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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Time dilation caused by a relative velocity</span> </div> </a> <button aria-controls="toc-Time_dilation_caused_by_a_relative_velocity-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 Time dilation caused by a relative velocity subsection</span> </button> <ul id="toc-Time_dilation_caused_by_a_relative_velocity-sublist" class="vector-toc-list"> <li id="toc-Simple_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Simple inference</span> </div> </a> <ul id="toc-Simple_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reciprocity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Reciprocity</span> </div> </a> <ul id="toc-Reciprocity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experimental_testing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experimental_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Experimental testing</span> </div> </a> <ul id="toc-Experimental_testing-sublist" class="vector-toc-list"> <li id="toc-Moving_particles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Moving_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Moving particles</span> </div> </a> <ul id="toc-Moving_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Doppler_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Doppler_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Doppler effect</span> </div> </a> <ul id="toc-Doppler_effect-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proper_time_and_Minkowski_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proper_time_and_Minkowski_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Proper time and Minkowski diagram</span> </div> </a> <ul id="toc-Proper_time_and_Minkowski_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_and_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_and_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Derivation and formulation</span> </div> </a> <ul id="toc-Derivation_and_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbolic_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperbolic_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Hyperbolic motion</span> </div> </a> <ul id="toc-Hyperbolic_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clock_hypothesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clock_hypothesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Clock hypothesis</span> </div> </a> <ul id="toc-Clock_hypothesis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Time_dilation_caused_by_gravity_or_acceleration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Time_dilation_caused_by_gravity_or_acceleration"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Time dilation caused by gravity or acceleration</span> </div> </a> <button aria-controls="toc-Time_dilation_caused_by_gravity_or_acceleration-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 Time dilation caused by gravity or acceleration subsection</span> </button> <ul id="toc-Time_dilation_caused_by_gravity_or_acceleration-sublist" class="vector-toc-list"> <li id="toc-Experimental_testing_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experimental_testing_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Experimental testing</span> </div> </a> <ul id="toc-Experimental_testing_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Combined_effect_of_velocity_and_gravitational_time_dilation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Combined_effect_of_velocity_and_gravitational_time_dilation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Combined effect of velocity and gravitational time dilation</span> </div> </a> <button aria-controls="toc-Combined_effect_of_velocity_and_gravitational_time_dilation-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 Combined effect of velocity and gravitational time dilation subsection</span> </button> <ul id="toc-Combined_effect_of_velocity_and_gravitational_time_dilation-sublist" class="vector-toc-list"> <li id="toc-Experimental_testing_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experimental_testing_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Experimental testing</span> </div> </a> <ul id="toc-Experimental_testing_3-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In popular culture</span> </div> </a> <ul id="toc-In_popular_culture-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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-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-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">10</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" title="Table of Contents" > <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">Time dilation</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" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 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-56" 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">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Tydsrekking" title="Tydsrekking – Afrikaans" lang="af" hreflang="af" data-title="Tydsrekking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Zeitdilatation" title="Zeitdilatation – Alemannic" lang="gsw" hreflang="gsw" data-title="Zeitdilatation" 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%AA%D9%85%D8%AF%D8%AF_%D8%A7%D9%84%D8%B2%D9%85%D9%86" title="تمدد الزمن – Arabic" lang="ar" hreflang="ar" data-title="تمدد الزمن" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A6%AF%E0%A6%BC%E0%A7%87%E0%A6%B0_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%B8%E0%A6%BE%E0%A6%B0%E0%A6%A3" title="সময়ের প্রসারণ – Bangla" lang="bn" hreflang="bn" data-title="সময়ের প্রসারণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BB%D0%B0%D1%82%D0%B8%D0%B2%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%BE_%D0%B7%D0%B0%D0%B1%D0%B0%D0%B2%D1%8F%D0%BD%D0%B5_%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%BC%D0%B5%D1%82%D0%BE" title="Релативистично забавяне на времето – Bulgarian" lang="bg" hreflang="bg" data-title="Релативистично забавяне на времето" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Vremenska_dilatacija" title="Vremenska dilatacija – Bosnian" lang="bs" hreflang="bs" data-title="Vremenska dilatacija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dilataci%C3%B3_del_temps" title="Dilatació del temps – Catalan" lang="ca" hreflang="ca" data-title="Dilatació del 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/Dilatace_%C4%8Dasu" title="Dilatace času – Czech" lang="cs" hreflang="cs" data-title="Dilatace času" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tidsforl%C3%A6ngelse" title="Tidsforlængelse – Danish" lang="da" hreflang="da" data-title="Tidsforlængelse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zeitdilatation" title="Zeitdilatation – German" lang="de" hreflang="de" data-title="Zeitdilatation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CF%83%CF%84%CE%BF%CE%BB%CE%AE_%CF%84%CE%BF%CF%85_%CF%87%CF%81%CF%8C%CE%BD%CE%BF%CF%85" title="Διαστολή του χρόνου – Greek" lang="el" hreflang="el" data-title="Διαστολή του χρόνου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dilataci%C3%B3n_del_tiempo" title="Dilatación del tiempo – Spanish" lang="es" hreflang="es" data-title="Dilatación del tiempo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tempodilati%C4%9Do" title="Tempodilatiĝo – Esperanto" lang="eo" hreflang="eo" data-title="Tempodilatiĝo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Denboraren_zabalkuntza" title="Denboraren zabalkuntza – Basque" lang="eu" hreflang="eu" data-title="Denboraren zabalkuntza" 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/%D8%A7%D8%AA%D8%B3%D8%A7%D8%B9_%D8%B2%D9%85%D8%A7%D9%86" title="اتساع زمان – Persian" lang="fa" hreflang="fa" data-title="اتساع زمان" 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/Dilatation_du_temps" title="Dilatation du temps – French" lang="fr" hreflang="fr" data-title="Dilatation du temps" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Iarmhairt_amleata" title="Iarmhairt amleata – Irish" lang="ga" hreflang="ga" data-title="Iarmhairt amleata" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Dilataci%C3%B3n_do_tempo" title="Dilatación do tempo – Galician" lang="gl" hreflang="gl" data-title="Dilatación do tempo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%9C%EA%B0%84_%ED%8C%BD%EC%B0%BD" title="시간 팽창 – Korean" lang="ko" hreflang="ko" data-title="시간 팽창" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BA%D5%A1%D5%B4%D5%A1%D5%B6%D5%A1%D5%AF%D5%AB_%D5%A4%D5%A1%D5%B6%D5%A4%D5%A1%D5%B2%D5%B8%D6%82%D5%B4" title="Ժամանակի դանդաղում – Armenian" lang="hy" hreflang="hy" data-title="Ժամանակի դանդաղում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%AB%E0%A4%BE%E0%A4%B0%E0%A4%A3" title="समय विस्फारण – Hindi" lang="hi" hreflang="hi" data-title="समय विस्फारण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vremenska_dilatacija" title="Vremenska dilatacija – Croatian" lang="hr" hreflang="hr" data-title="Vremenska dilatacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dilatasi_waktu" title="Dilatasi waktu – Indonesian" lang="id" hreflang="id" data-title="Dilatasi waktu" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/T%C3%ADmabj%C3%B6gun" title="Tímabjögun – Icelandic" lang="is" hreflang="is" data-title="Tímabjögun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dilatazione_del_tempo" title="Dilatazione del tempo – Italian" lang="it" hreflang="it" data-title="Dilatazione del tempo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%90%D7%A8%D7%9B%D7%95%D7%AA_%D7%94%D7%96%D7%9E%D7%9F" title="התארכות הזמן – Hebrew" lang="he" hreflang="he" data-title="התארכות הזמן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Dilasi_wektu" title="Dilasi wektu – Javanese" lang="jv" hreflang="jv" data-title="Dilasi wektu" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Id%C5%91dilat%C3%A1ci%C3%B3" title="Idődilatáció – Hungarian" lang="hu" hreflang="hu" data-title="Idődilatáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D1%81%D0%BA%D0%BE_%D0%B8%D0%B7%D0%B4%D0%BE%D0%BB%D0%B6%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Временско издолжување – Macedonian" lang="mk" hreflang="mk" data-title="Временско издолжување" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Dilasi_masa" title="Dilasi masa – Malay" lang="ms" hreflang="ms" data-title="Dilasi masa" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tijddilatatie" title="Tijddilatatie – Dutch" lang="nl" hreflang="nl" data-title="Tijddilatatie" 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%E9%96%93%E3%81%AE%E9%81%85%E3%82%8C" title="時間の遅れ – Japanese" lang="ja" hreflang="ja" data-title="時間の遅れ" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tidsdilatasjon" title="Tidsdilatasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tidsdilatasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tidsdilatasjon" title="Tidsdilatasjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tidsdilatasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dylatacja_czasu" title="Dylatacja czasu – Polish" lang="pl" hreflang="pl" data-title="Dylatacja czasu" 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/Dilata%C3%A7%C3%A3o_do_tempo" title="Dilatação do tempo – Portuguese" lang="pt" hreflang="pt" data-title="Dilatação do tempo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Dilatarea_timpului" title="Dilatarea timpului – Romanian" lang="ro" hreflang="ro" data-title="Dilatarea timpului" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BC%D0%B5%D0%B4%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%B8" title="Замедление времени – Russian" lang="ru" hreflang="ru" data-title="Замедление времени" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Time_dilation" title="Time dilation – Scots" lang="sco" hreflang="sco" data-title="Time dilation" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8F%E0%B6%BD_%E0%B7%80%E0%B7%92%E0%B7%83%E0%B7%8A%E0%B6%AD%E0%B7%8F%E0%B6%BB%E0%B6%AB%E0%B6%BA" title="කාල විස්තාරණය – Sinhala" lang="si" hreflang="si" data-title="කාල විස්තාරණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Time_dilation" title="Time dilation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Time dilation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Dilat%C3%A1cia_%C4%8Dasu" title="Dilatácia času – Slovak" lang="sk" hreflang="sk" data-title="Dilatácia času" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Podalj%C5%A1anje_%C4%8Dasa" title="Podaljšanje časa – Slovenian" lang="sl" hreflang="sl" data-title="Podaljšanje časa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D0%BB%D0%B0%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0_%D0%B2%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%B0" title="Дилатација времена – Serbian" lang="sr" hreflang="sr" data-title="Дилатација времена" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vremenska_dilatacija" title="Vremenska dilatacija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Vremenska dilatacija" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tidsdilatation" title="Tidsdilatation – Swedish" lang="sv" hreflang="sv" data-title="Tidsdilatation" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%B2_%E0%AE%B5%E0%AE%BF%E0%AE%B0%E0%AE%BF%E0%AE%B5%E0%AF%81" title="கால விரிவு – Tamil" lang="ta" hreflang="ta" data-title="கால விரிவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%92%D0%B0%D0%BA%D1%8B%D1%82%D0%BD%D1%8B%D2%A3_%D1%80%D0%B5%D0%BB%D1%8F%D1%82%D0%B8%D0%B2%D0%B8%D1%81%D1%82%D0%B8%D0%BA_%D3%99%D0%BA%D1%80%D0%B5%D0%BD%D3%99%D1%8E%D0%B5" title="Вакытның релятивистик әкренәюе – Tatar" lang="tt" hreflang="tt" data-title="Вакытның релятивистик әкренәюе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%82%E0%B8%A2%E0%B8%B2%E0%B8%A2%E0%B8%82%E0%B8%99%E0%B8%B2%E0%B8%94%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%80%E0%B8%A7%E0%B8%A5%E0%B8%B2" title="การขยายขนาดของเวลา – Thai" lang="th" hreflang="th" data-title="การขยายขนาดของเวลา" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Zaman_geni%C5%9Flemesi" title="Zaman genişlemesi – Turkish" lang="tr" hreflang="tr" data-title="Zaman genişlemesi" 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%A0%D0%B5%D0%BB%D1%8F%D1%82%D0%B8%D0%B2%D1%96%D1%81%D1%82%D1%81%D1%8C%D0%BA%D0%B5_%D1%83%D0%BF%D0%BE%D0%B2%D1%96%D0%BB%D1%8C%D0%BD%D0%B5%D0%BD%D0%BD%D1%8F_%D1%87%D0%B0%D1%81%D1%83" title="Релятивістське уповільнення часу – Ukrainian" lang="uk" hreflang="uk" data-title="Релятивістське уповільнення часу" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%B9%D8%A7%D8%A6%D9%85_%DA%88%D8%A7%D8%A6%DB%8C%D9%84%DB%8C%D8%B4%D9%86" title="ٹائم ڈائیلیشن – Urdu" lang="ur" hreflang="ur" data-title="ٹائم ڈائیلیشن" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Th%E1%BB%9Di_gian_gi%C3%A3n_n%E1%BB%9F" title="Thời gian giãn nở – Vietnamese" lang="vi" hreflang="vi" data-title="Thời gian giãn nở" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%97%B6%E9%97%B4%E8%86%A8%E8%83%80" title="时间膨胀 – Wu" lang="wuu" hreflang="wuu" data-title="时间膨胀" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%99%82%E9%96%93%E8%86%A8%E8%84%B9" title="時間膨脹 – Cantonese" lang="yue" hreflang="yue" data-title="時間膨脹" 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/%E6%99%82%E9%96%93%E8%86%A8%E8%84%B9" title="時間膨脹 – Chinese" lang="zh" hreflang="zh" data-title="時間膨脹" 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/Q185918#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> 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searchaux" style="display:none">Measured time difference as explained by relativity theory</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about a physical concept. For the term used in psychology, see <a href="/wiki/Time_perception" title="Time perception">Time perception</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1282087196">.mw-parser-output .special-relativity-sidebar .sidebar-list-title{background:#ddf;border-radius:3px;padding-bottom:0}.mw-parser-output .special-relativity-sidebar .sidebar-list{padding-top:0}.mw-parser-output .special-relativity-sidebar-content1{padding-bottom:0.6em}.mw-parser-output .special-relativity-sidebar .sidebar-below{background-color:transparent;border-color:#A2B8BF}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist special-relativity-sidebar"><tbody><tr><th class="sidebar-title"><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:World_line.svg" class="mw-file-description" title="The world line: a diagrammatic representation of spacetime"><img alt="The world line: a diagrammatic representation of spacetime" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/200px-World_line.svg.png" decoding="async" width="200" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/300px-World_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/400px-World_line.svg.png 2x" data-file-width="481" data-file-height="491" /></a></span></td></tr><tr><td class="sidebar-content .special-relativity-sidebar-content1"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Formulations_of_special_relativity" title="Formulations of special relativity">Formulations</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><span style="font-size:120%">Foundations</span></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Postulates_of_special_relativity" title="Postulates of special relativity">Einstein's postulates</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><span style="font-size:120%">Consequences</span></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a class="mw-selflink selflink">Time dilation</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">Relativistic disk</a></li> <li><a href="/wiki/Bell%27s_spaceship_paradox" title="Bell's spaceship paradox">Bell's spaceship paradox</a></li> <li><a href="/wiki/Ehrenfest_paradox" title="Ehrenfest paradox">Ehrenfest paradox</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><span style="font-size:120%"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></span></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Spacetime diagram</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><span style="font-size:120%"><a href="/wiki/Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</a></span></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Invariant_mass" title="Invariant mass">Proper mass</a></li> <li><a href="/wiki/Four-momentum" title="Four-momentum">Four-momentum</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist" style="font-size:110%;"><ul><li><a href="/wiki/History_of_special_relativity" title="History of special relativity">History</a></li><li>Precursors</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a></li> <li><a href="/wiki/Aether_theories" title="Aether theories">Aether theories</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><span style="font-size:120%">People</span></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/Albert_A._Michelson" title="Albert A. Michelson">Michelson</a></li> <li><a href="/wiki/Edward_W._Morley" title="Edward W. Morley">Morley</a></li> <li><a href="/wiki/George_Francis_FitzGerald" title="George Francis FitzGerald">FitzGerald</a></li> <li><a href="/wiki/Gustav_Herglotz" title="Gustav Herglotz">Herglotz</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Max_Abraham" title="Max Abraham">Abraham</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Richard_C._Tolman" title="Richard C. Tolman">Tolman</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below 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/20px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="13" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Special_relativity" title="Category:Special relativity">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Special_relativity_sidebar" title="Template:Special relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Special_relativity_sidebar" title="Template talk:Special relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Special_relativity_sidebar" title="Special:EditPage/Template:Special relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Time dilation</b> is the difference in elapsed <a href="/wiki/Time_in_physics" title="Time in physics">time</a> as measured by two <a href="/wiki/Clocks" class="mw-redirect" title="Clocks">clocks</a>, either because of a relative <a href="/wiki/Velocity" title="Velocity">velocity</a> between them (<a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>), or a difference in <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a> between their locations (<a href="/wiki/General_relativity" title="General relativity">general relativity</a>). When unspecified, "time dilation" usually refers to the effect due to velocity. </p><p>After compensating for varying signal delays resulting from the changing distance between an <a href="/wiki/Observer_(special_relativity)" title="Observer (special relativity)">observer</a> and a moving clock (i.e. <a href="/wiki/Doppler_effect" title="Doppler effect">Doppler effect</a>), the observer will measure the moving clock as ticking more slowly than a clock at rest in the observer's own <a href="/wiki/Reference_frame" class="mw-redirect" title="Reference frame">reference frame</a>. There is a difference between observed and measured relativistic time dilation - the observer does not visually perceive time dilation in the same way that they measure it.<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 addition, a clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated farther from the same massive body (and which is at a higher gravitational potential). </p><p>These predictions of the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of <a href="/wiki/Satellite_navigation" title="Satellite navigation">satellite navigation</a> systems such as <a href="/wiki/GPS" class="mw-redirect" title="GPS">GPS</a> and <a href="/wiki/Galileo_(satellite_navigation)" title="Galileo (satellite navigation)">Galileo</a>.<sup id="cite_ref-Ashby_2-0" class="reference"><a href="#cite_note-Ashby-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_special_relativity" title="History of special relativity">History of special relativity</a></div> <p>Time dilation by the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> was predicted by several authors at the turn of the 20th century.<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><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/Joseph_Larmor" title="Joseph Larmor">Joseph Larmor</a> (1897) wrote that, at least for those orbiting a nucleus, individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio: <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 {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8218447e8f4e64c006c3e040533d1737022128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.792ex; height:4.676ex;" alt="{\textstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}" /></span>.<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> <a href="/wiki/Emil_Cohn" title="Emil Cohn">Emil Cohn</a> (1904) specifically related this formula to the rate of clocks.<sup id="cite_ref-cohn_6-0" class="reference"><a href="#cite_note-cohn-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In the context of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> it was shown by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> (1905) that this effect concerns the nature of time itself, and he was also the first to point out its reciprocity or symmetry.<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> Subsequently, <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> (1907) introduced the concept of <a href="/wiki/Proper_time" title="Proper time">proper time</a> which further clarified the meaning of time dilation.<sup id="cite_ref-mink2_8-0" class="reference"><a href="#cite_note-mink2-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <span class="anchor" id="Velocity_time_dilation"></span><span class="anchor" id="Relation_to_velocity"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Time_dilation_caused_by_a_relative_velocity">Time dilation caused by a relative velocity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=2" title="Edit section: Time dilation caused by a relative velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Special_relativity#Time_dilation" title="Special relativity">Special relativity § Time dilation</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Nonsymmetric_velocity_time_dilation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Nonsymmetric_velocity_time_dilation.gif/220px-Nonsymmetric_velocity_time_dilation.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/72/Nonsymmetric_velocity_time_dilation.gif 1.5x" data-file-width="256" data-file-height="256" /></a><figcaption>From the local frame of reference of the blue clock, the red clock, being in motion, is measured as ticking slower.<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></figcaption></figure> <p><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a> indicates that, for an observer in an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>, a clock that is moving relative to the observer will be measured to tick more slowly than a clock at rest in the observer's frame of reference. This is sometimes called special relativistic time dilation. The faster the <a href="/wiki/Relative_velocity" title="Relative velocity">relative velocity</a>, the greater the time dilation between them, with time slowing to a stop as one clock approaches the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> (299,792,458 m/s). </p><p>In theory, time dilation would make it possible for passengers in a fast-moving vehicle to advance into the future in a short period of their own time. With sufficiently high speeds, the effect would be dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 <a href="/wiki/Standard_gravity" title="Standard gravity">g</a> acceleration would permit humans to travel through <a href="/wiki/Observable_universe" title="Observable universe">the entire known Universe</a> in one human lifetime.<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> </p><p>With current technology severely limiting the velocity of space travel, the differences experienced in practice are minuscule. After 6 months on the <a href="/wiki/International_Space_Station" title="International Space Station">International Space Station</a> (ISS), orbiting Earth at a speed of about 7,700 m/s, an astronaut would have aged about 0.005 seconds less than he would have on Earth.<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> The cosmonauts <a href="/wiki/Sergei_Krikalev" title="Sergei Krikalev">Sergei Krikalev</a> and <a href="/wiki/Sergey_Avdeev" title="Sergey Avdeev">Sergey Avdeev</a> both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.<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><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <span class="anchor" id="Simple_inference_of_velocity_time_dilation"></span><span class="anchor" id="Simple_inference"></span> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Simple_inference">Simple inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=3" title="Edit section: Simple inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Time-dilation-002-mod.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Time-dilation-002-mod.svg/660px-Time-dilation-002-mod.svg.png" decoding="async" width="660" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Time-dilation-002-mod.svg/990px-Time-dilation-002-mod.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Time-dilation-002-mod.svg/1320px-Time-dilation-002-mod.svg.png 2x" data-file-width="1424" data-file-height="503" /></a><figcaption><b>Left</b>: Observer at rest measures time 2<i>L</i>/<i>c</i> between co-local events of light signal generation at A and arrival at A.<br /><b>Right</b>: Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at time <i>t'=</i>0, top mirror B when signal gets reflected at time <i>t'=D/c</i>, bottom mirror A when signal returns at time <i>t'=2D/c</i></figcaption></figure> <p>Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the <a href="/wiki/Special_relativity#Postulates" title="Special relativity">second postulate of special relativity</a>. This constancy of the speed of light means that, counter to intuition, the speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source.<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><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Consider then, a simple vertical clock consisting of two mirrors <span class="texhtml">A</span> and <span class="texhtml">B</span>, between which a light pulse is bouncing. The separation of the mirrors is <span class="texhtml"><i>L</i></span> and the clock ticks once each time the light pulse hits mirror <span class="texhtml">A</span>. </p><p>In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length <span class="texhtml">2<i>L</i></span> and the time period between the ticks of the clock <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 \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" /></span> is equal to <span class="texhtml">2<i>L</i></span> divided by the speed of light <span class="texhtml"><i>c</i></span>: </p> <dl><dd><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 \Delta t={\frac {2L}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t={\frac {2L}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60aa91e3bdae4aa24d9d3ee8adffe11b70c7c22d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.455ex; height:5.176ex;" alt="{\displaystyle \Delta t={\frac {2L}{c}}}" /></span></dd></dl> <p>From the frame of reference of a moving observer traveling at the speed <span class="texhtml"><i>v</i></span> relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path <span class="texhtml">2<i>D</i></span>. Keeping the speed of light constant for all inertial observers requires a lengthening (that is dilation) of the time period between the ticks of this clock <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 \Delta t'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ea24905e052383d75e10e87e33a3d805d39b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:2.509ex;" alt="{\displaystyle \Delta t'}" /></span> from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running (that is ticking) more slowly, since tick rate equals one over the time period between ticks 1/<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 \Delta t'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ea24905e052383d75e10e87e33a3d805d39b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:2.509ex;" alt="{\displaystyle \Delta t'}" /></span>. </p><p>Straightforward application of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> leads to the well-known prediction of special relativity: </p><p>The total time for the light pulse to trace its path is given by: </p> <dl><dd><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 \Delta t'={\frac {2D}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>D</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'={\frac {2D}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b038ef0d4557cde64066c2f93f0ad80a7d4c9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.482ex; height:5.176ex;" alt="{\displaystyle \Delta t'={\frac {2D}{c}}}" /></span></dd></dl> <p>The length of the half path can be calculated as a function of known quantities as: </p> <dl><dd><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 D={\sqrt {\left({\frac {1}{2}}v\Delta t'\right)^{2}+L^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>v</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\sqrt {\left({\frac {1}{2}}v\Delta t'\right)^{2}+L^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7901378847f6cff51f7f68ab065e64d9df2ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.886ex; height:7.676ex;" alt="{\displaystyle D={\sqrt {\left({\frac {1}{2}}v\Delta t'\right)^{2}+L^{2}}}}" /></span></dd></dl> <p>Elimination of the variables <span class="texhtml"><i>D</i></span> and <span class="texhtml"><i>L</i></span> from these three equations results in: </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 10px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Time dilation equation</b> <p><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 \Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\gamma }{\Delta t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\gamma }{\Delta t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c53ce76ac62cf8c7621811a6218bc7dc37f5839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.323ex; height:8.176ex;" alt="{\displaystyle \Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\gamma }{\Delta t}}" /></span> </p> </div> <p>which expresses the fact that the moving observer's period of the clock <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 \Delta t'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ea24905e052383d75e10e87e33a3d805d39b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:2.509ex;" alt="{\displaystyle \Delta t'}" /></span> is longer than the period <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 \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" /></span> in the frame of the clock itself. The <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> gamma (<span class="texhtml"><i>γ</i></span>) is defined as<sup id="cite_ref-Forshaw_2014_18-0" class="reference"><a href="#cite_note-Forshaw_2014-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abacf009fa4db767f015172584c8fc89e7725745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:13.989ex; height:8.009ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}" /></span></dd></dl> <p>Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks—mechanical, electronic, optical (such as an identical horizontal version of the clock in the example)—should exhibit the same velocity-dependent time dilation.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Reciprocity">Reciprocity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=4" title="Edit section: Reciprocity"><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:Time_dilation02.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Time_dilation02.gif/220px-Time_dilation02.gif" decoding="async" width="220" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/eb/Time_dilation02.gif 1.5x" data-file-width="279" data-file-height="352" /></a><figcaption>Transversal time dilation. The blue dots represent a pulse of light. Each pair of dots with light "bouncing" between them is a clock. In the frame of each group of clocks, the other group is measured to tick more slowly, because the moving clock's light pulse has to travel a larger distance than the stationary clock's light pulse. That is so, even though the clocks are identical and their relative motion is perfectly reciprocal.</figcaption></figure> <p>Given a certain frame of reference, and the "stationary" observer described earlier, if a second observer accompanied the "moving" clock, each of the observers would measure the other's clock as ticking at a <i>slower</i> rate than their own local clock, due to them both measure the other to be the one that is in motion relative to their own stationary frame of reference. </p><p>Common sense would dictate that, if the passage of time has slowed for a moving object, said object would observe the external world's time to be correspondingly sped up. Counterintuitively, special relativity predicts the opposite. When two observers are in motion relative to each other, each will measure the other's clock slowing down, in concordance with them being in motion relative to the observer's frame of reference. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Eigenzeit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Eigenzeit.svg/250px-Eigenzeit.svg.png" decoding="async" width="220" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Eigenzeit.svg/330px-Eigenzeit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Eigenzeit.svg/500px-Eigenzeit.svg.png 2x" data-file-width="640" data-file-height="790" /></a><figcaption>Time UV of a clock in S is shorter compared to Ux′ in S′, and time UW of a clock in S′ is shorter compared to Ux in S.</figcaption></figure> <p>While this seems self-contradictory, a similar oddity occurs in everyday life. If two persons A and B observe each other from a distance, B will appear small to A, but at the same time, A will appear small to B. Being familiar with the effects of <a href="/wiki/Perspective_(visual)" class="mw-redirect" title="Perspective (visual)">perspective</a>, there is no contradiction or paradox in this situation.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The reciprocity of the phenomenon also leads to the so-called <a href="/wiki/Twin_paradox" title="Twin paradox">twin paradox</a> where the aging of twins, one staying on Earth and the other embarking on space travel, is compared, and where the reciprocity suggests that both persons should have the same age when they reunite. On the contrary, at the end of the round-trip, the traveling twin will be younger than the sibling on Earth. The dilemma posed by the paradox can be explained by the fact that situation is not symmetric. The twin staying on Earth is in a single inertial frame, and the traveling twin is in two different inertial frames: one on the way out and another on the way back. See also <a href="/wiki/Twin_paradox#Role_of_acceleration" title="Twin paradox">Twin paradox § Role of acceleration</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Experimental_testing">Experimental testing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=5" title="Edit section: Experimental testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Experimental_testing_of_time_dilation" title="Experimental testing of time dilation">Experimental testing of time dilation</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Tests_of_special_relativity" title="Tests of special relativity">Tests of special relativity</a></div> <div class="mw-heading mw-heading4"><h4 id="Moving_particles">Moving particles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=6" title="Edit section: Moving particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A comparison of <a href="/wiki/Muon" title="Muon">muon</a> lifetimes at different speeds is possible. In the laboratory, slow muons are produced; and in the atmosphere, very fast-moving muons are introduced by cosmic rays. Taking the muon lifetime at rest as the laboratory value of 2.197 μs, the lifetime of a cosmic-ray-produced muon traveling at 98% of the speed of light is about five times longer, in agreement with observations. An example is Rossi and Hall (1941), who compared the population of cosmic-ray-produced muons at the top of a mountain to that observed at sea level.<sup id="cite_ref-Stewart_21-0" class="reference"><a href="#cite_note-Stewart-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li>The lifetime of particles produced in particle accelerators are longer due to time dilation. In such experiments, the "clock" is the time taken by processes leading to muon decay, and these processes take place in the moving muon at its own "clock rate", which is much slower than the laboratory clock. This is routinely taken into account in particle physics, and many dedicated measurements have been performed. For instance, in the muon storage ring at CERN the lifetime of muons circulating with γ = 29.327 was found to be dilated to 64.378 μs, confirming time dilation to an accuracy of 0.9 ± 0.4 parts per thousand.<sup id="cite_ref-Bailey_22-0" class="reference"><a href="#cite_note-Bailey-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="Doppler_effect">Doppler effect</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=7" title="Edit section: Doppler effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ives%E2%80%93Stilwell_experiment" title="Ives–Stilwell experiment">Ives–Stilwell experiment</a></div> <ul><li>The stated purpose by Ives and Stilwell (1938, 1941) of these experiments was to verify the time dilation effect, predicted by Larmor–Lorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect in <a href="/wiki/Canal_ray" class="mw-redirect" title="Canal ray">canal rays</a> would provide a suitable experiment. These experiments measured the <a href="/wiki/Doppler_shift" class="mw-redirect" title="Doppler shift">Doppler shift</a> of the radiation emitted from <a href="/wiki/Cathode_ray" title="Cathode ray">cathode rays</a>, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classically predicted values:<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 {\frac {f_{0}}{1-v/c}}\qquad {\text{and}}\qquad {\frac {f_{0}}{1+v/c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> </mfrac> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f_{0}}{1-v/c}}\qquad {\text{and}}\qquad {\frac {f_{0}}{1+v/c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a564eb467afd14e546960b72945caaca325925f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.31ex; height:6.176ex;" alt="{\displaystyle {\frac {f_{0}}{1-v/c}}\qquad {\text{and}}\qquad {\frac {f_{0}}{1+v/c}}}" /></span>The high and low frequencies of the radiation from the moving sources were measured as:<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><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 {\sqrt {\frac {1+v/c}{1-v/c}}}f_{0}=\gamma \left(1+v/c\right)f_{0}\qquad {\text{and}}\qquad {\sqrt {\frac {1-v/c}{1+v/c}}}f_{0}=\gamma \left(1-v/c\right)f_{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> </mfrac> </msqrt> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {1+v/c}{1-v/c}}}f_{0}=\gamma \left(1+v/c\right)f_{0}\qquad {\text{and}}\qquad {\sqrt {\frac {1-v/c}{1+v/c}}}f_{0}=\gamma \left(1-v/c\right)f_{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115d84036bb70211139fa7b5872a2fe6c52690bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.606ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\frac {1+v/c}{1-v/c}}}f_{0}=\gamma \left(1+v/c\right)f_{0}\qquad {\text{and}}\qquad {\sqrt {\frac {1-v/c}{1+v/c}}}f_{0}=\gamma \left(1-v/c\right)f_{0}\,}" /></span>as deduced by Einstein (1905) from the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>, when the source is running slow by the Lorentz factor.</li> <li>Hasselkamp, Mondry, and Scharmann<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> (1979) measured the Doppler shift from a source moving at right angles to the line of sight. The most general relationship between frequencies of the radiation from the moving sources is given by:<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 f_{\mathrm {detected} }=f_{\mathrm {rest} }{\left(1-{\frac {v}{c}}\cos \phi \right)/{\sqrt {1-{v^{2}}/{c^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {detected} }=f_{\mathrm {rest} }{\left(1-{\frac {v}{c}}\cos \phi \right)/{\sqrt {1-{v^{2}}/{c^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087b0ceb126c51f4b2b2c18ae73067cdf5142643" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.825ex; height:5.009ex;" alt="{\displaystyle f_{\mathrm {detected} }=f_{\mathrm {rest} }{\left(1-{\frac {v}{c}}\cos \phi \right)/{\sqrt {1-{v^{2}}/{c^{2}}}}}}" /></span>as deduced by Einstein (1905).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For <span class="nowrap"><i>ϕ</i> = 90°</span> (<span class="nowrap">cos <i>ϕ</i> = 0</span>) this reduces to <span class="nowrap"><i>f</i><sub>detected</sub> = <i>f</i><sub>rest</sub>γ</span>. This lower frequency from the moving source can be attributed to the time dilation effect and is often called the <a href="/wiki/Transverse_Doppler_effect" class="mw-redirect" title="Transverse Doppler effect">transverse Doppler effect</a> and was predicted by relativity.</li> <li>In 2010 time dilation was observed at speeds of less than 10 metres per second using optical atomic clocks connected by 75 metres of optical fiber.<sup id="cite_ref-Chou_26-0" class="reference"><a href="#cite_note-Chou-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Proper_time_and_Minkowski_diagram">Proper time and Minkowski diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=8" title="Edit section: Proper time and Minkowski diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/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 span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="theader">Minkowski diagram and twin paradox</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Zeitdilatation3.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Zeitdilatation3.svg/250px-Zeitdilatation3.svg.png" decoding="async" width="200" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Zeitdilatation3.svg/330px-Zeitdilatation3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Zeitdilatation3.svg/500px-Zeitdilatation3.svg.png 2x" data-file-width="656" data-file-height="759" /></a></span></div><div class="thumbcaption">Clock C in relative motion between two synchronized clocks A and B. C meets A at <i>d</i>, and B at <i>f</i>.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:EigenzeitZwill.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/EigenzeitZwill.svg/200px-EigenzeitZwill.svg.png" decoding="async" width="200" height="343" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/EigenzeitZwill.svg/300px-EigenzeitZwill.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/EigenzeitZwill.svg/400px-EigenzeitZwill.svg.png 2x" data-file-width="379" data-file-height="650" /></a></span></div><div class="thumbcaption"><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a>. One twin has to change frames, leading to different <a href="/wiki/Proper_time" title="Proper time">proper times</a> in the twin's world lines.</div></div></div></div></div> <p>In the <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a> from the first image on the right, clock C resting in inertial frame S′ meets clock A at <i>d</i> and clock B at <i>f</i> (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersecting <i>f</i> is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous with <i>d</i> in S are on the x-axis, in S′ on the x′-axis. </p><p>The <a href="/wiki/Proper_time" title="Proper time">proper time</a> between two events is indicated by a clock present at both events.<sup id="cite_ref-taylor_27-0" class="reference"><a href="#cite_note-taylor-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Interval <i>df</i> is, therefore, the proper time of clock C, and is shorter with respect to the coordinate times <i>ef=dg</i> of clocks B and A in S. Conversely, also proper time <i>ef</i> of B is shorter with respect to time <i>if</i> in S′, because event <i>e</i> was measured in S′ already at time <i>i</i> due to relativity of simultaneity, long before C started to tick. </p><p>From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always the <i>minimal</i> time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock is <i>maximal</i>, which is the solution to the <a href="/wiki/Twin_paradox" title="Twin paradox">twin paradox</a>.<sup id="cite_ref-taylor_27-1" class="reference"><a href="#cite_note-taylor-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_and_formulation">Derivation and formulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=9" title="Edit section: Derivation and formulation"><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:Time_dilation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Time_dilation.svg/310px-Time_dilation.svg.png" decoding="async" width="310" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Time_dilation.svg/465px-Time_dilation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Time_dilation.svg/620px-Time_dilation.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption><a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> as a function of speed (in natural units where <i>c</i> = 1). Notice that for small speeds (as v tends to zero), γ is approximately 1.</figcaption></figure> <p>In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>.<sup id="cite_ref-born_28-0" class="reference"><a href="#cite_note-born-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Let there be two events at which the moving clock indicates <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 t_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1f89579c6b54711128de7bad0f0f760e116180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.941ex; height:2.343ex;" alt="{\displaystyle t_{a}}" /></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 t_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca101f3ac73e0d38184d4aec1bdbb5dcad7cbb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.777ex; height:2.343ex;" alt="{\displaystyle t_{b}}" /></span>, thus: </p> <dl><dd><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 t_{a}^{\prime }={\frac {t_{a}-{\frac {vx_{a}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ t_{b}^{\prime }={\frac {t_{b}-{\frac {vx_{b}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}^{\prime }={\frac {t_{a}-{\frac {vx_{a}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ t_{b}^{\prime }={\frac {t_{b}-{\frac {vx_{b}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acdf610d73367d95ae2df6167f35a28822d6c3ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:30.786ex; height:9.509ex;" alt="{\displaystyle t_{a}^{\prime }={\frac {t_{a}-{\frac {vx_{a}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ t_{b}^{\prime }={\frac {t_{b}-{\frac {vx_{b}}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}" /></span></dd></dl> <p>Since the clock remains at rest in its inertial frame, it follows <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 x_{a}=x_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{a}=x_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6511f31d81ff2d0b16242ef2132ef577fe981ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.797ex; height:2.009ex;" alt="{\displaystyle x_{a}=x_{b}}" /></span>, thus the interval <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 \Delta t^{\prime }=t_{b}^{\prime }-t_{a}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t^{\prime }=t_{b}^{\prime }-t_{a}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7819b64fe630e424473dd5490fb9a1180bcbf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.118ex; height:3.176ex;" alt="{\displaystyle \Delta t^{\prime }=t_{b}^{\prime }-t_{a}^{\prime }}" /></span> is given by: </p> <dl><dd><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 \Delta t'=\gamma \,\Delta t={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>γ</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t'=\gamma \,\Delta t={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64dbd28073a6abed57c9d547e8380572a1c00e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:24.097ex; height:8.176ex;" alt="{\displaystyle \Delta t'=\gamma \,\Delta t={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}" /></span></dd></dl> <p>where Δ<i>t</i> is the time interval between <i>two co-local events</i> (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on their clock), known as the <i><a href="/wiki/Proper_time" title="Proper time">proper time</a></i>, Δ<var>t′</var> is the time interval between those same events, as measured by another observer, inertially moving with velocity <i>v</i> with respect to the former observer, <i>v</i> is the relative velocity between the observer and the moving clock, <i>c</i> is the speed of light, and the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> (conventionally denoted by the Greek letter <a href="/wiki/Gamma" title="Gamma">gamma</a> or γ) is: </p> <dl><dd><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 \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf389bd536348831f10ad35131b1fea07b76ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:14.376ex; height:8.009ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,}" /></span></dd></dl> <p>Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, where <span class="nowrap"><i>v</i> ≪ <i>c</i>,</span> even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. As an approximate threshold, time dilation of 0.5% may become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).<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> </p> <div class="mw-heading mw-heading3"><h3 id="Hyperbolic_motion">Hyperbolic motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=10" title="Edit section: Hyperbolic motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hyperbolic_motion_(relativity)" title="Hyperbolic motion (relativity)">Hyperbolic motion (relativity)</a></div> <p>In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculate <a href="/wiki/Proper_time" title="Proper time">proper time</a> and movement in space for the simple case of a spaceship which is applied with a force per unit mass, relative to some reference object in uniform (i.e. constant velocity) motion, equal to <i>g</i> throughout the period of measurement. </p><p>Let <i>t</i> be the time in an inertial frame subsequently called the rest frame. Let <i>x</i> be a spatial coordinate, and let the direction of the constant acceleration as well as the spaceship's velocity (relative to the rest frame) be parallel to the <i>x</i>-axis. Assuming the spaceship's position at time <span class="nowrap"><i>t</i> = 0</span> being <span class="nowrap"><i>x</i> = 0</span> and the velocity being <i>v</i><sub>0</sub> and defining the following abbreviation: </p> <dl><dd><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 \gamma _{0}={\frac {1}{\sqrt {1-v_{0}^{2}/c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}={\frac {1}{\sqrt {1-v_{0}^{2}/c^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5959bcf2e2777c1fa8139b415da7cd21771d0e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:17.925ex; height:8.009ex;" alt="{\displaystyle \gamma _{0}={\frac {1}{\sqrt {1-v_{0}^{2}/c^{2}}}}}" /></span></dd></dl> <p>the following formulas hold:<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>Position: </p> <dl><dd><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 x(t)={\frac {c^{2}}{g}}\left({\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}-\gamma _{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)={\frac {c^{2}}{g}}\left({\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}-\gamma _{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682c84ef96349ecbc2f48b06b767f9eafcd2d51f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:38.79ex; height:8.509ex;" alt="{\displaystyle x(t)={\frac {c^{2}}{g}}\left({\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}-\gamma _{0}\right)}" /></span></dd></dl> <p>Velocity: </p> <dl><dd><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 v(t)={\frac {gt+v_{0}\gamma _{0}}{\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)={\frac {gt+v_{0}\gamma _{0}}{\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d45a90a610b55f28b902bc39eae25e2ff0f028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:22.958ex; height:9.343ex;" alt="{\displaystyle v(t)={\frac {gt+v_{0}\gamma _{0}}{\sqrt {1+{\frac {\left(gt+v_{0}\gamma _{0}\right)^{2}}{c^{2}}}}}}}" /></span></dd></dl> <p>Proper time as function of coordinate time: </p> <dl><dd><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 \tau (t)=\tau _{0}+\int _{0}^{t}{\sqrt {1-\left({\frac {v(t')}{c}}\right)^{2}}}dt'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (t)=\tau _{0}+\int _{0}^{t}{\sqrt {1-\left({\frac {v(t')}{c}}\right)^{2}}}dt'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad2547bb8535787e9debcabcaf6d4ea0c351823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.377ex; height:7.509ex;" alt="{\displaystyle \tau (t)=\tau _{0}+\int _{0}^{t}{\sqrt {1-\left({\frac {v(t')}{c}}\right)^{2}}}dt'}" /></span></dd></dl> <p>In the case where <i>v</i>(0) = <i>v</i><sub>0</sub> = 0 and <i>τ</i>(0) = <i>τ</i><sub>0</sub> = 0 the integral can be expressed as a logarithmic function or, equivalently, as an <a href="/wiki/Hyperbolic_function#Inverse_functions_as_logarithms" class="mw-redirect" title="Hyperbolic function">inverse hyperbolic function</a>: </p> <dl><dd><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 \tau (t)={\frac {c}{g}}\ln \left({\frac {gt}{c}}+{\sqrt {1+\left({\frac {gt}{c}}\right)^{2}}}\right)={\frac {c}{g}}\operatorname {arsinh} \left({\frac {gt}{c}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>g</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>t</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>t</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>g</mi> </mfrac> </mrow> <mi>arsinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>t</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (t)={\frac {c}{g}}\ln \left({\frac {gt}{c}}+{\sqrt {1+\left({\frac {gt}{c}}\right)^{2}}}\right)={\frac {c}{g}}\operatorname {arsinh} \left({\frac {gt}{c}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc38e349328afe5c73a3e88460dd4a638d39c8cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.008ex; height:7.843ex;" alt="{\displaystyle \tau (t)={\frac {c}{g}}\ln \left({\frac {gt}{c}}+{\sqrt {1+\left({\frac {gt}{c}}\right)^{2}}}\right)={\frac {c}{g}}\operatorname {arsinh} \left({\frac {gt}{c}}\right)}" /></span></dd></dl> <p>As functions of the proper time <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span> of the ship, the following formulae hold:<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> </p><p>Position: </p> <dl><dd><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 x(\tau )={\frac {c^{2}}{g}}\left(\cosh {\frac {g\tau }{c}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>τ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\tau )={\frac {c^{2}}{g}}\left(\cosh {\frac {g\tau }{c}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22706e2fc9c62b4c13f2ea16092ac1e1e8615a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.447ex; height:6.176ex;" alt="{\displaystyle x(\tau )={\frac {c^{2}}{g}}\left(\cosh {\frac {g\tau }{c}}-1\right)}" /></span></dd></dl> <p>Velocity: </p> <dl><dd><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 v(\tau )=c\tanh {\frac {g\tau }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>τ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(\tau )=c\tanh {\frac {g\tau }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fcb81029c4b544c29fe6706bf9fbf5837e11be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.825ex; height:4.843ex;" alt="{\displaystyle v(\tau )=c\tanh {\frac {g\tau }{c}}}" /></span></dd></dl> <p>Coordinate time as function of proper time: </p> <dl><dd><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 t(\tau )={\frac {c}{g}}\sinh {\frac {g\tau }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>g</mi> </mfrac> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>τ</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(\tau )={\frac {c}{g}}\sinh {\frac {g\tau }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0528a8116015b762d0debf67d9a4923c78ba30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.978ex; height:5.343ex;" alt="{\displaystyle t(\tau )={\frac {c}{g}}\sinh {\frac {g\tau }{c}}}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Clock_hypothesis">Clock hypothesis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=11" title="Edit section: Clock hypothesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>clock hypothesis</b> is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. This is equivalent to stating that a clock moving along a path <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> measures the <a href="/wiki/Proper_time" title="Proper time">proper time</a>, defined by: </p> <dl><dd><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 \tau =\int _{P}{\sqrt {dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\int _{P}{\sqrt {dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47e74b17a96164648426d76b0a195be418c431a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.463ex; height:5.676ex;" alt="{\displaystyle \tau =\int _{P}{\sqrt {dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}}}" /></span></dd></dl> <p>The clock hypothesis was implicitly (but not explicitly) included in Einstein's original 1905 formulation of special relativity. Since then, it has become a standard assumption and is usually included in the axioms of special relativity, especially in light of experimental verification up to very high accelerations in <a href="/wiki/Particle_accelerator" title="Particle accelerator">particle accelerators</a>.<sup id="cite_ref-Bailey_1977_32-0" class="reference"><a href="#cite_note-Bailey_1977-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> <span class="anchor" id="Gravitational_time_dilation"></span><span class="anchor" id="Relation_to_gravity"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Time_dilation_caused_by_gravity_or_acceleration">Time dilation caused by gravity or acceleration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=12" title="Edit section: Time dilation caused by gravity or acceleration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Soyuz_TMA-1_at_the_ISS.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Soyuz_TMA-1_at_the_ISS.jpg/260px-Soyuz_TMA-1_at_the_ISS.jpg" decoding="async" width="260" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Soyuz_TMA-1_at_the_ISS.jpg/390px-Soyuz_TMA-1_at_the_ISS.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Soyuz_TMA-1_at_the_ISS.jpg/520px-Soyuz_TMA-1_at_the_ISS.jpg 2x" data-file-width="3032" data-file-height="2008" /></a><figcaption>Time dilation explains why two working clocks will report different times after different accelerations. For example, time goes slower at the <a href="/wiki/International_Space_Station" title="International Space Station">ISS</a>, lagging approximately 0.01 seconds for every 12 Earth months passed. For <a href="/wiki/Global_Positioning_System" title="Global Positioning System">GPS</a> satellites to work, they must adjust for similar bending of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> to coordinate properly with systems on Earth.<sup id="cite_ref-Ashby_2-1" class="reference"><a href="#cite_note-Ashby-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:The_Earth_seen_from_Apollo_17.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/The_Earth_seen_from_Apollo_17.jpg/260px-The_Earth_seen_from_Apollo_17.jpg" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/The_Earth_seen_from_Apollo_17.jpg/390px-The_Earth_seen_from_Apollo_17.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/The_Earth_seen_from_Apollo_17.jpg/520px-The_Earth_seen_from_Apollo_17.jpg 2x" data-file-width="3000" data-file-height="3002" /></a><figcaption>Time passes more quickly further from a center of gravity, as is witnessed with massive objects (like the Earth).</figcaption></figure> <p>Gravitational time dilation is experienced by an observer that, at a certain altitude within a gravitational potential well, finds that their local clocks measure less elapsed time than identical clocks situated at higher altitude (and which are therefore at higher gravitational potential). </p><p>Gravitational time dilation is at play e.g. for ISS astronauts. While the astronauts' <a href="/wiki/Relative_velocity" title="Relative velocity">relative velocity</a> slows down their time, the reduced gravitational influence at their location speeds it up, although to a lesser degree. Also, a climber's time is theoretically passing slightly faster at the top of a mountain compared to people at sea level. It has also been calculated that due to time dilation, the <a href="/wiki/Structure_of_the_Earth" class="mw-redirect" title="Structure of the Earth">core of the Earth</a> is 2.5 years younger than the <a href="/wiki/Earth%27s_crust" title="Earth's crust">crust</a>.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> "A clock used to time a full rotation of the Earth will measure the day to be approximately an extra 10 ns/day longer for every km of altitude above the reference geoid."<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Travel to regions of space where extreme gravitational time dilation is taking place, such as near (but not beyond the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> of) a <a href="/wiki/Black_hole" title="Black hole">black hole</a>, could yield time-shifting results analogous to those of near-lightspeed space travel. </p><p>Contrarily to velocity time dilation, in which both observers measure the other as aging slower (a reciprocal effect), gravitational time dilation is not reciprocal. This means that with gravitational time dilation both observers agree that the clock nearer the center of the gravitational field is slower in rate, and they agree on the ratio of the difference. </p> <div class="mw-heading mw-heading3"><h3 id="Experimental_testing_2">Experimental testing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=13" title="Edit section: Experimental testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Experimental_testing_of_time_dilation" title="Experimental testing of time dilation">Experimental testing of time dilation</a></div> <ul><li>In 1959, <a href="/wiki/Robert_Pound" title="Robert Pound">Robert Pound</a> and <a href="/wiki/Glen_Rebka" title="Glen Rebka">Glen Rebka</a> measured the very slight <a href="/wiki/Gravitational_redshift" title="Gravitational redshift">gravitational redshift</a> in the frequency of light emitted at a lower height, where Earth's gravitational field is relatively more intense. The results were within 10% of the predictions of general relativity. In 1964, Pound and J. L. Snider measured a result within 1% of the value predicted by gravitational time dilation.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> (See <a href="/wiki/Pound%E2%80%93Rebka_experiment" title="Pound–Rebka experiment">Pound–Rebka experiment</a>)</li> <li>In 2010, gravitational time dilation was measured at the Earth's surface with a height difference of only one meter, using optical atomic clocks.<sup id="cite_ref-Chou_26-1" class="reference"><a href="#cite_note-Chou-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Combined_effect_of_velocity_and_gravitational_time_dilation">Combined effect of velocity and gravitational time dilation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=14" title="Edit section: Combined effect of velocity and gravitational time dilation"><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:Daily_satellite_time_dilation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Daily_satellite_time_dilation.png/350px-Daily_satellite_time_dilation.png" decoding="async" width="350" height="385" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/f9/Daily_satellite_time_dilation.png 1.5x" data-file-width="364" data-file-height="400" /></a><figcaption>Daily time dilation (gain or loss if negative) in microseconds as a function of (circular) orbit radius <i>r</i> = <i>rs</i>/<i>re</i>, where <i>rs</i> is satellite orbit radius and <i>re</i> is the equatorial Earth radius, calculated using the Schwarzschild metric. At <i>r</i> ≈ 1.497<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> there is no time dilation. Here the effects of motion and reduced gravity cancel. ISS astronauts fly below, whereas GPS and geostationary satellites fly above.<sup id="cite_ref-Ashby_2-2" class="reference"><a href="#cite_note-Ashby-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>High-accuracy timekeeping, low-Earth-orbit satellite tracking, and <a href="/wiki/Pulsar_timing" class="mw-redirect" title="Pulsar timing">pulsar timing</a> are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include the <a href="/wiki/International_Atomic_Time" title="International Atomic Time">International Atomic Time</a> standard and its relationship with the <a href="/wiki/Barycentric_Coordinate_Time" title="Barycentric Coordinate Time">Barycentric Coordinate Time</a> standard used for interplanetary objects. </p><p>Relativistic time dilation effects for the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> and the Earth can be modeled very precisely by the <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a> to the Einstein field equations. In the Schwarzschild metric, the interval <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 dt_{\text{E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt_{\text{E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6def0dfbc454013c31dae77e3bfbfe9c1c186dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.407ex; height:2.509ex;" alt="{\displaystyle dt_{\text{E}}}" /></span> is given by:<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ashby02_40-0" class="reference"><a href="#cite_note-ashby02-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 dt_{\text{E}}^{2}=\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)dt_{\text{c}}^{2}-\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)^{-1}{\frac {dx^{2}+dy^{2}+dz^{2}}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt_{\text{E}}^{2}=\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)dt_{\text{c}}^{2}-\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)^{-1}{\frac {dx^{2}+dy^{2}+dz^{2}}{c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fbd04106b03c9a1ddd8060d85312383856aaa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.469ex; height:6.509ex;" alt="{\displaystyle dt_{\text{E}}^{2}=\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)dt_{\text{c}}^{2}-\left(1-{\frac {2GM_{\text{i}}}{r_{\text{i}}c^{2}}}\right)^{-1}{\frac {dx^{2}+dy^{2}+dz^{2}}{c^{2}}}}" /></span></dd></dl> <p>where: </p> <ul><li><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 dt_{\text{E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt_{\text{E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6def0dfbc454013c31dae77e3bfbfe9c1c186dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.407ex; height:2.509ex;" alt="{\displaystyle dt_{\text{E}}}" /></span> is a small increment of proper time <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 t_{\text{E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf8a7b0eb05e0c7c80d72ddc99393aed7a8b810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.343ex;" alt="{\displaystyle t_{\text{E}}}" /></span> (an interval that could be recorded on an atomic clock),</li> <li><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 dt_{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt_{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc982e42ef3b4c8f59ddf2c09ec3e06e51e57162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="{\displaystyle dt_{\text{c}}}" /></span> is a small increment in the coordinate <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 t_{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c854c6efc4da872d5d9380eaf968512923b6bf42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.343ex;" alt="{\displaystyle t_{\text{c}}}" /></span> (<a href="/wiki/Coordinate_time" title="Coordinate time">coordinate time</a>),</li> <li><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 dx,dy,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx,dy,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0df5a109b0087620b32449bd5ea7b4c1fffd9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.289ex; height:2.509ex;" alt="{\displaystyle dx,dy,dz}" /></span> are small increments in the three coordinates <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 x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" /></span> of the clock's position,</li> <li><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 {\frac {-GM_{i}}{r_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {-GM_{i}}{r_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f52581c1db538c1f5fef4bd415a2f32c13d725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.525ex; height:5.676ex;" alt="{\displaystyle {\frac {-GM_{i}}{r_{i}}}}" /></span> represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances <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 r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}" /></span> from the clock. This sum includes any tidal potentials.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:Spacetime_lattice_analogy.svg" class="mw-file-description" title="Spacetime curvature schematic"><img alt="Spacetime curvature schematic" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/220px-Spacetime_lattice_analogy.svg.png" decoding="async" width="220" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/330px-Spacetime_lattice_analogy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/440px-Spacetime_lattice_analogy.svg.png 2x" data-file-width="1260" data-file-height="469" /></a></span><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><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 G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}" /></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <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/20px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="13" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:General_relativity" title="Category:General relativity">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><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:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The coordinate velocity of the clock is given by: </p> <dl><dd><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 v^{2}={\frac {dx^{2}+dy^{2}+dz^{2}}{dt_{\text{c}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={\frac {dx^{2}+dy^{2}+dz^{2}}{dt_{\text{c}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597f1b203962a863359b6931d572c7959227571f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.188ex; height:6.343ex;" alt="{\displaystyle v^{2}={\frac {dx^{2}+dy^{2}+dz^{2}}{dt_{\text{c}}^{2}}}}" /></span></dd></dl> <p>The coordinate time <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 t_{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c854c6efc4da872d5d9380eaf968512923b6bf42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.343ex;" alt="{\displaystyle t_{\text{c}}}" /></span> is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (<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 U=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d868ee7902023d29169252eb059f8faff9f08fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.044ex; height:2.176ex;" alt="{\displaystyle U=0}" /></span>), and stationary in the system of coordinates (<span class="nowrap"><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 v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}" /></span></span>). The exact relation between the rate of proper time and the rate of coordinate time for a clock with a radial component of velocity is: </p> <dl><dd><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 {\frac {dt_{\text{E}}}{dt_{\text{c}}}}={\sqrt {1+{\frac {2U}{c^{2}}}-{\frac {v^{2}}{c^{2}}}+\left({\frac {c^{2}}{2U}}+1\right)^{-1}{\frac {{v_{\shortparallel }}^{2}}{c^{2}}}}}={\sqrt {1-\left(\beta ^{2}+\beta _{e}^{2}+{\frac {\beta _{\shortparallel }^{2}\beta _{e}^{2}}{1-\beta _{e}^{2}}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>U</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dt_{\text{E}}}{dt_{\text{c}}}}={\sqrt {1+{\frac {2U}{c^{2}}}-{\frac {v^{2}}{c^{2}}}+\left({\frac {c^{2}}{2U}}+1\right)^{-1}{\frac {{v_{\shortparallel }}^{2}}{c^{2}}}}}={\sqrt {1-\left(\beta ^{2}+\beta _{e}^{2}+{\frac {\beta _{\shortparallel }^{2}\beta _{e}^{2}}{1-\beta _{e}^{2}}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9181c1d9767db5649ee4e409c9855465bd2d0e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.116ex; height:7.843ex;" alt="{\displaystyle {\frac {dt_{\text{E}}}{dt_{\text{c}}}}={\sqrt {1+{\frac {2U}{c^{2}}}-{\frac {v^{2}}{c^{2}}}+\left({\frac {c^{2}}{2U}}+1\right)^{-1}{\frac {{v_{\shortparallel }}^{2}}{c^{2}}}}}={\sqrt {1-\left(\beta ^{2}+\beta _{e}^{2}+{\frac {\beta _{\shortparallel }^{2}\beta _{e}^{2}}{1-\beta _{e}^{2}}}\right)}}}" /></span></dd></dl> <p>where: </p> <ul><li><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 v_{\shortparallel }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\shortparallel }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f8fbaf730acf0b9c2e676178e419013c67ec14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2ex; height:2.009ex;" alt="{\displaystyle v_{\shortparallel }}" /></span> is the radial velocity,</li> <li><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 v_{e}={\sqrt {\frac {2GM_{i}}{r_{i}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{e}={\sqrt {\frac {2GM_{i}}{r_{i}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ed241b5a6b32c57333194ef026c191968a3327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.427ex; height:7.676ex;" alt="{\displaystyle v_{e}={\sqrt {\frac {2GM_{i}}{r_{i}}}}}" /></span> is the escape speed,</li> <li><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 \beta =v/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =v/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c09197ac61cf6c55baab7eaaf25cbde57010efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.727ex; height:2.843ex;" alt="{\displaystyle \beta =v/c}" /></span>, <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 \beta _{e}=v_{e}/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{e}=v_{e}/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ef1e5d4c39ba03d4e6b9d2411a577279223176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.708ex; height:2.843ex;" alt="{\displaystyle \beta _{e}=v_{e}/c}" /></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 \beta _{\shortparallel }=v_{\shortparallel }/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{\shortparallel }=v_{\shortparallel }/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2308d5e460eca74109cdf62e3d7308f6d7de58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.455ex; height:2.843ex;" alt="{\displaystyle \beta _{\shortparallel }=v_{\shortparallel }/c}" /></span> are velocities as a percentage of speed of light <i>c</i>,</li> <li><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 U={\frac {-GM_{i}}{r_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {-GM_{i}}{r_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae27bb2a7446936a533189b6b21d2c1ed3d0202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.406ex; height:5.676ex;" alt="{\displaystyle U={\frac {-GM_{i}}{r_{i}}}}" /></span> is the Newtonian potential; hence <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 -U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed76156191d991f942061551fbf9283984d24c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.591ex; height:2.343ex;" alt="{\displaystyle -U}" /></span> equals half the square of the escape speed.</li></ul> <p>The above equation is exact under the assumptions of the Schwarzschild solution. It reduces to velocity time dilation equation in the presence of motion and absence of gravity, i.e. <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 \beta _{e}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{e}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b83217a076b757048c40052749d261fd87fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.575ex; height:2.509ex;" alt="{\displaystyle \beta _{e}=0}" /></span>. It reduces to gravitational time dilation equation in the absence of motion and presence of gravity, i.e. <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 \beta =0=\beta _{\shortparallel }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-variant">∥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =0=\beta _{\shortparallel }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/297b37cc0a743f489073a31a2a78723e59da21bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.879ex; height:2.509ex;" alt="{\displaystyle \beta =0=\beta _{\shortparallel }}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Experimental_testing_3">Experimental testing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=15" title="Edit section: Experimental testing"><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:Time_Dilation_vs_Orbital_Height.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Time_Dilation_vs_Orbital_Height.png/350px-Time_Dilation_vs_Orbital_Height.png" decoding="async" width="350" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Time_Dilation_vs_Orbital_Height.png/525px-Time_Dilation_vs_Orbital_Height.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Time_Dilation_vs_Orbital_Height.png/700px-Time_Dilation_vs_Orbital_Height.png 2x" data-file-width="1139" data-file-height="760" /></a><figcaption>Daily time dilation over circular orbit height split into its components. On this chart, only <a href="/wiki/Gravity_Probe_A" title="Gravity Probe A">Gravity Probe A</a> was launched specifically to <i>test</i> general relativity. The other spacecraft on this chart (except for the ISS, whose range of points is marked "theory") carry atomic clocks whose proper operation <i>depend on</i> the validity of general relativity.</figcaption></figure> <ul><li><a href="/wiki/Hafele%E2%80%93Keating_experiment" title="Hafele–Keating experiment">Hafele and Keating</a>, in 1971, flew <a href="/wiki/Caesium" title="Caesium">caesium</a> atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at the <a href="/wiki/U.S._Naval_Observatory" class="mw-redirect" title="U.S. Naval Observatory">U.S. Naval Observatory</a>. Two opposite effects came into play. The clocks were expected to age more quickly (show a larger elapsed time) than the reference clock since they were in a higher (weaker) gravitational potential for most of the trip (c.f. <a href="/wiki/Pound%E2%80%93Rebka_experiment" title="Pound–Rebka experiment">Pound–Rebka experiment</a>). But also, contrastingly, the moving clocks were expected to age more slowly because of the speed of their travel. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40±23 nanoseconds during the eastward trip and should have gained 275±21 nanoseconds during the westward trip. Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59±10 nanoseconds during the eastward trip and gained 273±7 nanoseconds during the westward trip (where the error bars represent standard deviation).<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> In 2005, the <a href="/wiki/National_Physical_Laboratory_(United_Kingdom)" title="National Physical Laboratory (United Kingdom)">National Physical Laboratory</a> in the United Kingdom reported their limited replication of this experiment.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> The NPL experiment differed from the original in that the caesium clocks were sent on a shorter trip (London–Washington, D.C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity, within the uncertainty of the measurements.</li> <li>The <a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Positioning System</a> can be considered a continuously operating experiment in both special and general relativity. The in-orbit clocks are corrected for both special and general relativistic time dilation effects <a href="#Combined_effect_of_velocity_and_gravitational_time_dilation">as described above</a>, so that (as observed from the Earth's surface) they run at the same rate as clocks on the surface of the Earth.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=16" title="Edit section: In popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Velocity and gravitational time dilation have been the subject of science fiction works in a variety of media. Some examples in film are the movies <i><a href="/wiki/Interstellar_(film)" title="Interstellar (film)">Interstellar</a></i> and <i><a href="/wiki/Planet_of_the_Apes_(1968_film)" title="Planet of the Apes (1968 film)">Planet of the Apes</a></i>.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> In <i>Interstellar</i>, a key plot point involves a planet, which is close to a <a href="/wiki/Rotating_black_hole" title="Rotating black hole">rotating black hole</a> and on the surface of which one hour is equivalent to seven years on Earth due to time dilation.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> Physicist <a href="/wiki/Kip_Thorne" title="Kip Thorne">Kip Thorne</a> collaborated in making the film and explained its scientific concepts in the book <i><a href="/wiki/The_Science_of_Interstellar" title="The Science of Interstellar">The Science of Interstellar</a></i>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Queen_(band)" title="Queen (band)">Queen</a> song <a href="/wiki/%2739" title="'39">'39</a> was written by astrophysicist as well as musician <a href="/wiki/Brian_May" title="Brian May">Brian May</a>, and is centred around the time dilation effect on spacefarers searching for a new home for mankind, as we gradually ruin planet Earth. They return successful, only to find that all and everything they knew has long since passed away. </p><p>Time dilation was used in the <i><a href="/wiki/Doctor_Who" title="Doctor Who">Doctor Who</a></i> episodes "<a href="/wiki/World_Enough_and_Time_(Doctor_Who)" title="World Enough and Time (Doctor Who)">World Enough and Time</a>" and "<a href="/wiki/The_Doctor_Falls" title="The Doctor Falls">The Doctor Falls</a>", which take place on a spaceship in the vicinity of a black hole. Due to the immense gravitational pull of the black hole and the ship's length (400 miles), time moves faster at one end than the other. When The Doctor's companion, Bill, gets taken away to the other end of the ship, she waits years for him to rescue her; in his time, only minutes pass.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Furthermore, the dilation allows the <a href="/wiki/Cybermen" class="mw-redirect" title="Cybermen">Cybermen</a> to evolve at a "faster" rate than previously seen in the show. </p><p><i><a href="/wiki/Tau_Zero" title="Tau Zero">Tau Zero</a></i>, a novel by <a href="/wiki/Poul_Anderson" title="Poul Anderson">Poul Anderson</a>, is an early example of the concept in science fiction literature. In the novel, a spacecraft uses a <a href="/wiki/Bussard_ramjet" title="Bussard ramjet">Bussard ramjet</a> to accelerate to high enough speeds that the crew spends five years on board, but thirty-three years pass on the Earth before they arrive at their destination. The velocity time dilation is explained by Anderson in terms of the <a href="/wiki/Tau_factor" class="mw-redirect" title="Tau factor">tau factor</a> which decreases closer and closer to zero as the ship approaches the speed of light—hence the title of the novel.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> Due to an accident, the crew is unable to stop accelerating the spacecraft, causing such extreme time dilation that the crew experiences the <a href="/wiki/Big_Crunch" title="Big Crunch">Big Crunch</a> at the end of the universe.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Other examples in literature, such as <i><a href="/wiki/Rocannon%27s_World" title="Rocannon's World">Rocannon's World</a></i>, <i><a href="/wiki/Hyperion_Cantos" title="Hyperion Cantos">Hyperion</a></i> and <i><a href="/wiki/The_Forever_War" title="The Forever War">The Forever War</a></i>, similarly make use of relativistic time dilation as a scientifically plausible literary device to have certain characters age slower than the rest of the universe.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <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=Time_dilation&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Mass in special relativity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=18" title="Edit section: Footnotes"><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"><ol class="references"> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Average time dilation has a weak dependence on the orbital inclination angle (Ashby 2003, p.32). The <i>r</i> ≈ 1.497 result corresponds to<sup id="cite_ref-Ashby3_37-0" class="reference"><a href="#cite_note-Ashby3-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> the orbital inclination of modern GPS satellites, which is 55 degrees.</span> </li> </ol></div></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=Time_dilation&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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 id="CITEREFHughesKersting2021" class="citation journal cs1">Hughes, Theo; Kersting, Magdalena (5 January 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-6552%2Fabce02">"The invisibility of time dilation"</a>. <i>Physics Education</i>. <b>56</b> (2): 025011. <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/2021PhyEd..56b5011H">2021PhyEd..56b5011H</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/10.1088%2F1361-6552%2Fabce02">10.1088/1361-6552/abce02</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Education&rft.atitle=The+invisibility+of+time+dilation&rft.volume=56&rft.issue=2&rft.pages=025011&rft.date=2021-01-05&rft_id=info%3Adoi%2F10.1088%2F1361-6552%2Fabce02&rft_id=info%3Abibcode%2F2021PhyEd..56b5011H&rft.aulast=Hughes&rft.aufirst=Theo&rft.au=Kersting%2C+Magdalena&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1361-6552%252Fabce02&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-Ashby-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ashby_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ashby_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Ashby_2-2"><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="CITEREFAshby2003" class="citation journal cs1">Ashby, Neil (2003). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5253894">"Relativity in the Global Positioning System"</a>. <i><a href="/wiki/Living_Reviews_in_Relativity" title="Living Reviews in Relativity">Living Reviews in Relativity</a></i>. <b>6</b> (1): 16. <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/2003LRR.....6....1A">2003LRR.....6....1A</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/10.12942%2Flrr-2003-1">10.12942/lrr-2003-1</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5253894">5253894</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28163638">28163638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Reviews+in+Relativity&rft.atitle=Relativity+in+the+Global+Positioning+System&rft.volume=6&rft.issue=1&rft.pages=16&rft.date=2003&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5253894%23id-name%3DPMC&rft_id=info%3Apmid%2F28163638&rft_id=info%3Adoi%2F10.12942%2Flrr-2003-1&rft_id=info%3Abibcode%2F2003LRR.....6....1A&rft.aulast=Ashby&rft.aufirst=Neil&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5253894&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMiller1981" class="citation book cs1">Miller, Arthur I. (1981). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/alberteinsteinss0000mill"><i>Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)</i></a></span>. Reading, Massachusetts: Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-04679-3" title="Special:BookSources/978-0-201-04679-3"><bdi>978-0-201-04679-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Albert+Einstein%27s+Special+Theory+of+Relativity%3A+Emergence+%281905%29+and+Early+Interpretation+%281905%E2%80%931911%29&rft.place=Reading%2C+Massachusetts&rft.pub=Addison%E2%80%93Wesley&rft.date=1981&rft.isbn=978-0-201-04679-3&rft.aulast=Miller&rft.aufirst=Arthur+I.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falberteinsteinss0000mill&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDarrigol2005" class="citation book cs1">Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity". <a rel="nofollow" class="external text" href="http://www.bourbaphy.fr/darrigol2.pdf"><i>Einstein, 1905–2005</i></a> <span class="cs1-format">(PDF)</span>. Vol. 1. pp. <span class="nowrap">1–</span>22. <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%2F3-7643-7436-5_1">10.1007/3-7643-7436-5_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-7435-8" title="Special:BookSources/978-3-7643-7435-8"><bdi>978-3-7643-7435-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Genesis+of+the+Theory+of+Relativity&rft.btitle=Einstein%2C+1905%E2%80%932005&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E22&rft.date=2005&rft_id=info%3Adoi%2F10.1007%2F3-7643-7436-5_1&rft.isbn=978-3-7643-7435-8&rft.aulast=Darrigol&rft.aufirst=Olivier&rft_id=http%3A%2F%2Fwww.bourbaphy.fr%2Fdarrigol2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|work=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></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="CITEREFLarmor1897" class="citation journal cs1">Larmor, Joseph (1897). <span class="cs1-ws-icon" title="s:Dynamical Theory of the Electric and Luminiferous Medium III"><a class="external text" href="https://en.wikisource.org/wiki/Dynamical_Theory_of_the_Electric_and_Luminiferous_Medium_III">"On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with Material Media" </a></span>. <i>Philosophical Transactions of the Royal Society</i>. <b>190</b>: <span class="nowrap">205–</span>300. <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/1897RSPTA.190..205L">1897RSPTA.190..205L</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/10.1098%2Frsta.1897.0020">10.1098/rsta.1897.0020</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=On+a+Dynamical+Theory+of+the+Electric+and+Luminiferous+Medium%2C+Part+3%2C+Relations+with+Material+Media&rft.volume=190&rft.pages=%3Cspan+class%3D%22nowrap%22%3E205-%3C%2Fspan%3E300&rft.date=1897&rft_id=info%3Adoi%2F10.1098%2Frsta.1897.0020&rft_id=info%3Abibcode%2F1897RSPTA.190..205L&rft.aulast=Larmor&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-cohn-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-cohn_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCohn1904" class="citation cs2 cs1-prop-long-vol cs1-prop-foreign-lang-source">Cohn, Emil (1904), <span class="cs1-ws-icon" title="s:de:Zur Elektrodynamik bewegter Systeme II"><a class="external text" href="https://en.wikisource.org/wiki/de:Zur_Elektrodynamik_bewegter_Systeme_II">"Zur Elektrodynamik bewegter Systeme II" </a></span> [<a href="https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Systems_II" class="extiw" title="s:Translation:On the Electrodynamics of Moving Systems II">On the Electrodynamics of Moving Systems II</a>], <i>Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften</i> (in German and English), vol. 1904/2, no. 43, pp. <span class="nowrap">1404–</span>1416</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften&rft.atitle=Zur+Elektrodynamik+bewegter+Systeme+II&rft.volume=1904%2F2&rft.issue=43&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1404-%3C%2Fspan%3E1416&rft.date=1904&rft.aulast=Cohn&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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 cs1-prop-foreign-lang-source">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="http://sedici.unlp.edu.ar/handle/10915/2786">"Zur Elektrodynamik bewegter Körper"</a>. <i>Annalen der Physik</i> (in German). <b>322</b> (10): <span class="nowrap">891–</span>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=%3Cspan+class%3D%22nowrap%22%3E891-%3C%2Fspan%3E921&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%2Fsedici.unlp.edu.ar%2Fhandle%2F10915%2F2786&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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-mink2-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-mink2_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMinkowski1908" class="citation cs2 cs1-prop-foreign-lang-source">Minkowski, Hermann (1908) [1907], <span class="cs1-ws-icon" title="s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern"><a class="external text" href="https://en.wikisource.org/wiki/de:Die_Grundgleichungen_f%C3%BCr_die_elektromagnetischen_Vorg%C3%A4nge_in_bewegten_K%C3%B6rpern">"Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" </a></span> [<a href="https://en.wikisource.org/wiki/Translation:The_Fundamental_Equations_for_Electromagnetic_Processes_in_Moving_Bodies" class="extiw" title="s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies">The Fundamental Equations for Electromagnetic Processes in Moving Bodies</a>], <i>Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse</i> (in German and English), pp. <span class="nowrap">53–</span>111</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nachrichten+von+der+Gesellschaft+der+Wissenschaften+zu+G%C3%B6ttingen%2C+Mathematisch-Physikalische+Klasse&rft.atitle=Die+Grundgleichungen+f%C3%BCr+die+elektromagnetischen+Vorg%C3%A4nge+in+bewegten+K%C3%B6rpern&rft.pages=%3Cspan+class%3D%22nowrap%22%3E53-%3C%2Fspan%3E111&rft.date=1908&rft.aulast=Minkowski&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></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="CITEREFHraskó2011" class="citation book cs1">Hraskó, Péter (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AEdvt1gc3eMC"><i>Basic Relativity: An Introductory Essay</i></a> (illustrated ed.). Springer Science & Business Media. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-17810-8" title="Special:BookSources/978-3-642-17810-8"><bdi>978-3-642-17810-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Relativity%3A+An+Introductory+Essay&rft.pages=60&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011&rft.isbn=978-3-642-17810-8&rft.aulast=Hrask%C3%B3&rft.aufirst=P%C3%A9ter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAEdvt1gc3eMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AEdvt1gc3eMC&pg=PA60">Extract of page 60</a></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="CITEREFCalder2006" class="citation book cs1">Calder, Nigel (2006). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/magicuniversegra0000cald"><i>Magic Universe: A grand tour of modern science</i></a></span>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/magicuniversegra0000cald/page/378">378</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-280669-7" title="Special:BookSources/978-0-19-280669-7"><bdi>978-0-19-280669-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Magic+Universe%3A+A+grand+tour+of+modern+science&rft.pages=378&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=978-0-19-280669-7&rft.aulast=Calder&rft.aufirst=Nigel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmagicuniversegra0000cald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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">-25 microseconds per day results in 0.00458 seconds per 183 days</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="CITEREFOverbye2005" class="citation news cs1">Overbye, Dennis (2005-06-28). <a rel="nofollow" class="external text" href="https://www.nytimes.com/2005/06/28/science/a-trip-forward-in-time-your-travel-agent-einstein.html?_r=0">"A Trip Forward in Time. Your Travel Agent: Einstein"</a>. <i><a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-12-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=A+Trip+Forward+in+Time.+Your+Travel+Agent%3A+Einstein.&rft.date=2005-06-28&rft.aulast=Overbye&rft.aufirst=Dennis&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2005%2F06%2F28%2Fscience%2Fa-trip-forward-in-time-your-travel-agent-einstein.html%3F_r%3D0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGott2002" class="citation book cs1">Gott, Richard J. (2002). <i>Time Travel in Einstein's Universe</i>. p. 75.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Travel+in+Einstein%27s+Universe&rft.pages=75&rft.date=2002&rft.aulast=Gott&rft.aufirst=Richard+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></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="CITEREFCassidyHoltonRutherford2002" class="citation book cs1">Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rpQo7f9F1xUC&pg=PA422"><i>Understanding Physics</i></a>. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 422. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98756-9" title="Special:BookSources/978-0-387-98756-9"><bdi>978-0-387-98756-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Physics&rft.pages=422&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-0-387-98756-9&rft.aulast=Cassidy&rft.aufirst=David+C.&rft.au=Holton%2C+Gerald+James&rft.au=Rutherford%2C+Floyd+James&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrpQo7f9F1xUC%26pg%3DPA422&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCutner2003" class="citation book cs1">Cutner, Mark Leslie (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2QVmiMW0O0MC&pg=PA128"><i>Astronomy, A Physical Perspective</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 128. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-82196-4" title="Special:BookSources/978-0-521-82196-4"><bdi>978-0-521-82196-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomy%2C+A+Physical+Perspective&rft.pages=128&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-82196-4&rft.aulast=Cutner&rft.aufirst=Mark+Leslie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2QVmiMW0O0MC%26pg%3DPA128&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLerner1996" class="citation book cs1">Lerner, Lawrence S. (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1051"><i>Physics for Scientists and Engineers, Volume 2</i></a>. <a href="/wiki/Jones_and_Bartlett" class="mw-redirect" title="Jones and Bartlett">Jones and Bartlett</a>. pp. <span class="nowrap">1051–</span>1052. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-0460-5" title="Special:BookSources/978-0-7637-0460-5"><bdi>978-0-7637-0460-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%2C+Volume+2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1051-%3C%2Fspan%3E1052&rft.pub=Jones+and+Bartlett&rft.date=1996&rft.isbn=978-0-7637-0460-5&rft.aulast=Lerner&rft.aufirst=Lawrence+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB8K_ym9rS6UC%26pg%3DPA1051&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEllisWilliams2000" class="citation book cs1">Ellis, George F. R.; Williams, Ruth M. 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New York: 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&rft.pub=W.+H.+Freeman&rft.date=1992&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%2Farchive.org%2Fdetails%2Fspacetimephysics00edwi_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-born-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-born_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBorn1964" class="citation cs2"><a href="/wiki/Max_Born" title="Max Born">Born, Max</a> (1964), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/einsteinstheoryo0000born"><i>Einstein's Theory of Relativity</i></a></span>, Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60769-6" title="Special:BookSources/978-0-486-60769-6"><bdi>978-0-486-60769-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+Theory+of+Relativity&rft.pub=Dover+Publications&rft.date=1964&rft.isbn=978-0-486-60769-6&rft.aulast=Born&rft.aufirst=Max&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feinsteinstheoryo0000born&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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="CITEREFPetkov2009" class="citation book cs1">Petkov, Vesselin (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AzfFo6A94WEC"><i>Relativity and the Nature of Spacetime</i></a> (2nd, illustrated ed.). Springer Science & Business Media. p. 87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-01962-3" title="Special:BookSources/978-3-642-01962-3"><bdi>978-3-642-01962-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity+and+the+Nature+of+Spacetime&rft.pages=87&rft.edition=2nd%2C+illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-3-642-01962-3&rft.aulast=Petkov&rft.aufirst=Vesselin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAzfFo6A94WEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AzfFo6A94WEC&pg=PA87">Extract of page 87</a></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">See equations 3, 4, 6 and 9 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIorio2005" class="citation journal cs1">Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". <i><a href="/wiki/Foundations_of_Physics_Letters" class="mw-redirect" title="Foundations of Physics Letters">Foundations of Physics Letters</a></i>. <b>18</b> (1): <span class="nowrap">1–</span>19. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0405038">physics/0405038</a></span>. <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/2005FoPhL..18....1I">2005FoPhL..18....1I</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%2Fs10702-005-2466-8">10.1007/s10702-005-2466-8</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:15081211">15081211</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics+Letters&rft.atitle=An+analytical+treatment+of+the+Clock+Paradox+in+the+framework+of+the+Special+and+General+Theories+of+Relativity&rft.volume=18&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E19&rft.date=2005&rft_id=info%3Aarxiv%2Fphysics%2F0405038&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15081211%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10702-005-2466-8&rft_id=info%3Abibcode%2F2005FoPhL..18....1I&rft.aulast=Iorio&rft.aufirst=Lorenzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" 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="CITEREFRindler1977" class="citation book cs1">Rindler, W. (1977). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/essentialrelativ00rind_279"><i>Essential Relativity</i></a></span>. Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/essentialrelativ00rind_279/page/n61">49</a>–50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3540079705" title="Special:BookSources/978-3540079705"><bdi>978-3540079705</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Relativity&rft.pages=49-50&rft.pub=Springer&rft.date=1977&rft.isbn=978-3540079705&rft.aulast=Rindler&rft.aufirst=W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fessentialrelativ00rind_279&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-Bailey_1977-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bailey_1977_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaileyBorerCombleyDrumm1977" class="citation journal cs1">Bailey, H.; Borer, K.; Combley, F.; Drumm, H.; Krienen, F.; Lange, F.; Picasso, E.; von Ruden, W.; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". <i>Nature</i>. <b>268</b> (5618): <span class="nowrap">301–</span>305. <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/1977Natur.268..301B">1977Natur.268..301B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F268301a0">10.1038/268301a0</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:4173884">4173884</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Measurements+of+relativistic+time+dilatation+for+positive+and+negative+muons+in+a+circular+orbit&rft.volume=268&rft.issue=5618&rft.pages=%3Cspan+class%3D%22nowrap%22%3E301-%3C%2Fspan%3E305&rft.date=1977&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4173884%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F268301a0&rft_id=info%3Abibcode%2F1977Natur.268..301B&rft.aulast=Bailey&rft.aufirst=H.&rft.au=Borer%2C+K.&rft.au=Combley%2C+F.&rft.au=Drumm%2C+H.&rft.au=Krienen%2C+F.&rft.au=Lange%2C+F.&rft.au=Picasso%2C+E.&rft.au=von+Ruden%2C+W.&rft.au=Farley+F.+J.+M.&rft.au=Field+J.+H.&rft.au=Flegel+W.&rft.au=Hattersley+P.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRoosMarraffinoReucroftWaters1980" class="citation journal cs1">Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration". <i>Nature</i>. <b>286</b> (5770): <span class="nowrap">244–</span>245. <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/1980Natur.286..244R">1980Natur.286..244R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F286244a0">10.1038/286244a0</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:4280317">4280317</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=%CF%83%2B%2F-+lifetimes+and+longitudinal+acceleration&rft.volume=286&rft.issue=5770&rft.pages=%3Cspan+class%3D%22nowrap%22%3E244-%3C%2Fspan%3E245&rft.date=1980&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4280317%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F286244a0&rft_id=info%3Abibcode%2F1980Natur.286..244R&rft.aulast=Roos&rft.aufirst=C.+E.&rft.au=Marraffino%2C+J.&rft.au=Reucroft%2C+S.&rft.au=Waters%2C+J.&rft.au=Webster%2C+M.+S.&rft.au=Williams%2C+E.+G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://phys.org/news/2016-05-earth-core-younger-thought.html">"New calculations show Earth's core is much younger than thought"</a>. Phys.org. 26 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=New+calculations+show+Earth%27s+core+is+much+younger+than+thought&rft.pub=Phys.org&rft.date=2016-05-26&rft_id=http%3A%2F%2Fphys.org%2Fnews%2F2016-05-earth-core-younger-thought.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBurnsLeveilleDominguezGebhard2017" class="citation journal cs1">Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>85</b> (10): <span class="nowrap">757–</span>762. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1707.00171">1707.00171</a></span>. <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/2017AmJPh..85..757B">2017AmJPh..85..757B</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%2F1.5000802">10.1119/1.5000802</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:119503665">119503665</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=Measurement+of+gravitational+time+dilation%3A+An+undergraduate+research+project&rft.volume=85&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E757-%3C%2Fspan%3E762&rft.date=2017-09-18&rft_id=info%3Aarxiv%2F1707.00171&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119503665%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.5000802&rft_id=info%3Abibcode%2F2017AmJPh..85..757B&rft.aulast=Burns&rft.aufirst=M.+Shane&rft.au=Leveille%2C+Michael+D.&rft.au=Dominguez%2C+Armand+R.&rft.au=Gebhard%2C+Brian+B.&rft.au=Huestis%2C+Samuel+E.&rft.au=Steele%2C+Jeffrey&rft.au=Patterson%2C+Brian&rft.au=Sell%2C+Jerry+F.&rft.au=Serna%2C+Mario&rft.au=Gearba%2C+M.+Alina&rft.au=Olesen%2C+Robert&rft.au=O%27Shea%2C+Patrick&rft.au=Schiller%2C+Jonathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPoundSnider_J._L.1964" class="citation journal cs1">Pound, R. V.; Snider J. L. (November 2, 1964). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.539">"Effect of Gravity on Nuclear Resonance"</a>. <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>13</b> (18): <span class="nowrap">539–</span>540. <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/1964PhRvL..13..539P">1964PhRvL..13..539P</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/10.1103%2FPhysRevLett.13.539">10.1103/PhysRevLett.13.539</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Effect+of+Gravity+on+Nuclear+Resonance&rft.volume=13&rft.issue=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E539-%3C%2Fspan%3E540&rft.date=1964-11-02&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.13.539&rft_id=info%3Abibcode%2F1964PhRvL..13..539P&rft.aulast=Pound&rft.aufirst=R.+V.&rft.au=Snider+J.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.13.539&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-Ashby3-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ashby3_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAshby2002" class="citation journal cs1">Ashby, Neil (2002). <a rel="nofollow" class="external text" href="http://www.physicstoday.org/resource/1/phtoad/v55/i5/p41_s1">"Relativity in the Global Positioning System"</a>. <i><a href="/wiki/Physics_Today" title="Physics Today">Physics Today</a></i>. <b>55</b> (5): 45. <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/2002PhT....55e..41A">2002PhT....55e..41A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1485583">10.1063/1.1485583</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5253894">5253894</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28163638">28163638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Relativity+in+the+Global+Positioning+System&rft.volume=55&rft.issue=5&rft.pages=45&rft.date=2002&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5253894%23id-name%3DPMC&rft_id=info%3Apmid%2F28163638&rft_id=info%3Adoi%2F10.1063%2F1.1485583&rft_id=info%3Abibcode%2F2002PhT....55e..41A&rft.aulast=Ashby&rft.aufirst=Neil&rft_id=http%3A%2F%2Fwww.physicstoday.org%2Fresource%2F1%2Fphtoad%2Fv55%2Fi5%2Fp41_s1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">See equations 2 & 3 (combined here and divided throughout by <i>c</i><sup>2</sup>) at pp. 35–36 in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMoyer1981" class="citation journal cs1">Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". <i><a href="/wiki/Celestial_Mechanics_(journal)" class="mw-redirect" title="Celestial Mechanics (journal)">Celestial Mechanics</a></i>. <b>23</b> (1): <span class="nowrap">33–</span>56. <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/1981CeMec..23...33M">1981CeMec..23...33M</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%2FBF01228543">10.1007/BF01228543</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F19770007221">2060/19770007221</a></span>. <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:118077433">118077433</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics&rft.atitle=Transformation+from+proper+time+on+Earth+to+coordinate+time+in+solar+system+barycentric+space-time+frame+of+reference&rft.volume=23&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E33-%3C%2Fspan%3E56&rft.date=1981&rft_id=info%3Ahdl%2F2060%2F19770007221&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118077433%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01228543&rft_id=info%3Abibcode%2F1981CeMec..23...33M&rft.aulast=Moyer&rft.aufirst=T.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-ashby02-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-ashby02_40-0">^</a></b></span> <span class="reference-text">A version of the same relationship can also be seen at equation 2 in<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAshbey2002" class="citation journal cs1">Ashbey, Neil (2002). <a rel="nofollow" class="external text" href="http://www.ipgp.fr/~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf">"Relativity and the Global Positioning System"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Physics_Today" title="Physics Today">Physics Today</a></i>. <b>55</b> (5): 45. <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/2002PhT....55e..41A">2002PhT....55e..41A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1485583">10.1063/1.1485583</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Relativity+and+the+Global+Positioning+System&rft.volume=55&rft.issue=5&rft.pages=45&rft.date=2002&rft_id=info%3Adoi%2F10.1063%2F1.1485583&rft_id=info%3Abibcode%2F2002PhT....55e..41A&rft.aulast=Ashbey&rft.aufirst=Neil&rft_id=http%3A%2F%2Fwww.ipgp.fr%2F~tarantola%2FFiles%2FProfessional%2FGPS%2FNeil_Ashby_Relativity_GPS.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNave2005" class="citation web cs1">Nave, C. R. (22 August 2005). <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html">"Hafele and Keating Experiment"</a>. <i><a href="/wiki/HyperPhysics" title="HyperPhysics">HyperPhysics</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=HyperPhysics&rft.atitle=Hafele+and+Keating+Experiment&rft.date=2005-08-22&rft.aulast=Nave&rft.aufirst=C.+R.&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Frelativ%2Fairtim.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.npl.co.uk/upload/pdf/metromnia_issue18.pdf">"Einstein"</a> <span class="cs1-format">(PDF)</span>. <i>Metromnia</i>. <a href="/wiki/National_Physical_Laboratory_(United_Kingdom)" title="National Physical Laboratory (United Kingdom)">National Physical Laboratory</a>. 2005. pp. <span class="nowrap">1–</span>4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Metromnia&rft.atitle=Einstein&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E4&rft.date=2005&rft_id=http%3A%2F%2Fwww.npl.co.uk%2Fupload%2Fpdf%2Fmetromnia_issue18.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaplanHegarty2005" class="citation book cs1">Kaplan, Elliott; Hegarty, Christopher (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-sPXPuOW7ggC"><i>Understanding GPS: Principles and Applications</i></a>. Artech House. p. 306. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58053-895-4" title="Special:BookSources/978-1-58053-895-4"><bdi>978-1-58053-895-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+GPS%3A+Principles+and+Applications&rft.pages=306&rft.pub=Artech+House&rft.date=2005&rft.isbn=978-1-58053-895-4&rft.aulast=Kaplan&rft.aufirst=Elliott&rft.au=Hegarty%2C+Christopher&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-sPXPuOW7ggC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-sPXPuOW7ggC&pg=PA306">Extract of page 306</a></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeiner2008" class="citation web cs1">Weiner, Adam (30 April 2008). <a rel="nofollow" class="external text" href="https://www.popsci.com/entertainment-gaming/article/2008-04/science-sci-fi/">"The Science of Sci-Fi"</a>. <i><a href="/wiki/Popular_Science" title="Popular Science">Popular Science</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Popular+Science&rft.atitle=The+Science+of+Sci-Fi&rft.date=2008-04-30&rft.aulast=Weiner&rft.aufirst=Adam&rft_id=https%3A%2F%2Fwww.popsci.com%2Fentertainment-gaming%2Farticle%2F2008-04%2Fscience-sci-fi%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLuminet2016" class="citation web cs1"><a href="/wiki/Jean-Pierre_Luminet" title="Jean-Pierre Luminet">Luminet, Jean-Pierre</a> (16 January 2016). <a rel="nofollow" class="external text" href="https://blogs.futura-sciences.com/e-luminet/2016/01/16/warped-science-interstellar-46-time-dilation-penrose-process/">"The Warped Science of Interstellar (4/6) : Time dilation and Penrose process"</a>. <i>e-LUMINESCIENCES</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=e-LUMINESCIENCES&rft.atitle=The+Warped+Science+of+Interstellar+%284%2F6%29+%3A+Time+dilation+and+Penrose+process&rft.date=2016-01-16&rft.aulast=Luminet&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Fblogs.futura-sciences.com%2Fe-luminet%2F2016%2F01%2F16%2Fwarped-science-interstellar-46-time-dilation-penrose-process%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKranking2019" class="citation web cs1">Kranking, Carlyn (31 May 2019). 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Interviewed by Marlow Stern.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Daily+Beast&rft.atitle=Neil+deGrasse+Tyson+Breaks+Down+%27Interstellar%27%3A+Black+Holes%2C+Time+Dilations%2C+and+Massive+Waves&rft.date=2017-07-12&rft.aulast=Tyson&rft.aufirst=Neil+deGrasse&rft_id=https%3A%2F%2Fwww.thedailybeast.com%2Fneil-degrasse-tyson-breaks-down-interstellar-black-holes-time-dilations-and-massive-waves&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCollins2017" class="citation web cs1">Collins, Frank (26 June 2017). <a rel="nofollow" class="external text" href="http://www.framerated.co.uk/doctor-who-10x11-world-time/">"DOCTOR WHO, 10.11 – 'World Enough and Time'<span class="cs1-kern-right"></span>"</a>. <i>Frame Rated</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Frame+Rated&rft.atitle=DOCTOR+WHO%2C+10.11+%E2%80%93+%27World+Enough+and+Time%27&rft.date=2017-06-26&rft.aulast=Collins&rft.aufirst=Frank&rft_id=http%3A%2F%2Fwww.framerated.co.uk%2Fdoctor-who-10x11-world-time%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMeaney2003" class="citation web cs1"><a href="/wiki/John_Meaney" title="John Meaney">Meaney, John</a> (17 December 2003). <a rel="nofollow" class="external text" href="http://johnmeaney.com/nopub/107169527892588754.html">"Time passages (2)"</a>. <i>John Meaney's WebLog</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=John+Meaney%27s+WebLog&rft.atitle=Time+passages+%282%29&rft.date=2003-12-17&rft.aulast=Meaney&rft.aufirst=John&rft_id=http%3A%2F%2Fjohnmeaney.com%2Fnopub%2F107169527892588754.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLangfordStableford2018" class="citation web cs1"><a href="/wiki/David_Langford" title="David Langford">Langford, David</a>; <a href="/wiki/Brian_Stableford" title="Brian Stableford">Stableford, Brian M</a> (20 August 2018). <a href="/wiki/John_Clute" title="John Clute">Clute, John</a>; Langford, David; <a href="/wiki/Peter_Nicholls_(writer)" title="Peter Nicholls (writer)">Nicholls, Peter</a>; <a href="/wiki/Graham_Sleight" title="Graham Sleight">Sleight, Graham</a> (eds.). <a rel="nofollow" class="external text" href="http://sf-encyclopedia.com/entry/relativity">"Relativity"</a>. <i><a href="/wiki/The_Encyclopedia_of_Science_Fiction" title="The Encyclopedia of Science Fiction">The Encyclopedia of Science Fiction</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Encyclopedia+of+Science+Fiction&rft.atitle=Relativity&rft.date=2018-08-20&rft.aulast=Langford&rft.aufirst=David&rft.au=Stableford%2C+Brian+M&rft_id=http%3A%2F%2Fsf-encyclopedia.com%2Fentry%2Frelativity&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCramer1989" class="citation magazine cs1"><a href="/wiki/John_G._Cramer" title="John G. Cramer">Cramer, John G.</a> (20 August 1989). <a rel="nofollow" class="external text" href="https://www.npl.washington.edu/AV/altvw38.html">"The Twin Paradox Revisited"</a>. <i><a href="/wiki/Analog_Science_Fiction_and_Fact" title="Analog Science Fiction and Fact">Analog Science Fiction and Fact</a></i>. No. March-1990 – via <a href="/wiki/University_of_Washington" title="University of Washington">University of Washington</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Analog+Science+Fiction+and+Fact&rft.atitle=The+Twin+Paradox+Revisited&rft.issue=March-1990&rft.date=1989-08-20&rft.aulast=Cramer&rft.aufirst=John+G.&rft_id=https%3A%2F%2Fwww.npl.washington.edu%2FAV%2Faltvw38.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWalter2018" class="citation web cs1">Walter, Damien (22 February 2018). <a rel="nofollow" class="external text" href="https://www.theguardian.com/books/2016/jan/15/sci-fi-general-relativity-einstein-planet-of-the-apes">"It's about time: how sci-fi has described Einstein's universe"</a>. <i><a href="/wiki/The_Guardian" title="The Guardian">The Guardian</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Guardian&rft.atitle=It%27s+about+time%3A+how+sci-fi+has+described+Einstein%27s+universe&rft.date=2018-02-22&rft.aulast=Walter&rft.aufirst=Damien&rft_id=https%3A%2F%2Fwww.theguardian.com%2Fbooks%2F2016%2Fjan%2F15%2Fsci-fi-general-relativity-einstein-planet-of-the-apes&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Time_dilation&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCallenderEdney2001" class="citation book cs1"><a href="/wiki/Craig_Callender" title="Craig Callender">Callender, C.</a>; Edney, R. (2001). <i>Introducing Time</i>. <a href="/wiki/Icon_Books" class="mw-redirect" title="Icon Books">Icon Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84046-592-1" title="Special:BookSources/978-1-84046-592-1"><bdi>978-1-84046-592-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Time&rft.pub=Icon+Books&rft.date=2001&rft.isbn=978-1-84046-592-1&rft.aulast=Callender&rft.aufirst=C.&rft.au=Edney%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEinstein1905" class="citation journal cs1">Einstein, A. (1905). <a rel="nofollow" class="external text" href="http://sedici.unlp.edu.ar/bitstream/handle/10915/2786/Documento_completo__.pdf?sequence=1">"Zur Elektrodynamik bewegter Körper"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i>. <b>322</b> (10): 891. <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&rft.date=1905&rft_id=info%3Adoi%2F10.1002%2Fandp.19053221004&rft_id=info%3Abibcode%2F1905AnP...322..891E&rft.aulast=Einstein&rft.aufirst=A.&rft_id=http%3A%2F%2Fsedici.unlp.edu.ar%2Fbitstream%2Fhandle%2F10915%2F2786%2FDocumento_completo__.pdf%3Fsequence%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEinstein1907" class="citation journal cs1">Einstein, A. (1907). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1424095">"Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips"</a>. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i>. <b>328</b> (6): <span class="nowrap">197–</span>198. <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/1907AnP...328..197E">1907AnP...328..197E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19073280613">10.1002/andp.19073280613</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=%C3%9Cber+die+M%C3%B6glichkeit+einer+neuen+Pr%C3%BCfung+des+Relativit%C3%A4tsprinzips&rft.volume=328&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E198&rft.date=1907&rft_id=info%3Adoi%2F10.1002%2Fandp.19073280613&rft_id=info%3Abibcode%2F1907AnP...328..197E&rft.aulast=Einstein&rft.aufirst=A.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1424095&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHasselkampMondryScharmann1979" class="citation journal cs1">Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct Observation of the Transversal Doppler-Shift". <i><a href="/wiki/Zeitschrift_f%C3%BCr_Physik_A" class="mw-redirect" title="Zeitschrift für Physik A">Zeitschrift für Physik A</a></i>. <b>289</b> (2): <span class="nowrap">151–</span>155. <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/1979ZPhyA.289..151H">1979ZPhyA.289..151H</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%2FBF01435932">10.1007/BF01435932</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:120963034">120963034</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+A&rft.atitle=Direct+Observation+of+the+Transversal+Doppler-Shift&rft.volume=289&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E151-%3C%2Fspan%3E155&rft.date=1979&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120963034%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01435932&rft_id=info%3Abibcode%2F1979ZPhyA.289..151H&rft.aulast=Hasselkamp&rft.aufirst=D.&rft.au=Mondry%2C+E.&rft.au=Scharmann%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIvesStilwell1938" class="citation journal cs1">Ives, H. E.; Stilwell, G. R. (1938). "An experimental study of the rate of a moving clock". <i><a href="/wiki/Journal_of_the_Optical_Society_of_America" title="Journal of the Optical Society of America">Journal of the Optical Society of America</a></i>. <b>28</b> (7): <span class="nowrap">215–</span>226. <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/1938JOSA...28..215I">1938JOSA...28..215I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FJOSA.28.000215">10.1364/JOSA.28.000215</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Optical+Society+of+America&rft.atitle=An+experimental+study+of+the+rate+of+a+moving+clock&rft.volume=28&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E215-%3C%2Fspan%3E226&rft.date=1938&rft_id=info%3Adoi%2F10.1364%2FJOSA.28.000215&rft_id=info%3Abibcode%2F1938JOSA...28..215I&rft.aulast=Ives&rft.aufirst=H.+E.&rft.au=Stilwell%2C+G.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIvesStilwell1941" class="citation journal cs1">Ives, H. E.; Stilwell, G. R. (1941). "An experimental study of the rate of a moving clock. II". <i><a href="/wiki/Journal_of_the_Optical_Society_of_America" title="Journal of the Optical Society of America">Journal of the Optical Society of America</a></i>. <b>31</b> (5): <span class="nowrap">369–</span>374. <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/1941JOSA...31..369I">1941JOSA...31..369I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FJOSA.31.000369">10.1364/JOSA.31.000369</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Optical+Society+of+America&rft.atitle=An+experimental+study+of+the+rate+of+a+moving+clock.+II&rft.volume=31&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E369-%3C%2Fspan%3E374&rft.date=1941&rft_id=info%3Adoi%2F10.1364%2FJOSA.31.000369&rft_id=info%3Abibcode%2F1941JOSA...31..369I&rft.aulast=Ives&rft.aufirst=H.+E.&rft.au=Stilwell%2C+G.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoos1959" class="citation book cs1">Joos, G. (1959). "Bewegte Bezugssysteme in der Akustik. Der Doppler-Effekt". <i>Lehrbuch der Theoretischen Physik, Zweites Buch</i> (11th ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bewegte+Bezugssysteme+in+der+Akustik.+Der+Doppler-Effekt&rft.btitle=Lehrbuch+der+Theoretischen+Physik%2C+Zweites+Buch&rft.edition=11th&rft.date=1959&rft.aulast=Joos&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLarmor1897" class="citation journal cs1">Larmor, J. (1897). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1897.0020">"On a dynamical theory of the electric and luminiferous medium"</a>. <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society" title="Philosophical Transactions of the Royal Society">Philosophical Transactions of the Royal Society</a></i>. <b>190</b>: <span class="nowrap">205–</span>300. <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/1897RSPTA.190..205L">1897RSPTA.190..205L</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/10.1098%2Frsta.1897.0020">10.1098/rsta.1897.0020</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=On+a+dynamical+theory+of+the+electric+and+luminiferous+medium&rft.volume=190&rft.pages=%3Cspan+class%3D%22nowrap%22%3E205-%3C%2Fspan%3E300&rft.date=1897&rft_id=info%3Adoi%2F10.1098%2Frsta.1897.0020&rft_id=info%3Abibcode%2F1897RSPTA.190..205L&rft.aulast=Larmor&rft.aufirst=J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frsta.1897.0020&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span> (third and last in a series of papers with the same name).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPoincaré1900" class="citation journal cs1">Poincaré, H. (1900). <a rel="nofollow" class="external text" href="http://www.biodiversitylibrary.org/pdf2/003524600031470.pdf">"La théorie de Lorentz et le principe de Réaction"</a> <span class="cs1-format">(PDF)</span>. <i>Archives Néerlandaises</i>. <b>5</b>: <span class="nowrap">253–</span>78.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archives+N%C3%A9erlandaises&rft.atitle=La+th%C3%A9orie+de+Lorentz+et+le+principe+de+R%C3%A9action&rft.volume=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E253-%3C%2Fspan%3E78&rft.date=1900&rft.aulast=Poincar%C3%A9&rft.aufirst=H.&rft_id=http%3A%2F%2Fwww.biodiversitylibrary.org%2Fpdf2%2F003524600031470.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPuri2015" class="citation journal cs1">Puri, A. (2015). "Einstein versus the simple pendulum formula: does gravity slow all clocks?". <i><a href="/wiki/Physics_Education" title="Physics Education">Physics Education</a></i>. <b>50</b> (4): 431. <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/2015PhyEd..50..431P">2015PhyEd..50..431P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9120%2F50%2F4%2F431">10.1088/0031-9120/50/4/431</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:118217730">118217730</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Education&rft.atitle=Einstein+versus+the+simple+pendulum+formula%3A+does+gravity+slow+all+clocks%3F&rft.volume=50&rft.issue=4&rft.pages=431&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118217730%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0031-9120%2F50%2F4%2F431&rft_id=info%3Abibcode%2F2015PhyEd..50..431P&rft.aulast=Puri&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReinhardt2007" class="citation journal cs1">Reinhardt, S.; et al. (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090712195322/http://www.mpq.mpg.de/~haensch/comb/people/thomas/NaturePhysics07.pdf">"Test of relativistic time dilation with fast optical atomic clocks at different velocities"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Nature_Physics" title="Nature Physics">Nature Physics</a></i>. <b>3</b> (12): <span class="nowrap">861–</span>864. <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/2007NatPh...3..861R">2007NatPh...3..861R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys778">10.1038/nphys778</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mpq.mpg.de/~haensch/comb/people/thomas/NaturePhysics07.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2009-07-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=Test+of+relativistic+time+dilation+with+fast+optical+atomic+clocks+at+different+velocities&rft.volume=3&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E861-%3C%2Fspan%3E864&rft.date=2007&rft_id=info%3Adoi%2F10.1038%2Fnphys778&rft_id=info%3Abibcode%2F2007NatPh...3..861R&rft.aulast=Reinhardt&rft.aufirst=S.&rft_id=http%3A%2F%2Fwww.mpq.mpg.de%2F~haensch%2Fcomb%2Fpeople%2Fthomas%2FNaturePhysics07.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRossiHall1941" class="citation journal cs1">Rossi, B.; Hall, D. B. (1941). "Variation of the Rate of Decay of Mesotrons with Momentum". <i><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></i>. <b>59</b> (3): 223. <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/1941PhRv...59..223R">1941PhRv...59..223R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.59.223">10.1103/PhysRev.59.223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Variation+of+the+Rate+of+Decay+of+Mesotrons+with+Momentum&rft.volume=59&rft.issue=3&rft.pages=223&rft.date=1941&rft_id=info%3Adoi%2F10.1103%2FPhysRev.59.223&rft_id=info%3Abibcode%2F1941PhRv...59..223R&rft.aulast=Rossi&rft.aufirst=B.&rft.au=Hall%2C+D.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeiss" class="citation web cs1">Weiss, M. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170529153938/http://tf.nist.gov/timefreq/time/twoway.htm">"Two way time transfer for satellites"</a>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>. Archived from <a rel="nofollow" class="external text" href="http://tf.nist.gov/timefreq/time/twoway.htm">the original</a> on 2017-05-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Two+way+time+transfer+for+satellites&rft.pub=National+Institute+of+Standards+and+Technology&rft.aulast=Weiss&rft.aufirst=M.&rft_id=http%3A%2F%2Ftf.nist.gov%2Ftimefreq%2Ftime%2Ftwoway.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVoigt1887" class="citation journal cs1">Voigt, W. (1887). "Über das Doppler'sche princip". <i>Nachrichten von der Königlicher Gesellschaft der Wissenschaften zu Göttingen</i>. <b>2</b>: <span class="nowrap">41–</span>51.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nachrichten+von+der+K%C3%B6niglicher+Gesellschaft+der+Wissenschaften+zu+G%C3%B6ttingen&rft.atitle=%C3%9Cber+das+Doppler%27sche+princip&rft.volume=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E41-%3C%2Fspan%3E51&rft.date=1887&rft.aulast=Voigt&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></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=Time_dilation&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Time_dilation" class="extiw" title="commons:Category:Time dilation">Time dilation</a> at Wikimedia Commons</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMerrifield" class="citation web cs1">Merrifield, Michael. <a rel="nofollow" class="external text" href="http://www.sixtysymbols.com/videos/lorentz.htm">"Lorentz Factor (and time dilation)"</a>. <i>Sixty Symbols</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=Lorentz+Factor+%28and+time+dilation%29&rft.aulast=Merrifield&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Florentz.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATime+dilation" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_topics" title="Template:Time topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_topics" title="Template talk:Time topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_topics" title="Special:EditPage/Template:Time topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time758" style="font-size:114%;margin:0 4em"><a href="/wiki/Time" title="Time">Time</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Past" title="Past">Past</a></li> <li><a href="/wiki/Present" title="Present">Present</a></li> <li><a href="/wiki/Future" title="Future">Future</a></li> <li><a href="/wiki/Eternity" title="Eternity">Eternity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Measurement</a><br />and <a href="/wiki/Time_standard" title="Time standard">standards</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></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/Coordinated_Universal_Time" title="Coordinated Universal Time">UTC</a></li> <li><a href="/wiki/Universal_Time" title="Universal Time">UT</a></li> <li><a href="/wiki/International_Atomic_Time" title="International Atomic Time">TAI</a></li> <li><a href="/wiki/Unit_of_time" title="Unit of time">Unit of time</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude (time)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/System_of_measurement" class="mw-redirect" title="System of measurement">Measurement<br />systems</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/Italian_six-hour_clock" title="Italian six-hour clock">Italian six-hour clock</a></li> <li><a href="/wiki/Thai_six-hour_clock" title="Thai six-hour clock">Thai six-hour clock</a></li> <li><a href="/wiki/12-hour_clock" title="12-hour clock">12-hour clock</a></li> <li><a href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Relative_hour" title="Relative hour">Relative hour</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Traditional_Chinese_timekeeping" title="Traditional Chinese timekeeping">Chinese</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal</a></li> <li><a href="/wiki/Hexadecimal_time" title="Hexadecimal time">Hexadecimal</a></li> <li><a href="/wiki/Hindu_units_of_time" title="Hindu units of time">Hindu</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric</a></li> <li><a href="/wiki/Roman_timekeeping" title="Roman timekeeping">Roman</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Calendar" title="Calendar">Calendars</a></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/Calendar#Systems" title="Calendar">Main types</a> <ul><li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li></ul></li> <li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Chinese_calendar" title="Chinese calendar">Chinese</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu Panchang</a></li> <li><a href="/wiki/Maya_calendar" title="Maya calendar">Maya</a></li> <li><i><a href="/wiki/List_of_calendars" title="List of calendars">List</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Clock" title="Clock">Clocks</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/Clock#Types" title="Clock">Main types</a> <ul><li><a href="/wiki/Astronomical_clock" title="Astronomical clock">astronomical</a> <ul><li><a href="/wiki/Astrarium" title="Astrarium">astrarium</a></li></ul></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">atomic</a> <ul><li><a href="/wiki/Quantum_clock" class="mw-redirect" title="Quantum clock">quantum</a></li></ul></li> <li><a href="/wiki/Hourglass" title="Hourglass">hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">marine</a></li> <li><a href="/wiki/Sundial" title="Sundial">sundial</a></li> <li><a href="/wiki/Watch" title="Watch">watch</a> <ul><li><a href="/wiki/Mechanical_watch" title="Mechanical watch">mechanical</a></li> <li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">water-based</a></li></ul></li> <li><a href="/wiki/Cuckoo_clock" title="Cuckoo clock">Cuckoo clock</a></li> <li><a href="/wiki/Digital_clock" title="Digital clock">Digital clock</a></li> <li><a href="/wiki/Grandfather_clock" title="Grandfather clock">Grandfather clock</a></li> <li><i><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History</a></i> <ul><li><i><a href="/wiki/Timeline_of_time_measurement_inventions" title="Timeline of time measurement inventions">Timeline</a></i></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Chronology" title="Chronology">Chronology</a></li><li><a href="/wiki/History" title="History">History</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></li> <li><a href="/wiki/Big_History" title="Big History">Big History</a></li> <li><a href="/wiki/Calendar_era" title="Calendar era">Calendar era</a></li> <li><a href="/wiki/Deep_time" title="Deep time">Deep time</a></li> <li><a href="/wiki/Periodization" title="Periodization">Periodization</a></li> <li><a href="/wiki/Regnal_year" title="Regnal year">Regnal year</a></li> <li><a href="/wiki/Timeline" title="Timeline">Timeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A_series_and_B_series" title="A series and B series">A series and B series</a></li> <li><a href="/wiki/B-theory_of_time" title="B-theory of time">B-theory of time</a></li> <li><a href="/wiki/Chronocentrism" title="Chronocentrism">Chronocentrism</a></li> <li><a href="/wiki/Duration_(philosophy)" title="Duration (philosophy)">Duration</a></li> <li><a href="/wiki/Endurantism" title="Endurantism">Endurantism</a></li> <li><a href="/wiki/Eternal_return" title="Eternal return">Eternal return</a></li> <li><a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">Eternalism</a></li> <li><a href="/wiki/Event_(philosophy)" title="Event (philosophy)">Event</a></li> <li><a href="/wiki/Perdurantism" title="Perdurantism">Perdurantism</a></li> <li><a href="/wiki/Philosophical_presentism" title="Philosophical presentism">Presentism</a></li> <li><a href="/wiki/Temporal_finitism" title="Temporal finitism">Temporal finitism</a></li> <li><a href="/wiki/Temporal_parts" title="Temporal parts">Temporal parts</a></li> <li><i><a href="/wiki/The_Unreality_of_Time" title="The Unreality of Time">The Unreality of Time</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Category:Time_in_religion" title="Category:Time in religion">Religion</a></li><li><a href="/wiki/Template:Time_in_religion_and_mythology" title="Template:Time in religion and mythology">Mythology</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ages_of_Man" title="Ages of Man">Ages of Man</a></li> <li><a href="/wiki/Destiny" title="Destiny">Destiny</a></li> <li><a href="/wiki/Immortality" title="Immortality">Immortality</a></li> <li><a href="/wiki/The_Dreaming" title="The Dreaming">Dreamtime</a></li> <li><a href="/wiki/K%C4%81la" title="Kāla">Kāla</a></li> <li><a href="/wiki/Time_and_fate_deities" title="Time and fate deities">Time and fate deities</a> <ul><li><a href="/wiki/Father_Time" title="Father Time">Father Time</a></li></ul></li> <li><a href="/wiki/Wheel_of_time" title="Wheel of time">Wheel of time</a> <ul><li><a href="/wiki/Kalachakra" title="Kalachakra">Kalachakra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_perception" title="Time perception">Human experience</a><br />and <a href="/wiki/Time-use_research" title="Time-use research">use of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronemics" title="Chronemics">Chronemics</a></li> <li><a href="/wiki/Generation_time" title="Generation time">Generation time</a></li> <li><a href="/wiki/Mental_chronometry" title="Mental chronometry">Mental chronometry</a></li> <li><a href="/wiki/Duration_(music)" title="Duration (music)">Music</a> <ul><li><a href="/wiki/Tempo" title="Tempo">tempo</a></li> <li><a href="/wiki/Time_signature" title="Time signature">time signature</a></li></ul></li> <li><a href="/wiki/Rosy_retrospection" title="Rosy retrospection">Rosy retrospection</a></li> <li><a href="/wiki/Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li> <li><a href="/wiki/Time_management" title="Time management">Time management</a></li> <li><a href="/wiki/Yesterday_(time)" title="Yesterday (time)">Yesterday</a> – <a href="/wiki/Present" title="Present">Today</a> – <a href="/wiki/Tomorrow_(time)" title="Tomorrow (time)">Tomorrow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Time in <a href="/wiki/Science" title="Science">science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Geology" title="Geology">Geology</a></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/Geologic_time_scale" title="Geologic time scale">Geological time</a> <ul><li><a href="/wiki/Age_(geology)" class="mw-redirect" title="Age (geology)">age</a></li> <li><a href="/wiki/Chronozone" title="Chronozone">chron</a></li> <li><a href="/wiki/Eon_(geology)" class="mw-redirect" title="Eon (geology)">eon</a></li> <li><a href="/wiki/Epoch_(geology)" class="mw-redirect" title="Epoch (geology)">epoch</a></li> <li><a href="/wiki/Era_(geology)" class="mw-redirect" title="Era (geology)">era</a></li> <li><a href="/wiki/Geological_period" class="mw-redirect" title="Geological period">period</a></li></ul></li> <li><a href="/wiki/Geochronology" title="Geochronology">Geochronology</a></li> <li><a href="/wiki/Geological_history_of_Earth" title="Geological history of Earth">Geological history of Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Time_in_physics" title="Time in physics">Physics</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/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">Time translation symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">Time reversal symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;">Other fields</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/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Chronobiology" title="Chronobiology">Chronobiology</a> <ul><li><a href="/wiki/Circadian_rhythm" title="Circadian rhythm">Circadian rhythms</a></li></ul></li> <li><a href="/wiki/Chemical_clock" title="Chemical clock">Clock reaction</a></li> <li><a href="/wiki/Glottochronology" title="Glottochronology">Glottochronology</a></li> <li><a href="/wiki/Time_geography" title="Time geography">Time geography</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Memory" title="Memory">Memory</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/System_time" title="System time">System time</a></li> <li><i><a href="/wiki/Tempus_fugit" title="Tempus fugit">Tempus fugit</a></i></li> <li><a href="/wiki/Time_capsule" title="Time capsule">Time capsule</a></li> <li><a href="/wiki/Time_immemorial" title="Time immemorial">Time immemorial</a></li> <li><a href="/wiki/Time_travel" title="Time travel">Time travel</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Time" title="Category:Time">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Time" class="extiw" title="commons:Category:Time">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Time_measurement_and_standards342" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Time_measurement_and_standards" title="Template:Time measurement and standards"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_measurement_and_standards" title="Template talk:Time measurement and standards"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_measurement_and_standards" title="Special:EditPage/Template:Time measurement and standards"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time_measurement_and_standards342" style="font-size:114%;margin:0 4em"><a href="/wiki/Time" title="Time">Time measurement</a> and <a href="/wiki/Time_standard" title="Time standard">standards</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude</a></li> <li><a href="/wiki/Time_metrology" class="mw-redirect" title="Time metrology">Metrology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">International standards</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinated_Universal_Time" title="Coordinated Universal Time">Coordinated Universal Time</a> <ul><li><a href="/wiki/UTC_offset" title="UTC offset">offset</a></li></ul></li> <li><a href="/wiki/Universal_Time" title="Universal Time">UT</a></li> <li><a href="/wiki/%CE%94T_(timekeeping)" title="ΔT (timekeeping)">ΔT</a></li> <li><a href="/wiki/DUT1" title="DUT1">DUT1</a></li> <li><a href="/wiki/International_Earth_Rotation_and_Reference_Systems_Service" title="International Earth Rotation and Reference Systems Service">International Earth Rotation and Reference Systems Service</a></li> <li><a href="/wiki/ISO_31-1" title="ISO 31-1">ISO 31-1</a></li> <li><a href="/wiki/ISO_8601" title="ISO 8601">ISO 8601</a></li> <li><a href="/wiki/International_Atomic_Time" title="International Atomic Time">International Atomic Time</a></li> <li><a href="/wiki/12-hour_clock" title="12-hour clock">12-hour clock</a></li> <li><a href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Barycentric_Coordinate_Time" title="Barycentric Coordinate Time">Barycentric Coordinate Time</a></li> <li><a href="/wiki/Barycentric_Dynamical_Time" title="Barycentric Dynamical Time">Barycentric Dynamical Time</a></li> <li><a href="/wiki/Civil_time" title="Civil time">Civil time</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Geocentric_Coordinate_Time" title="Geocentric Coordinate Time">Geocentric Coordinate Time</a></li> <li><a href="/wiki/International_Date_Line" title="International Date Line">International Date Line</a></li> <li><a href="/wiki/IERS_Reference_Meridian" title="IERS Reference Meridian">IERS Reference Meridian</a></li> <li><a href="/wiki/Leap_second" title="Leap second">Leap second</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar time</a></li> <li><a href="/wiki/Terrestrial_Time" title="Terrestrial Time">Terrestrial Time</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li> <li><a href="/wiki/180th_meridian" title="180th meridian">180th meridian</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Hourglass" title="Hourglass"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/120px-Marine_sandglass_MMM.jpg" decoding="async" width="75" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/250px-Marine_sandglass_MMM.jpg 2x" data-file-width="1637" data-file-height="3819" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Time_zone" title="Time zone"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/75px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png" decoding="async" width="75" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/113px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/150px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 2x" data-file-width="496" data-file-height="1007" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Obsolete standards</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ephemeris_time" title="Ephemeris time">Ephemeris time</a></li> <li><a href="/wiki/Greenwich_Mean_Time" title="Greenwich Mean Time">Greenwich Mean Time</a></li> <li><a href="/wiki/Prime_meridian" title="Prime meridian">Prime meridian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">Continuous signal</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Cosmological_decade" class="mw-redirect" title="Cosmological decade">Cosmological decade</a></li> <li><a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete time and continuous time</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a class="mw-selflink selflink">Time dilation</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">Time-translation symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Horology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clock" title="Clock">Clock</a></li> <li><a href="/wiki/Astrarium" title="Astrarium">Astrarium</a></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">Atomic clock</a></li> <li><a href="/wiki/Complication_(horology)" title="Complication (horology)">Complication</a></li> <li><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History of timekeeping devices</a></li> <li><a href="/wiki/Hourglass" title="Hourglass">Hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">Marine chronometer</a></li> <li><a href="/wiki/Marine_sandglass" title="Marine sandglass">Marine sandglass</a></li> <li><a href="/wiki/Radio_clock" title="Radio clock">Radio clock</a></li> <li><a href="/wiki/Watch" title="Watch">Watch</a> <ul><li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">Water clock</a></li> <li><a href="/wiki/Sundial" title="Sundial">Sundial</a></li> <li><a href="/wiki/Dialing_scales" title="Dialing scales">Dialing scales</a></li> <li><a href="/wiki/Equation_of_time" title="Equation of time">Equation of time</a></li> <li><a href="/wiki/History_of_sundials" title="History of sundials">History of sundials</a></li> <li><a href="/wiki/Schema_for_horizontal_dials" title="Schema for horizontal dials">Sundial markup schema</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Calendar" title="Calendar">Calendar</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu</a></li> <li><a href="/wiki/Holocene_calendar" title="Holocene calendar">Holocene</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a> (lunar Hijri)</li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Astronomical_year_numbering" title="Astronomical year numbering">Astronomical</a></li> <li><a href="/wiki/Dominical_letter" title="Dominical letter">Dominical letter</a></li> <li><a href="/wiki/Epact" title="Epact">Epact</a></li> <li><a href="/wiki/Equinox" title="Equinox">Equinox</a></li> <li><a href="/wiki/Intercalation_(timekeeping)" title="Intercalation (timekeeping)">Intercalation</a></li> <li><a href="/wiki/Julian_day" title="Julian day">Julian day</a></li> <li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li> <li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Solstice" title="Solstice">Solstice</a></li> <li><a href="/wiki/Tropical_year" title="Tropical year">Tropical year</a></li> <li><a href="/wiki/Determination_of_the_day_of_the_week" title="Determination of the day of the week">Weekday determination</a></li> <li><a href="/wiki/Names_of_the_days_of_the_week" title="Names of the days of the week">Weekday names</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Archaeology and geology</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Geologic_time_scale" title="Geologic time scale">Geologic time scale</a></li> <li><a href="/wiki/International_Commission_on_Stratigraphy" title="International Commission on Stratigraphy">International Commission on Stratigraphy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Galactic_year" title="Galactic year">Galactic year</a></li> <li><a href="/wiki/Nuclear_timescale" title="Nuclear timescale">Nuclear timescale</a></li> <li><a href="/wiki/Precession" title="Precession">Precession</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Unit_of_time" title="Unit of time">units of time</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a href="/wiki/Flick_(time)" title="Flick (time)">Flick</a></li> <li><a href="/wiki/Shake_(unit)" title="Shake (unit)">Shake</a></li> <li><a href="/wiki/Jiffy_(time)" title="Jiffy (time)">Jiffy</a></li> <li><a href="/wiki/Second" title="Second">Second</a></li> <li><a href="/wiki/Minute" title="Minute">Minute</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Hour" title="Hour">Hour</a></li> <li><a href="/wiki/Day" title="Day">Day</a></li> <li><a href="/wiki/Week" title="Week">Week</a></li> <li><a href="/wiki/Fortnight" title="Fortnight">Fortnight</a></li> <li><a href="/wiki/Month" title="Month">Month</a></li> <li><a href="/wiki/Year" title="Year">Year</a></li> <li><a href="/wiki/Olympiad" title="Olympiad">Olympiad</a></li> <li><a href="/wiki/Lustrum" title="Lustrum">Lustrum</a></li> <li><a href="/wiki/Decade" title="Decade">Decade</a></li> <li><a href="/wiki/Century" title="Century">Century</a></li> <li><a href="/wiki/Saeculum" title="Saeculum">Saeculum</a></li> <li><a href="/wiki/Millennium" title="Millennium">Millennium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronology" title="Chronology">Chronology</a></li> <li><a href="/wiki/Duration_(philosophy)" title="Duration (philosophy)">Duration</a> <ul><li><a href="/wiki/Duration_(music)" title="Duration (music)">music</a></li></ul></li> <li><a href="/wiki/Mental_chronometry" title="Mental chronometry">Mental chronometry</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal time</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric time</a></li> <li><a href="/wiki/System_time" title="System time">System time</a></li> <li><a href="/wiki/Time_metrology" class="mw-redirect" title="Time metrology">Time metrology</a></li> <li><a href="/wiki/Time_value_of_money" title="Time value of money">Time value of money</a></li> <li><a href="/wiki/Timekeeper" title="Timekeeper">Timekeeper</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Relativity254" 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="Relativity254" 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 class="mw-selflink selflink">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> <li><a href="/wiki/Ellis_wormhole" title="Ellis wormhole">Ellis</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /></div><div role="navigation" class="navbox" aria-labelledby="Science_fiction458" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Science_fiction" title="Template:Science fiction"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Science_fiction" title="Template talk:Science fiction"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Science_fiction" title="Special:EditPage/Template:Science fiction"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Science_fiction458" style="font-size:114%;margin:0 4em"><a href="/wiki/Science_fiction" title="Science fiction">Science fiction</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_science_fiction" title="Outline of science fiction">Outline</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_science-fiction_authors" title="List of science-fiction authors">Authors</a></li> <li><a href="/wiki/Definitions_of_science_fiction" title="Definitions of science fiction">Definitions</a> <ul><li><a href="/wiki/Anthropological_science_fiction" title="Anthropological science fiction">Anthropological</a></li> <li><a href="/wiki/Hard_science_fiction" title="Hard science fiction">Hard</a></li> <li><a href="/wiki/Scientific_romance" title="Scientific romance">Scientific romance</a></li> <li><a href="/wiki/Soft_science_fiction" title="Soft science fiction">Soft</a></li></ul></li> <li><a href="/wiki/Golden_Age_of_Science_Fiction" title="Golden Age of Science Fiction">Golden Age</a></li> <li><a href="/wiki/History_of_science_fiction" title="History of science fiction">History</a></li> <li><a href="/wiki/New_Wave_science_fiction" class="mw-redirect" title="New Wave science fiction">New Wave</a></li> <li><a href="/wiki/Timeline_of_science_fiction" title="Timeline of science fiction">Timeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subgenres</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><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apocalyptic_and_post-apocalyptic_fiction" title="Apocalyptic and post-apocalyptic fiction">Apocalyptic and post-apocalyptic</a></li> <li><a href="/wiki/Science_fiction_comedy" title="Science fiction comedy">Comedy</a> <ul><li><a href="/wiki/List_of_science_fiction_sitcoms" class="mw-redirect" title="List of science fiction sitcoms">Sitcoms</a></li></ul></li> <li><a href="/wiki/Feminist_science_fiction" title="Feminist science fiction">Feminist</a></li> <li><a href="/wiki/Grimdark" title="Grimdark">Grimdark</a></li> <li><a href="/wiki/Inner_space_(science_fiction)" title="Inner space (science fiction)">Inner space</a></li> <li><a href="/wiki/Mecha" title="Mecha">Mecha</a> <ul><li><a href="/wiki/Mecha_anime_and_manga" title="Mecha anime and manga">Anime and manga</a></li></ul></li> <li><a href="/wiki/Mundane_science_fiction" title="Mundane science fiction">Mundane</a></li> <li><a href="/wiki/Space_warfare_in_science_fiction" title="Space warfare in science fiction">Space warfare</a> <ul><li><a href="/wiki/Military_science_fiction" title="Military science fiction">Military</a></li> <li><a href="/wiki/Space_opera" title="Space opera">Space opera</a></li> <li><a href="/wiki/Space_Western" title="Space Western">Space Western</a></li></ul></li> <li><a href="/wiki/Parallel_universes_in_fiction" title="Parallel universes in fiction">Parallel universes</a> <ul><li><a href="/wiki/Isekai" title="Isekai">Isekai</a></li></ul></li> <li><a href="/wiki/Science_fantasy" title="Science fantasy">Science fantasy</a> <ul><li><a href="/wiki/Dying_Earth_(genre)" title="Dying Earth (genre)">Dying Earth</a></li> <li><a href="/wiki/Planetary_romance" title="Planetary romance">Planetary romance</a></li> <li><a href="/wiki/Superhero_fiction" title="Superhero fiction">Superhero</a></li> <li><a href="/wiki/Sword_and_planet" class="mw-redirect" title="Sword and planet">Sword and planet</a></li></ul></li> <li><a href="/wiki/Social_science_fiction" title="Social science fiction">Social</a> <ul><li><a href="/wiki/Climate_fiction" title="Climate fiction">Climate fiction</a></li> <li><a href="/wiki/Christian_science_fiction" title="Christian science fiction">Christian</a></li> <li><a href="/wiki/Libertarian_science_fiction" title="Libertarian science fiction">Libertarian</a></li> <li><a href="/wiki/Utopian_and_dystopian_fiction" title="Utopian and dystopian fiction">Utopian and dystopian</a></li></ul></li> <li><a href="/wiki/Tech_noir" title="Tech noir">Tech noir</a> <ul><li><a href="/wiki/Spy-Fi_(subgenre)" class="mw-redirect" title="Spy-Fi (subgenre)">Spy-Fi</a></li> <li><a href="/wiki/Techno-thriller" title="Techno-thriller">Techno-thriller</a></li></ul></li> <li><a href="/wiki/Tokusatsu" title="Tokusatsu">Tokusatsu</a> <ul><li><a href="/wiki/Kaiju" title="Kaiju">Kaiju</a></li></ul></li> <li><a href="/wiki/List_of_underwater_science_fiction_works" title="List of underwater science fiction works">Underwater</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cyberpunk_derivatives" title="Cyberpunk derivatives">Cyberpunk derivatives</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyberpunk" title="Cyberpunk">Cyberpunk</a> <ul><li><a href="/wiki/Japanese_cyberpunk" title="Japanese cyberpunk">Japanese</a></li></ul></li> <li><a href="/wiki/Biopunk" title="Biopunk">Biopunk</a></li> <li><a href="/wiki/Dieselpunk" title="Dieselpunk">Dieselpunk</a></li> <li><a href="/wiki/Nanopunk" title="Nanopunk">Nanopunk</a></li> <li><a href="/wiki/Solarpunk" title="Solarpunk">Solarpunk</a></li> <li><a href="/wiki/Steampunk" title="Steampunk">Steampunk</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Culture</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><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_fiction_convention" title="Science fiction convention">Conventions</a></li> <li><a href="/wiki/Science_fiction_fandom" title="Science fiction fandom">Fandom</a></li> <li><a href="/wiki/Science-fiction_fanzine" title="Science-fiction fanzine">Fanzines</a></li> <li><a href="/wiki/Internet_Speculative_Fiction_Database" title="Internet Speculative Fiction Database">ISFDB</a></li> <li><a href="/wiki/Science_fiction_libraries_and_museums" title="Science fiction libraries and museums">Libraries and museums</a></li> <li><a href="/wiki/EMP_Museum#Science_Fiction_Museum" class="mw-redirect" title="EMP Museum">Science Fiction Museum</a></li> <li><a href="/wiki/Science_fiction_studies" title="Science fiction studies">Studies</a></li> <li><a href="/wiki/Women_in_speculative_fiction" title="Women in speculative fiction">Women in SF</a></li> <li><a href="/wiki/Worldcon" title="Worldcon">Worldcon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Region</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Australian_science_fiction" title="Australian science fiction">Australian</a></li> <li><a href="/wiki/Bengali_science_fiction" title="Bengali science fiction">Bengali</a></li> <li><a href="/wiki/Brazilian_science_fiction" title="Brazilian science fiction">Brazilian</a></li> <li><a href="/wiki/Canadian_science_fiction" title="Canadian science fiction">Canadian</a></li> <li><a href="/wiki/Chilean_science_fiction" title="Chilean science fiction">Chilean</a></li> <li><a href="/wiki/Chinese_science_fiction" title="Chinese science fiction">Chinese</a></li> <li><a href="/wiki/Croatian_science_fiction" title="Croatian science fiction">Croatian</a></li> <li><a href="/wiki/Czech_science_fiction_and_fantasy" class="mw-redirect" title="Czech science fiction and fantasy">Czech</a></li> <li><a href="/wiki/Estonian_science_fiction" title="Estonian science fiction">Estonian</a></li> <li><a href="/wiki/French_science_fiction" title="French science fiction">French</a></li> <li><a href="/wiki/Hungarian_science_fiction" title="Hungarian science fiction">Hungarian</a></li> <li><a href="/wiki/Japanese_science_fiction" title="Japanese science fiction">Japanese</a></li> <li><a href="/wiki/Norwegian_science_fiction" title="Norwegian science fiction">Norwegian</a></li> <li><a href="/wiki/Science_fiction_and_fantasy_in_Poland" class="mw-redirect" title="Science fiction and fantasy in Poland">Polish</a></li> <li><a href="/wiki/Romanian_science_fiction" title="Romanian science fiction">Romanian</a></li> <li><a href="/wiki/Russian_science_fiction_and_fantasy" class="mw-redirect" title="Russian science fiction and fantasy">Russian</a></li> <li><a href="/wiki/Serbian_science_fiction" title="Serbian science fiction">Serbian</a></li> <li><a href="/wiki/Spanish_science_fiction" title="Spanish science fiction">Spanish</a></li> <li><a href="/wiki/Yugoslav_science_fiction" title="Yugoslav science fiction">Yugoslav</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Awards</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Cinematic</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jules_Verne_Awards" title="Jules Verne Awards">Jules Verne</a></li> <li><a href="/wiki/Saturn_Awards" title="Saturn Awards">Saturn</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Literary, art,<br />and audio</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astounding_Award_for_Best_New_Writer" title="Astounding Award for Best New Writer">Astounding</a></li> <li><a href="/wiki/Aurealis_Award" title="Aurealis Award">Aurealis</a></li> <li><a href="/wiki/BSFA_Award" title="BSFA Award">BSFA</a></li> <li><a href="/wiki/John_W._Campbell_Memorial_Award_for_Best_Science_Fiction_Novel" title="John W. Campbell Memorial Award for Best Science Fiction Novel">Campbell Memorial</a></li> <li><a href="/wiki/Chesley_Awards" title="Chesley Awards">Chesley</a></li> <li><a href="/wiki/Arthur_C._Clarke_Award" title="Arthur C. Clarke Award">Clarke</a></li> <li><a href="/wiki/Compton_Crook_Award" title="Compton Crook Award">Crook</a></li> <li><a href="/wiki/Deutscher_Science_Fiction_Preis" title="Deutscher Science Fiction Preis">Deutscher</a></li> <li><a href="/wiki/Philip_K._Dick_Award" title="Philip K. Dick Award">Dick</a></li> <li><a href="/wiki/Ditmar_Award" title="Ditmar Award">Ditmar</a></li> <li><a href="/wiki/Endeavour_Award" title="Endeavour Award">Endeavor</a></li> <li><a href="/wiki/FantLab%27s_Book_of_the_Year_Award" title="FantLab's Book of the Year Award">FantLab</a></li> <li><a href="/wiki/Galaxy_Award_(China)" title="Galaxy Award (China)">Galaxy</a></li> <li><a href="/wiki/Jack_Gaughan_Award" title="Jack Gaughan Award">Gaughan</a></li> <li><a href="/wiki/Geffen_Award" title="Geffen Award">Geffen</a></li> <li><a href="/wiki/Golden_Duck_Award" title="Golden Duck Award">Golden Duck</a></li> <li><a href="/wiki/Damon_Knight_Memorial_Grand_Master_Award" title="Damon Knight Memorial Grand Master Award">Grand Master</a></li> <li><a href="/wiki/Grand_prix_de_l%27Imaginaire" title="Grand prix de l'Imaginaire">Grand Prix</a></li> <li><a href="/wiki/Paul_Harland_Prize" title="Paul Harland Prize">Harland</a></li> <li><a href="/wiki/Robert_A._Heinlein_Award" title="Robert A. Heinlein Award">Heinlein</a></li> <li><a href="/wiki/Premio_Ignotus" title="Premio Ignotus">Ignotus</a></li> <li><a href="/wiki/Kitschies" title="Kitschies">Kitschies</a></li> <li><a href="/wiki/List_of_Lambda_Literary_Awards_winners_and_nominees_for_science_fiction,_fantasy_and_horror" class="mw-redirect" title="List of Lambda Literary Awards winners and nominees for science fiction, fantasy and horror">Lambda</a></li> <li><a href="/wiki/Kurd_La%C3%9Fwitz_Award" title="Kurd Laßwitz Award">Laßwitz</a></li> <li><a href="/wiki/Locus_Award" title="Locus Award">Locus</a></li> <li><a href="/wiki/Nautilus_Award" title="Nautilus Award">Nautilus</a></li> <li><a href="/wiki/Nebula_Award" title="Nebula Award">Nebula</a></li> <li><a href="/wiki/Nommo_Awards" title="Nommo Awards">Nommo</a></li> <li><a href="/wiki/Andre_Norton_Award" title="Andre Norton Award">Norton</a></li> <li><a href="/wiki/Parsec_Awards" title="Parsec Awards">Parsec</a></li> <li><a href="/wiki/Prometheus_Award" title="Prometheus Award">Prometheus</a></li> <li><a href="/wiki/Rhysling_Award" title="Rhysling Award">Rhysling</a></li> <li><a href="/wiki/SFERA_Award" title="SFERA Award">SFERA</a></li> <li><a href="/wiki/Sidewise_Award_for_Alternate_History" title="Sidewise Award for Alternate History">Sidewise</a></li> <li><a href="/wiki/Edward_E._Smith_Memorial_Award" title="Edward E. Smith Memorial Award">Skylark</a></li> <li><a href="/wiki/Theodore_Sturgeon_Award" title="Theodore Sturgeon Award">Sturgeon</a></li> <li><a href="/wiki/Sunburst_Award" title="Sunburst Award">Sunburst</a></li> <li><a href="/wiki/T%C3%A4htivaeltaja_Award" title="Tähtivaeltaja Award">Tähtivaeltaja</a></li> <li><a href="/wiki/TBD_Science_Fiction_Story_Award" title="TBD Science Fiction Story Award">TBD</a></li> <li><a href="/wiki/James_Tiptree_Jr._Award" class="mw-redirect" title="James Tiptree Jr. Award">Tiptree</a></li> <li><a href="/wiki/Prix_Apollo_Award" title="Prix Apollo Award">Tour-Apollo</a></li> <li><a href="/wiki/Science_Fiction_%26_Fantasy_Translation_Awards" title="Science Fiction & Fantasy Translation Awards">Translation</a></li> <li><a href="/wiki/Urania_Award" title="Urania Award">Urania</a></li> <li><a href="/wiki/Sir_Julius_Vogel_Award" title="Sir Julius Vogel Award">Vogel</a></li> <li><a href="/wiki/Writers_of_the_Future" class="mw-redirect" title="Writers of the Future">Writers and Illustrators of the Future</a></li> <li><a href="/wiki/Janusz_A._Zajdel_Award" title="Janusz A. Zajdel Award">Zajdel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Multimedia</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aurora_Awards" title="Aurora Awards">Aurora</a></li> <li><a href="/wiki/Chandler_Award" title="Chandler Award">Chandler</a></li> <li><a href="/wiki/Dragon_Awards" title="Dragon Awards">Dragon</a></li> <li><a href="/wiki/Hugo_Award" title="Hugo Award">Hugo</a></li> <li><a href="/wiki/Seiun_Award" title="Seiun Award">Seiun</a></li> <li><a href="/wiki/Gaylactic_Spectrum_Awards" title="Gaylactic Spectrum Awards">Spectrum</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Media</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:1%"><a href="/wiki/Science_fiction_film" title="Science fiction film">Film</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_science_fiction_films" title="History of science fiction films">Film history</a></li> <li><a href="/wiki/Lists_of_science_fiction_films" title="Lists of science fiction films">Films</a></li> <li><a href="/wiki/Science_fiction_films_in_India" title="Science fiction films in India">Indian</a></li> <li><a href="/wiki/Japanese_science_fiction" title="Japanese science fiction">Japanese</a> <ul><li><a href="/wiki/List_of_science_fiction_anime" title="List of science fiction anime">Anime</a></li> <li><a href="/wiki/Tokusatsu" title="Tokusatsu">Tokusatsu</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Literature</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_fiction_comics" title="Science fiction comics">Comics</a></li> <li><a href="/wiki/Science_fiction_magazine" title="Science fiction magazine">Magazines</a></li> <li><a href="/wiki/List_of_science_fiction_novels" title="List of science fiction novels">Novels</a></li> <li><a href="/wiki/List_of_science_fiction_publishers" title="List of science fiction publishers">Publishers</a></li> <li><a href="/wiki/List_of_science_fiction_short_stories" title="List of science fiction short stories">Short stories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Stage</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_fiction_opera" title="Science fiction opera">Opera</a></li> <li><a href="/wiki/Science_fiction_theatre" title="Science fiction theatre">Theatre</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Science_fiction_on_television" title="Science fiction on television">Television</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_science_fiction_television_programs" title="List of science fiction television programs">List of TV shows</a></li> <li><a href="/wiki/Australian_science_fiction_television" title="Australian science fiction television">Australasian</a></li> <li><a href="/wiki/British_television_science_fiction" title="British television science fiction">British</a></li> <li><a href="/wiki/Canadian_science_fiction_television" title="Canadian science fiction television">Canadian</a></li> <li><a href="/wiki/Science_fiction_on_television#Continental_European_science_fiction_series" title="Science fiction on television">European</a></li> <li><a href="/wiki/Japanese_science_fiction" title="Japanese science fiction">Japanese</a> <ul><li><a href="/wiki/List_of_science_fiction_anime" title="List of science fiction anime">Anime</a></li> <li><a href="/wiki/Tokusatsu" title="Tokusatsu">Live-action</a></li></ul></li> <li><a href="/wiki/U.S._television_science_fiction" title="U.S. television science fiction">U.S.</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_science_fiction_themes" title="List of science fiction themes">Themes</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:1%"><a href="/wiki/Space_architecture" title="Space architecture">Architectural</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Space_colonization" title="Space colonization">Colonization</a></li> <li><a href="/wiki/Dyson_sphere" title="Dyson sphere">Dyson sphere</a></li> <li><a href="/wiki/Matrioshka_brain" title="Matrioshka brain">Matrioshka brain</a></li> <li><a href="/wiki/Space_stations_and_habitats_in_fiction" title="Space stations and habitats in fiction">Space stations and habitats</a></li> <li><a href="/wiki/Stellar_engine" title="Stellar engine">Stellar engine</a></li> <li><a href="/wiki/Terraforming_in_popular_culture" title="Terraforming in popular culture">Terraforming</a></li> <li><a href="/wiki/Topopolis" title="Topopolis">Topopolis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Biology_in_fiction" title="Biology in fiction">Biological</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biological_warfare_in_popular_culture" title="Biological warfare in popular culture">Biological warfare</a></li> <li><a href="/wiki/Energy_being" title="Energy being">Energy being</a></li> <li><a href="/wiki/Evolution_in_fiction" title="Evolution in fiction">Evolution</a></li> <li><a href="/wiki/Extraterrestrials_in_fiction" title="Extraterrestrials in fiction">Extraterrestrials</a> <ul><li><a href="/wiki/Lists_of_fictional_extraterrestrials" title="Lists of fictional extraterrestrials">List</a></li></ul></li> <li><a href="/wiki/Gender_in_speculative_fiction" title="Gender in speculative fiction">Gender</a></li> <li><a href="/wiki/Genetic_engineering_in_science_fiction" class="mw-redirect" title="Genetic engineering in science fiction">Genetic engineering</a></li> <li><a href="/wiki/Invisibility_in_fiction" title="Invisibility in fiction">Invisibility</a></li> <li><a href="/wiki/Nanotechnology_in_fiction" title="Nanotechnology in fiction">Nanotechnology</a></li> <li><a href="/wiki/Organ_transplantation_in_fiction" title="Organ transplantation in fiction">Organ transplantation</a></li> <li><a href="/wiki/Parasites_in_fiction" title="Parasites in fiction">Parasites</a></li> <li><a href="/wiki/Prosthetics_in_fiction" title="Prosthetics in fiction">Prosthetics</a></li> <li><a href="/wiki/Sex_and_sexuality_in_speculative_fiction" title="Sex and sexuality in speculative fiction">Sex and sexuality</a></li> <li><a href="/wiki/Symbiosis_in_fiction" title="Symbiosis in fiction">Symbiosis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ansible" title="Ansible">Ansible</a></li> <li><a href="/wiki/Black_holes_in_fiction" title="Black holes in fiction">Black holes</a></li> <li><a href="/wiki/Extrasolar_planets_in_fiction" title="Extrasolar planets in fiction">Extrasolar planets</a></li> <li><a href="/wiki/Force_field_(technology)" title="Force field (technology)">Force field</a></li> <li><a href="/wiki/Hyperspace" title="Hyperspace">Hyperspace</a></li> <li><a href="/wiki/Inertialess_drive" class="mw-redirect" title="Inertialess drive">Inertialess</a></li> <li><a href="/wiki/Multiverse" title="Multiverse">Multiverse</a></li> <li><a href="/wiki/Parallel_universes_in_fiction" title="Parallel universes in fiction">Parallel universes</a></li> <li><a href="/wiki/Portable_hole" title="Portable hole">Portable hole</a></li> <li><a href="/wiki/Space_travel_in_science_fiction" title="Space travel in science fiction">Space travel</a></li> <li><a href="/wiki/Stargate_(device)" title="Stargate (device)">Stargate</a></li> <li><a href="/wiki/Stars_in_fiction" title="Stars in fiction">Stars</a></li> <li><a href="/wiki/Teleportation_in_fiction" title="Teleportation in fiction">Teleportation</a></li> <li><a href="/wiki/Time_travel_in_fiction" title="Time travel in fiction">Time travel</a> (<a href="/wiki/Time_viewer" title="Time viewer">Viewer</a>)</li> <li><a href="/wiki/Warp_drive" title="Warp drive">Warp drive</a></li> <li><a href="/wiki/Wormholes_in_fiction" title="Wormholes in fiction">Wormhole</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Psychological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Group_mind_(science_fiction)" title="Group mind (science fiction)">Group mind</a></li> <li><a href="/wiki/Mind_uploading_in_fiction" title="Mind uploading in fiction">Mind uploading</a></li> <li><a href="/wiki/Psionics" title="Psionics">Psionics</a></li> <li><a href="/wiki/Simulated_consciousness_in_fiction" title="Simulated consciousness in fiction">Simulated consciousness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Social</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Africanfuturism" title="Africanfuturism">Africanfuturism</a></li> <li><a href="/wiki/Afrofuturism" title="Afrofuturism">Afrofuturism</a></li> <li><a href="/wiki/Alien_invasion" title="Alien invasion">Alien invasion</a></li> <li><a href="/wiki/Alien_language" title="Alien language">Alien language</a></li> <li><a href="/wiki/Ancient_astronauts_in_popular_culture" title="Ancient astronauts in popular culture">Ancient astronauts</a></li> <li><a href="/wiki/Black_science_fiction" title="Black science fiction">Black</a></li> <li><a href="/wiki/Evil_corporation" title="Evil corporation">Evil corporation</a></li> <li><a href="/wiki/First_contact_(science_fiction)" title="First contact (science fiction)">First contact</a></li> <li><a href="/wiki/Frankenstein_complex" title="Frankenstein complex">Frankenstein complex</a></li> <li><a href="/wiki/Galactic_empire" title="Galactic empire">Galactic empire</a></li> <li><a href="/wiki/LGBT_themes_in_speculative_fiction" class="mw-redirect" title="LGBT themes in speculative fiction">LGBT</a></li> <li><a href="/wiki/Message_from_space_(science_fiction)" class="mw-redirect" title="Message from space (science fiction)">Message from space</a></li> <li><a href="/wiki/Transhumanism_in_fiction" title="Transhumanism in fiction">Transhumanism</a></li> <li><a href="/wiki/Uplift_(science_fiction)" title="Uplift (science fiction)">Uplift</a></li> <li><a href="/wiki/Xenoarchaeology" title="Xenoarchaeology">Xenoarchaeology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Technological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Artificial_intelligence_in_fiction" title="Artificial intelligence in fiction">Artificial intelligence</a> <ul><li><a href="/wiki/AI_takeover_in_popular_culture" title="AI takeover in popular culture">AI takeover</a></li></ul></li> <li><a href="/wiki/Astronomical_engineering" title="Astronomical engineering">Astroengineering</a></li> <li><a href="/wiki/Holography_in_fiction" title="Holography in fiction">Holography</a></li> <li><a href="/wiki/List_of_fictional_robots_and_androids" title="List of fictional robots and androids">Robots</a> and <a href="/wiki/Cyborgs_in_fiction" class="mw-redirect" title="Cyborgs in fiction">Cyborgs</a></li> <li><a href="/wiki/Self-replicating_machines_in_fiction" class="mw-redirect" title="Self-replicating machines in fiction">Self-replicating machines</a></li> <li><a href="/wiki/Simulated_reality" title="Simulated reality">Simulated reality</a></li> <li><a href="/wiki/List_of_fictional_spacecraft" title="List of fictional spacecraft">Spacecraft</a></li> <li><a href="/wiki/Tachyons_in_fiction" title="Tachyons in fiction">Tachyons</a></li> <li><a href="/wiki/Weapons_in_science_fiction" title="Weapons in science fiction">Weapons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Religious</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Christian_science_fiction" title="Christian science fiction">Christian science fiction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternate_history" title="Alternate history">Alternate history</a></li> <li><a href="/wiki/Fantasy" title="Fantasy">Fantasy</a></li> <li><a href="/wiki/Lists_of_fictional_astronauts" title="Lists of fictional astronauts">Fictional 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