>An introduction to the theory of relativity</i> (3rd improved, revised ed.). London: Butterworth. page 256, Figure 6.4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-408-55700-9" title="Special:BookSources/978-0-408-55700-9"><bdi>978-0-408-55700-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+relativity&rft.place=London&rft.pages=page+256%2C+Figure+6.4&rft.edition=3rd+improved%2C+revised&rft.pub=Butterworth&rft.date=1971&rft.isbn=978-0-408-55700-9&rft.aulast=Rosser&rft.aufirst=William+G.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorWheeler2003" class="citation book cs1">Taylor, Edwin F.; Wheeler, John Archibald (2003). <a rel="nofollow" class="external text" href="https://ia600503.us.archive.org/22/items/SpacetimePhysicsIntroductionToSpecialRelativityTaylorWheelerPDF/Spacetime%20Physics%20-%20Introduction%20to%20Special%20Relativity%20%5BTaylor-Wheeler%5DPDF.pdf"><i>Spacetime physics: introduction to special relativity</i></a> <span class="cs1-format">(PDF)</span> (2nd ed.). New York City: W.H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-2327-1" title="Special:BookSources/978-0-7167-2327-1"><bdi>978-0-7167-2327-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+physics%3A+introduction+to+special+relativity&rft.place=New+York+City&rft.edition=2nd&rft.pub=W.H.+Freeman&rft.date=2003&rft.isbn=978-0-7167-2327-1&rft.aulast=Taylor&rft.aufirst=Edwin+F.&rft.au=Wheeler%2C+John+Archibald&rft_id=https%3A%2F%2Fia600503.us.archive.org%2F22%2Fitems%2FSpacetimePhysicsIntroductionToSpecialRelativityTaylorWheelerPDF%2FSpacetime%2520Physics%2520-%2520Introduction%2520to%2520Special%2520Relativity%2520%255BTaylor-Wheeler%255DPDF.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter1999" class="citation book cs1">Walter, Scott (1999-07-22). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131016142709/http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf">"The non-Euclidean style of Minkowskian relativity"</a> <span class="cs1-format">(PDF)</span>. In Gray, Jeremy (ed.). <a rel="nofollow" class="external text" href="https://academic.oup.com/book/52904/chapter/421936750"><i>The Symbolic Universe</i></a>. Oxford University Press. pp. 91–127. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Foso%2F9780198500889.003.0007">10.1093/oso/9780198500889.003.0007</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850088-9" title="Special:BookSources/978-0-19-850088-9"><bdi>978-0-19-850088-9</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-10-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+non-Euclidean+style+of+Minkowskian+relativity&rft.btitle=The+Symbolic+Universe&rft.pages=91-127&rft.pub=Oxford+University+Press&rft.date=1999-07-22&rft_id=info%3Adoi%2F10.1093%2Foso%2F9780198500889.003.0007&rft.isbn=978-0-19-850088-9&rft.aulast=Walter&rft.aufirst=Scott&rft_id=http%3A%2F%2Fwww.univ-nancy2.fr%2FDepPhilo%2Fwalter%2Fpapers%2Fnes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpacetime+diagram" class="Z3988"></span> (see page 10 of e-link)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spacetime_diagram&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Relativity" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. 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