Einstein field equations

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cosmological constant</span> </div> </a> <ul id="toc-The_cosmological_constant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Features" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Features"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Features</span> </div> </a> <button aria-controls="toc-Features-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 Features subsection</span> </button> <ul id="toc-Features-sublist" class="vector-toc-list"> <li id="toc-Conservation_of_energy_and_momentum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conservation_of_energy_and_momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Conservation of energy and momentum</span> </div> </a> <ul id="toc-Conservation_of_energy_and_momentum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinearity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonlinearity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Nonlinearity</span> </div> </a> <ul id="toc-Nonlinearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_correspondence_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_correspondence_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>The correspondence principle</span> </div> </a> <ul id="toc-The_correspondence_principle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vacuum_field_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vacuum_field_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vacuum field equations</span> </div> </a> <ul id="toc-Vacuum_field_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Einstein–Maxwell_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Einstein–Maxwell_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Einstein–Maxwell equations</span> </div> </a> <ul id="toc-Einstein–Maxwell_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Solutions</span> </div> </a> <ul id="toc-Solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_linearized_EFE" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_linearized_EFE"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>The linearized EFE</span> </div> </a> <ul id="toc-The_linearized_EFE-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polynomial_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Polynomial form</span> </div> </a> <ul id="toc-Polynomial_form-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">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-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 External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-External_images" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#External_images"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>External images</span> </div> </a> <ul id="toc-External_images-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><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">Einstein field equations</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 47 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-47" 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">47 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%A7%D9%84%D8%AD%D9%82%D9%84_%D9%84%D8%A3%D9%8A%D9%86%D8%B4%D8%AA%D8%A7%D9%8A%D9%86" title="معادلات الحقل لأينشتاين – Arabic" lang="ar" hreflang="ar" data-title="معادلات الحقل لأينشتاين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Eyn%C5%9Fteyn_sah%C9%99_t%C9%99nlikl%C9%99ri" title="Eynşteyn sahə tənlikləri – Azerbaijani" lang="az" hreflang="az" data-title="Eynşteyn sahə tənlikləri" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%87%E0%A6%A8%E0%A6%B8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="আইনস্টাইন ক্ষেত্র সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="আইনস্টাইন ক্ষেত্র সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D1%96_%D0%AD%D0%B9%D0%BD%D1%88%D1%82%D1%8D%D0%B9%D0%BD%D0%B0" title="Ураўненні Эйнштэйна – Belarusian" lang="be" hreflang="be" data-title="Ураўненні Эйнштэйна" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%90%D0%B9%D0%BD%D1%89%D0%B0%D0%B9%D0%BD" title="Уравнения на Айнщайн – Bulgarian" lang="bg" hreflang="bg" data-title="Уравнения на Айнщайн" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kevatalenn_Einstein" title="Kevatalenn Einstein – Breton" lang="br" hreflang="br" data-title="Kevatalenn Einstein" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equacions_de_camp_d%27Einstein" title="Equacions de camp d'Einstein – Catalan" lang="ca" hreflang="ca" data-title="Equacions de camp d'Einstein" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%C4%95%D1%81%D0%B5%D0%BC" title="Эйнштейн танлăхĕсем – Chuvash" lang="cv" hreflang="cv" data-title="Эйнштейн танлăхĕсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Einsteinovy_rovnice_gravita%C4%8Dn%C3%ADho_pole" title="Einsteinovy rovnice gravitačního pole – Czech" lang="cs" hreflang="cs" data-title="Einsteinovy rovnice gravitačního pole" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Einsteinsche_Feldgleichungen" title="Einsteinsche Feldgleichungen – German" lang="de" hreflang="de" data-title="Einsteinsche Feldgleichungen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Einsteini_v%C3%A4ljav%C3%B5rrandid" title="Einsteini väljavõrrandid – Estonian" lang="et" hreflang="et" data-title="Einsteini väljavõrrandid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaciones_del_campo_de_Einstein" title="Ecuaciones del campo de Einstein – Spanish" lang="es" hreflang="es" data-title="Ecuaciones del campo de Einstein" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Einsteinen_eremu-ekuazioak" title="Einsteinen eremu-ekuazioak – Basque" lang="eu" hreflang="eu" data-title="Einsteinen eremu-ekuazioak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D8%A7%DB%8C%D9%86%D8%B4%D8%AA%DB%8C%D9%86" title="معادلات میدان اینشتین – Persian" lang="fa" hreflang="fa" data-title="معادلات میدان اینشتین" 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/%C3%89quation_d%27Einstein" title="Équation d'Einstein – French" lang="fr" hreflang="fr" data-title="Équation d'Einstein" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="아인슈타인 방정식 – Korean" lang="ko" hreflang="ko" data-title="아인슈타인 방정식" 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%B1%D5%B5%D5%B6%D5%B7%D5%BF%D5%A1%D5%B5%D5%B6%D5%AB_%D5%A4%D5%A1%D5%B7%D5%BF%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Այնշտայնի դաշտի հավասարումներ – Armenian" lang="hy" hreflang="hy" data-title="Այնշտայնի դաշտի հավասարումներ" 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%86%E0%A4%87%E0%A4%A8%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%BE%E0%A4%87%E0%A4%A8_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="आइनस्टाइन क्षेत्र समीकरण – Hindi" lang="hi" hreflang="hi" data-title="आइनस्टाइन क्षेत्र समीकरण" 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/Einsteinove_jednad%C5%BEbe_polja" title="Einsteinove jednadžbe polja – Croatian" lang="hr" hreflang="hr" data-title="Einsteinove jednadžbe polja" 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/Persamaan_medan_Einstein" title="Persamaan medan Einstein – Indonesian" lang="id" hreflang="id" data-title="Persamaan medan Einstein" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_campo_di_Einstein" title="Equazione di campo di Einstein – Italian" lang="it" hreflang="it" data-title="Equazione di campo di Einstein" 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%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%90%D7%99%D7%99%D7%A0%D7%A9%D7%98%D7%99%D7%99%D7%9F" title="משוואת איינשטיין – Hebrew" lang="he" hreflang="he" data-title="משוואת איינשטיין" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Einstein-egyenletek" title="Einstein-egyenletek – Hungarian" lang="hu" hreflang="hu" data-title="Einstein-egyenletek" 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%90%D1%98%D0%BD%D1%88%D1%82%D0%B0%D1%98%D0%BD%D0%BE%D0%B2%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8_%D0%B7%D0%B0_%D0%BF%D0%BE%D0%BB%D0%B5%D1%82%D0%BE" title="Ајнштајнови равенки за полето – Macedonian" lang="mk" hreflang="mk" data-title="Ајнштајнови равенки за полето" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%86%E0%A4%87%E0%A4%A8%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%BE%E0%A4%87%E0%A4%A8%E0%A4%9A%E0%A5%80_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3%E0%A5%87" title="आइनस्टाइनची क्षेत्र समीकरणे – Marathi" lang="mr" hreflang="mr" data-title="आइनस्टाइनची क्षेत्र समीकरणे" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Einstein-vergelijking" title="Einstein-vergelijking – Dutch" lang="nl" hreflang="nl" data-title="Einstein-vergelijking" 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/%E3%82%A2%E3%82%A4%E3%83%B3%E3%82%B7%E3%83%A5%E3%82%BF%E3%82%A4%E3%83%B3%E6%96%B9%E7%A8%8B%E5%BC%8F" title="アインシュタイン方程式 – Japanese" lang="ja" hreflang="ja" data-title="アインシュタイン方程式" 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 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field equations</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Field-equations in general relativity</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">"Einstein equation" redirects here. For the equation <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 E=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f73dbd37a0cac34406ee89057fa1b36a1e6a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}"></span>, see <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a>.</div> <p class="mw-empty-elt"> </p> <style 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles: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 ); 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"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a class="mw-selflink selflink">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. 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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/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></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"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In the <a href="/wiki/General_relativity" title="General relativity">general theory of relativity</a>, the <b>Einstein field equations</b> (<b>EFE</b>; also known as <b>Einstein's equations</b>) relate the geometry of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> to the distribution of <a href="/wiki/Matter#In_general_relativity_and_cosmology" title="Matter">matter</a> within it.<sup id="cite_ref-ein_1-0" class="reference"><a href="#cite_note-ein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The equations were published by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> in 1915 in the form of a <a href="/wiki/Tensor" title="Tensor">tensor equation</a><sup id="cite_ref-Ein1915_2-0" class="reference"><a href="#cite_note-Ein1915-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which related the local <i><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="spacetime_[[curvature]]"></span><span id="SPACETIME_CURVATURE"></span><span class="vanchor-text">spacetime <a href="/wiki/Curvature" title="Curvature">curvature</a></span></span></i> (expressed by the <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a>) with the local energy, <a href="/wiki/Momentum" title="Momentum">momentum</a> and stress within that spacetime (expressed by the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>).<sup id="cite_ref-FOOTNOTEMisnerThorneWheeler1973916_[ch._34]_3-0" class="reference"><a href="#cite_note-FOOTNOTEMisnerThorneWheeler1973916_[ch._34]-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Analogously to the way that <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a> are related to the distribution of <a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a> and <a href="/wiki/Electric_current" title="Electric current">currents</a> via <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, the EFE relate the <a href="/wiki/Spacetime_geometry" class="mw-redirect" title="Spacetime geometry">spacetime geometry</a> to the distribution of mass–energy, momentum and stress, that is, they determine the <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a> of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> when used in this way. The solutions of the EFE are the components of the metric tensor. The <a href="/wiki/Inertia" title="Inertia">inertial</a> trajectories of particles and radiation (<a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">geodesics</a>) in the resulting geometry are then calculated using the <a href="/wiki/Geodesic_equation" class="mw-redirect" title="Geodesic equation">geodesic equation</a>. </p><p>As well as implying local energy–momentum conservation, the EFE reduce to <a href="/wiki/Newton%27s_law_of_gravitation" class="mw-redirect" title="Newton's law of gravitation">Newton's law of gravitation</a> in the limit of a weak gravitational field and velocities that are much less than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>.<sup id="cite_ref-Carroll_4-0" class="reference"><a href="#cite_note-Carroll-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Exact solutions for the EFE can only be found under simplifying assumptions such as <a href="/wiki/Spacetime_symmetries" title="Spacetime symmetries">symmetry</a>. Special classes of <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solutions</a> are most often studied since they model many gravitational phenomena, such as <a href="/wiki/Rotating_black_hole" title="Rotating black hole">rotating black holes</a> and the <a href="/wiki/Metric_expansion_of_space" class="mw-redirect" title="Metric expansion of space">expanding universe</a>. Further simplification is achieved in approximating the spacetime as having only small deviations from <a href="/wiki/Minkowski_space" title="Minkowski space">flat spacetime</a>, leading to the <a href="/wiki/Linearized_gravity#Linearized_Einstein_field_equations" title="Linearized gravity">linearized EFE</a>. These equations are used to study phenomena such as <a href="/wiki/Gravitational_waves" class="mw-redirect" title="Gravitational waves">gravitational waves</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_form">Mathematical form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=1" title="Edit section: Mathematical form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><style data-mw-deduplicate="TemplateStyles:r1247671788">.mw-parser-output .spacetime .sidebar-list-title{background:transparent;border-top:1px solid #aaa;text-align:center}.mw-parser-output .spacetime .sidebar-below{background-color:transparent;border-color:#A2B8BF}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><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:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks spacetime"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:GPB_circling_earth.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/240px-GPB_circling_earth.jpg" decoding="async" width="240" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/360px-GPB_circling_earth.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/GPB_circling_earth.jpg/480px-GPB_circling_earth.jpg 2x" data-file-width="1200" data-file-height="900" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li></ul> </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)">Spacetime concepts</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Spacetime" title="Spacetime">Spacetime manifold</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">General relativity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction to general relativity</a></li> <li><a href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Mathematics of general relativity</a></li> <li><a class="mw-selflink selflink">Einstein field equations</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Classical gravity</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Gravity" title="Gravity">Introduction to gravitation</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Relevant mathematics</div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Four-vector" title="Four-vector">Four-vector</a></li> <li><a href="/wiki/Derivations_of_the_Lorentz_transformations" title="Derivations of the Lorentz transformations">Derivations of relativity</a></li> <li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Curved_space" title="Curved space">Curved space</a></li> <li><a href="/wiki/Curved_spacetime" title="Curved spacetime">Curved spacetime</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Spacetime_topology" title="Spacetime topology">Spacetime topology</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below"> <div class="hlist"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></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:Spacetime" title="Category:Spacetime">Category</a></span></li></ul> </div></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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:Spacetime" title="Template:Spacetime"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Spacetime" title="Template talk:Spacetime"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Spacetime" title="Special:EditPage/Template:Spacetime"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The Einstein field equations (EFE) may be written in the form:<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><sup id="cite_ref-ein_1-1" class="reference"><a href="#cite_note-ein-1"><span class="cite-bracket">[</span>1<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 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> <mi>κ</mi> <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/9665a78b0ebdb2359cb6b31072a6314bec06e2c5" 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></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EinsteinLeiden4.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/EinsteinLeiden4.jpg/300px-EinsteinLeiden4.jpg" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/EinsteinLeiden4.jpg/450px-EinsteinLeiden4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/EinsteinLeiden4.jpg/600px-EinsteinLeiden4.jpg 2x" data-file-width="2816" data-file-height="2112" /></a><figcaption>EFE on a wall in <a href="/wiki/Leiden" title="Leiden">Leiden</a>, Netherlands</figcaption></figure> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce4a5b59de7eda449c1f08ed7a84ae5de88884a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.921ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }}"></span> is the Einstein tensor, <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 }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf4140993a891f5782167dc8a0c236dc7667b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}"></span> is the metric tensor, <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_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463ab8cef859ece28e33b8460ebd4a6699834dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.452ex; height:2.843ex;" alt="{\displaystyle T_{\mu \nu }}"></span> is the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> is the <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> 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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> is the Einstein gravitational constant. </p><p>The Einstein tensor is defined 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 G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4857ef9554aad8fa33d364067e558744724ff1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.332ex; height:5.176ex;" alt="{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ee22d1a052bee0115efb8b5ffdaf10b04e42aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.859ex; height:2.843ex;" alt="{\displaystyle R_{\mu \nu }}"></span> is the <a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature tensor</a>, 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</a>. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. </p><p>The <b>Einstein gravitational constant</b> is defined as<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><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> </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 \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665\times 10^{-43}\,{\textrm {N}}^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>2.07665</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>43</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665\times 10^{-43}\,{\textrm {N}}^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57ec3755d3347f4647438863644cd1da7b2bcf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.745ex; height:5.676ex;" alt="{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665\times 10^{-43}\,{\textrm {N}}^{-1},}"></span> or <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 {\textrm {m/J}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m/J</mtext> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {m/J}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc9ea81aa96a061bbae6b21a26f784a77c8b429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.94ex; height:2.843ex;" alt="{\displaystyle {\textrm {m/J}},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">Newtonian constant of gravitation</a> and <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>. </p><p>The EFE can thus also be written 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 R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <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> <mi>κ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\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/e723dc04eeadf2f28ac04fe59daabac43bdef929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.859ex; height:5.176ex;" alt="{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.}"></span></dd></dl> <p>In standard units, each term on the left has units of 1/length<sup>2</sup>. </p><p>The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime. </p><p>These equations, together with the <a href="/wiki/Geodesic_(general_relativity)" class="mw-redirect" title="Geodesic (general relativity)">geodesic equation</a>,<sup id="cite_ref-SW1993_8-0" class="reference"><a href="#cite_note-SW1993-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> which dictates how freely falling matter moves through spacetime, form the core of the <a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">mathematical formulation</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p><p>The EFE is a tensor equation relating a set of <a href="/wiki/Symmetric_tensor" title="Symmetric tensor">symmetric 4 × 4 tensors</a>. Each tensor has 10 independent components. The four <a href="/wiki/Bianchi_identities" class="mw-redirect" title="Bianchi identities">Bianchi identities</a> reduce the number of independent equations from 10 to 6, leaving the metric with four <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge-fixing</a> <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>, which correspond to the freedom to choose a coordinate system. </p><p>Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions.<sup id="cite_ref-Stephani_et_al_9-0" class="reference"><a href="#cite_note-Stephani_et_al-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when <span class="texhtml"><i>T</i><sub><i>μν</i></sub></span> is everywhere zero) define <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifolds</a>. </p><p>The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor <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 }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf4140993a891f5782167dc8a0c236dc7667b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}"></span>, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.<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> <div class="mw-heading mw-heading3"><h3 id="Sign_convention">Sign convention</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=2" title="Edit section: Sign convention"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above form of the EFE is the standard established by <a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Misner, Thorne, and Wheeler</a> (MTW).<sup id="cite_ref-FOOTNOTEMisnerThorneWheeler1973501ff_11-0" class="reference"><a href="#cite_note-FOOTNOTEMisnerThorneWheeler1973501ff-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times \kappa T_{\mu \nu }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mn>2</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>γ</mi> <mo>,</mo> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mn>3</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mi>κ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times \kappa T_{\mu \nu }\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed40b82aae2918a6e4a83a431da5227daec5786a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:52.499ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times \kappa T_{\mu \nu }\end{aligned}}}"></span> </p><p>The third sign above is related to the choice of convention for the Ricci tensor: <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 R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mn>2</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mn>3</mn> <mo stretchy="false">]</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>α</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a370597ca453317d8e0203c784eac935102a37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.743ex; height:3.009ex;" alt="{\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}"></span> </p><p>With these definitions <a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Misner, Thorne, and Wheeler</a> classify themselves as <span class="texhtml">(+ + +)</span>, whereas Weinberg (1972)<sup id="cite_ref-FOOTNOTEWeinberg1972_12-0" class="reference"><a href="#cite_note-FOOTNOTEWeinberg1972-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> is <span class="texhtml">(+ − −)</span>, Peebles (1980)<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> and Efstathiou et al. (1990)<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> are <span class="texhtml">(− + +)</span>, Rindler (1977),<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2014)">citation needed</span></a></i>]</sup> Atwater (1974),<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2014)">citation needed</span></a></i>]</sup> Collins Martin & Squires (1989)<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> and Peacock (1999)<sup id="cite_ref-FOOTNOTEPeacock1999_16-0" class="reference"><a href="#cite_note-FOOTNOTEPeacock1999-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> are <span class="texhtml">(− + −)</span>. </p><p>Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative: <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 R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <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> <mo>−</mo> <mi>κ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\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/40acbd6807cb3db90e529fccc66de0cc6263f311" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.667ex; height:5.176ex;" alt="{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.}"></span> </p><p>The sign of the cosmological term would change in both these versions if the <span class="texhtml">(+ − − −)</span> metric <a href="/wiki/Sign_convention" title="Sign convention">sign convention</a> is used rather than the MTW <span class="texhtml">(− + + +)</span> metric sign convention adopted here. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_formulations">Equivalent formulations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=3" title="Edit section: Equivalent formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Taking the <a href="/wiki/Scalar_curvature#Definition" title="Scalar curvature">trace with respect to the metric</a> of both sides of the EFE one gets <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 R-{\frac {D}{2}}R+D\Lambda =\kappa T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mo>+</mo> <mi>D</mi> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mi>κ</mi> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58379098ae1ecdadae7d5bb53e3e6e2828367ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.227ex; height:5.176ex;" alt="{\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,}"></span> where <span class="texhtml mvar" style="font-style:italic;">D</span> is the spacetime dimension. Solving for <span class="texhtml"><i>R</i></span> and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form: <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 R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>D</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>κ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>D</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mi>T</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789b4c6a6aed1fa42cccb072187ae6f09d336468" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.454ex; height:6.176ex;" alt="{\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).}"></span> </p><p>In <span class="texhtml"><i>D</i> = 4</span> dimensions this reduces to <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 R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>κ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>T</mi> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75513940fceac01a42b573d404b5c24305bf3960" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.314ex; height:6.176ex;" alt="{\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).}"></span> </p><p>Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace <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 }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf4140993a891f5782167dc8a0c236dc7667b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}"></span> in the expression on the right with the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> without significant loss of accuracy). </p> <div class="mw-heading mw-heading2"><h2 id="The_cosmological_constant">The cosmological constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=4" title="Edit section: The cosmological constant"><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/Cosmological_constant" title="Cosmological constant">Cosmological constant</a></div> <p>In the Einstein field equations <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 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> <mi>κ</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </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/126cf1b0a0c4c4aaee36bfeeda85fc29f2fa220c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.502ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,}"></span> the term containing the cosmological constant <span class="texhtml">Λ</span> was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a <a href="/wiki/Static_universe" title="Static universe">universe that is not expanding or contracting</a>. This effort was unsuccessful because: </p> <ul><li>any desired steady state solution described by this equation is unstable, and</li> <li>observations by <a href="/wiki/Edwin_Hubble" title="Edwin Hubble">Edwin Hubble</a> showed that our universe is <a href="/wiki/Expanding_universe" class="mw-redirect" title="Expanding universe">expanding</a>.</li></ul> <p>Einstein then abandoned <span class="texhtml">Λ</span>, remarking to <a href="/wiki/George_Gamow" title="George Gamow">George Gamow</a> "that the introduction of the cosmological term was the biggest blunder of his life".<sup id="cite_ref-gamow_17-0" class="reference"><a href="#cite_note-gamow-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent <a href="/wiki/Astronomy" title="Astronomy">astronomical</a> observations have shown an <a href="/wiki/Accelerating_expansion_of_the_universe" title="Accelerating expansion of the universe">accelerating expansion of the universe</a>, and to explain this a positive value of <span class="texhtml">Λ</span> is needed.<sup id="cite_ref-wahl_18-0" class="reference"><a href="#cite_note-wahl-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-turner_19-0" class="reference"><a href="#cite_note-turner-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The effect of the cosmological constant is negligible at the scale of a galaxy or smaller. </p><p>Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor: <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 T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Λ</mi> <mi>κ</mi> </mfrac> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19fa7341c71cd81b16d8a899b20c445c43acf21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.245ex; height:5.343ex;" alt="{\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.}"></span> </p><p>This tensor describes a <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a> with an <a href="/wiki/Vacuum_energy" title="Vacuum energy">energy density</a> <span class="texhtml"><i>ρ</i><sub>vac</sub></span> and isotropic pressure <span class="texhtml"><i>p</i><sub>vac</sub></span> that are fixed constants and 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 \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Λ</mi> <mi>κ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e9439ca71cde6f7016c16ac789b7d4ed4e2b0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.777ex; height:5.343ex;" alt="{\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},}"></span> where it is assumed that <span class="texhtml">Λ</span> has SI unit m<sup>−2</sup> and <span class="texhtml"><i>κ</i></span> is defined as above. </p><p>The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity. </p> <div class="mw-heading mw-heading2"><h2 id="Features">Features</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=5" title="Edit section: Features"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Conservation_of_energy_and_momentum">Conservation of energy and momentum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=6" title="Edit section: Conservation of energy and momentum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>General relativity is consistent with the local conservation of energy and momentum expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8549738a49c13aadec1f3a249065e33a0650781" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.64ex; height:3.343ex;" alt="{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}"></span> </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Derivation of local energy–momentum conservation</strong> <p>Contracting the <a href="/wiki/Riemann_curvature_tensor#Symmetries_and_identities" title="Riemann curvature tensor">differential Bianchi identity</a> <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 R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mo stretchy="false">[</mo> <mi>γ</mi> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9757cd5645324e3171aec9e9d628e29bca6d2083" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.023ex; height:3.009ex;" alt="{\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0}"></span> with <span class="texhtml mvar" style="font-style:italic;">g<sup>αβ</sup></span> gives, using the fact that the metric tensor is covariantly constant, i.e. <span class="texhtml"><i>g<sup>αβ</sup><sub>;γ</sub></i> = 0</span>, <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 {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>ε</mi> <mi>γ</mi> <mo>;</mo> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb06337eb36cfb186d3a590284f7e4cc16ac48ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.704ex; height:3.009ex;" alt="{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"></span> </p><p>The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mi>ε</mi> <mo>;</mo> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6431f07bf6938881f8996867bf1df432248bd94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.704ex; height:3.009ex;" alt="{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"></span> which is equivalent to <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 R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>ε</mi> <mo>;</mo> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2852f95fc405374dd275b11d280f84dec8c9377f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.669ex; height:3.009ex;" alt="{\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}"></span> using the definition of the <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a>. </p><p>Next, contract again with the metric <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 g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>ε</mi> <mo>;</mo> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c85f1a6f016e1117d4440921e1b48621eb30c78e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.219ex; height:3.343ex;" alt="{\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0}"></span> to get <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 {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo>;</mo> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1118c7a7a1efadfcd5aae6d559f4bf2de645eb8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.532ex; height:3.343ex;" alt="{\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0}"></span> </p><p>The definitions of the Ricci curvature tensor and the scalar curvature then show that <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 R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a8facad79c109b8dd06b21b08a8dd14131332c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.721ex; height:3.009ex;" alt="{\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0}"></span> which can be rewritten as <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 \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0ac36b83630896fb13ee19c52d2f7488f63d6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.364ex; height:4.009ex;" alt="{\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0}"></span> </p><p>A final contraction with <span class="texhtml mvar" style="font-style:italic;">g<sup>εδ</sup></span> gives <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 \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3115111394fd9b246cc886c14bdac2700d374e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.85ex; height:4.009ex;" alt="{\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0}"></span> which by the symmetry of the bracketed term and the definition of the <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a>, gives, after relabelling the indices, <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 {G^{\alpha \beta }}_{;\beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {G^{\alpha \beta }}_{;\beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9707e1ee8e24aa438c83f7f4982fd8720167f51a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.945ex; height:3.343ex;" alt="{\displaystyle {G^{\alpha \beta }}_{;\beta }=0}"></span> </p><p>Using the EFE, this immediately gives, <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 \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19128f179a7bf9967a8b88022729d23bf16fad16" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.993ex; height:3.343ex;" alt="{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0}"></span> </p> </div> <p>which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition. </p> <div class="mw-heading mw-heading3"><h3 id="Nonlinearity">Nonlinearity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=7" title="Edit section: Nonlinearity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> are linear in the <a href="/wiki/Electric_field" title="Electric field">electric</a> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a>, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a> of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, which is linear in the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_correspondence_principle">The correspondence principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=8" title="Edit section: The correspondence principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The EFE reduce to <a href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</a> by using both the <a href="/wiki/Weak-field_approximation" class="mw-redirect" title="Weak-field approximation">weak-field approximation</a> and the <a href="/wiki/Slow-motion_approximation" class="mw-redirect" title="Slow-motion approximation">slow-motion approximation</a>. In fact, the constant <span class="texhtml mvar" style="font-style:italic;">G</span> appearing in the EFE is determined by making these two approximations. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Derivation of Newton's law of gravity</strong> <p>Newtonian gravitation can be written as the theory of a scalar field, <span class="texhtml">Φ</span>, which is the gravitational potential in joules per kilogram of the gravitational field <span class="texhtml"><i>g</i> = −∇Φ</span>, see <a href="/wiki/Gauss%27s_law_for_gravity#Poisson's_equation_and_gravitational_potential" title="Gauss's law for gravity">Gauss's law for gravity</a> <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 \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6704a5b5534a2a85bc0e7209ed75689d99d207e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.089ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)}"></span> where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the mass density. The orbit of a <a href="/wiki/Free-fall" class="mw-redirect" title="Free-fall">free-falling</a> particle satisfies <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 {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264c9e83c01ab3eab1d540aea8a26b45865e5e32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.241ex; height:3.343ex;" alt="{\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.}"></span> </p><p>In tensor notation, these become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861bb632c03e50949bc7973ed6570f3ab5142e50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.865ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}}"></span> </p><p>In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form <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 R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1db80b0c4ccdea28c9689ed58be5239ef94dbc53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.331ex; height:3.509ex;" alt="{\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)}"></span> for some constant, <span class="texhtml mvar" style="font-style:italic;">K</span>, and the <a href="/wiki/Geodesic_equation" class="mw-redirect" title="Geodesic equation">geodesic equation</a> <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 {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3b3c8e8efe83ebc3e138964f21371cead75ba0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.245ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}"></span> </p><p>To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero <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 {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>≈</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9796efbd0de39e63781698a9c8acb806eaa6609c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.919ex; height:6.343ex;" alt="{\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}"></span> and thus <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 {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45bd4450c067dca35db8be4f2171fabc876e7fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.215ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0}"></span> and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives <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 {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≈</mo> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a689e346ab394d0e04320dc70c0ae3d103647fdf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.473ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}}"></span> where two factors of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>dt</i></span><span class="sr-only">/</span><span class="den"><i>dτ</i></span></span>⁠</span></span> have been divided out. This will reduce to its Newtonian counterpart, provided <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 \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>≈</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>α</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8daff88d13bfcd77e4bec85090e214b0ede6153" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.871ex; height:3.509ex;" alt="{\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.}"></span> </p><p>Our assumptions force <span class="texhtml"><i>α</i> = <i>i</i></span> and the time (0) derivatives to be zero. So this simplifies to <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 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>≈</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d68af87cb69c674c1d4ff38b486b835734ae89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.22ex; height:3.343ex;" alt="{\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,}"></span> which is satisfied by letting <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 g_{00}\approx -c^{2}-2\Phi \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>≈</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">Φ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e085a0a64d25fc1fad64c16244ac555710102a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.668ex; height:3.009ex;" alt="{\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}"></span> </p><p>Turning to the Einstein equations, we only need the time-time component <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 R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>=</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24cc7d3e4216d308333a47eb46ef24099b147cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.675ex; height:3.509ex;" alt="{\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)}"></span> the low speed and static field assumptions imply that <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 T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>≈</mo> <mi>diag</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mi>diag</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c79213a1b9330d27427dbb5e017abbbf5ce775" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.826ex; height:3.343ex;" alt="{\displaystyle T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.}"></span> </p><p>So <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 T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>≈</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f6ad05255350f78ea7404b47de5e0c3c860742" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.613ex; height:5.509ex;" alt="{\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,}"></span> and thus <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 K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>K</mi> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27715193f8df668ff4b036b9f16bfcc78c656592" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:57.473ex; height:3.509ex;" alt="{\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.}"></span> </p><p>From the definition of the Ricci tensor <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 R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fbde78a228c2b9dbd65086b9a58fa9a9f88cc8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.824ex; height:3.509ex;" alt="{\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}"></span> </p><p>Our simplifying assumptions make the squares of <span class="texhtml mvar" style="font-style:italic;">Γ</span> disappear together with the time derivatives <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 R_{00}\approx \Gamma _{00,i}^{i}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>≈</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8162cc2ad2f0fcb84b2d5694899dd53d22fc033" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.126ex; height:3.509ex;" alt="{\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.}"></span> </p><p>Combining the above equations together <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 \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>≈</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>≈</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>=</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>K</mi> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a3188b069f342fcb95982774a1e8fc2412b4c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:47.814ex; height:3.676ex;" alt="{\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}}"></span> which reduces to the Newtonian field equation provided <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 {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>K</mi> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aac7ba9a7f62cce08d730dff41bd247f122398e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.996ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,}"></span> which will occur if <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 K={\frac {8\pi G}{c^{4}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {8\pi G}{c^{4}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dedcf7a3ea9c6c3c4bbe9ee6fbb49ef2d34b6af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.356ex; height:5.676ex;" alt="{\displaystyle K={\frac {8\pi G}{c^{4}}}\,.}"></span> </p> </div> <div class="mw-heading mw-heading2"><h2 id="Vacuum_field_equations">Vacuum field equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=9" title="Edit section: Vacuum field equations"><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:Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/fe/Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png" decoding="async" width="204" height="204" class="mw-file-element" data-file-width="204" data-file-height="204" /></a><figcaption>A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).</figcaption></figure> <p>If the energy–momentum tensor <span class="texhtml mvar" style="font-style:italic;">T<sub>μν</sub></span> is zero in the region under consideration, then the field equations are also referred to as the <a href="/wiki/Field_equation#Vacuum_field_equations" title="Field equation">vacuum field equations</a>. By setting <span class="texhtml"><i>T<sub>μν</sub></i> = 0</span> in the <a href="#Equivalent_formulations">trace-reversed field equations</a>, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as <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 R_{\mu \nu }=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db727b43a659ee0d1d490db14e77b179d7772d19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.154ex; height:2.843ex;" alt="{\displaystyle R_{\mu \nu }=0\,.}"></span> </p><p>In the case of nonzero cosmological constant, the equations are <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 R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Λ</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abb5f9f5aae1cb3c369afa45c5412f709c76f46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:18.231ex; height:7.009ex;" alt="{\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.}"></span> </p><p>The solutions to the vacuum field equations are called <a href="/wiki/Vacuum_solution_(general_relativity)" title="Vacuum solution (general relativity)">vacuum solutions</a>. Flat <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> is the simplest example of a vacuum solution. Nontrivial examples include the <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a> and the <a href="/wiki/Kerr_solution" class="mw-redirect" title="Kerr solution">Kerr solution</a>. </p><p><a href="/wiki/Manifold" title="Manifold">Manifolds</a> with a vanishing <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a>, <span class="texhtml"><i>R<sub>μν</sub></i> = 0</span>, are referred to as <a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifolds</a> and manifolds with a Ricci tensor proportional to the metric as <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifolds</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Einstein–Maxwell_equations"><span id="Einstein.E2.80.93Maxwell_equations"></span>Einstein–Maxwell equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=10" title="Edit section: Einstein–Maxwell equations"><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/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell's equations in curved spacetime</a></div> <p>If the energy–momentum tensor <span class="texhtml mvar" style="font-style:italic;">T<sub>μν</sub></span> is that of an <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> in <a href="/wiki/Free_space" class="mw-redirect" title="Free space">free space</a>, i.e. if the <a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">electromagnetic stress–energy tensor</a> <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 T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mi>τ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mi>τ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974687e791c2a7b02c9903527769bdb4e5891b64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.523ex; height:5.676ex;" alt="{\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)}"></span> is used, then the Einstein field equations are called the <i>Einstein–Maxwell equations</i> (with <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> <span class="texhtml">Λ</span>, taken to be zero in conventional relativity theory): <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 G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>κ</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mi>τ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> <mi>τ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239135d194bb8b12b6e7112ef755e44b2d4bd7a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.266ex; height:5.343ex;" alt="{\displaystyle G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}"></span> </p><p>Additionally, the <a href="/wiki/Electromagnetic_tensor#The_field_tensor_and_relativity" title="Electromagnetic tensor">covariant Maxwell equations</a> are also applicable in free space: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{[\alpha \beta ;\gamma ]}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>β</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>α</mi> <mi>β</mi> <mo>;</mo> <mi>γ</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mo>;</mo> <mi>γ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mo>;</mo> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> <mo>;</mo> <mi>β</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{[\alpha \beta ;\gamma ]}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3151562dc243162c7975eb508847246bd11a013" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.733ex; margin-bottom: -0.272ex; width:67.336ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{[\alpha \beta ;\gamma ]}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}}"></span> where the semicolon represents a <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>, and the brackets denote <a href="/wiki/Exterior_algebra#Alternating_tensor_algebra" title="Exterior algebra">anti-symmetrization</a>. The first equation asserts that the 4-<a href="/wiki/Divergence" title="Divergence">divergence</a> of the <a href="/wiki/2-form" class="mw-redirect" title="2-form">2-form</a> <span class="texhtml mvar" style="font-style:italic;">F</span> is zero, and the second that its <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> is zero. From the latter, it follows by the <a href="/wiki/Poincar%C3%A9_lemma" title="Poincaré lemma">Poincaré lemma</a> that in a coordinate chart it is possible to introduce an electromagnetic field potential <span class="texhtml mvar" style="font-style:italic;">A<sub>α</sub></span> such that <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_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>;</mo> <mi>β</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>;</mo> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1366362c75b12c4802bc762ba6733510511ba838" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.304ex; height:2.843ex;" alt="{\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }}"></span> in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.<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> However, there are global solutions of the equation that may lack a globally defined potential.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Solutions">Solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=11" title="Edit section: Solutions"><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/Solutions_of_the_Einstein_field_equations" title="Solutions of the Einstein field equations">Solutions of the Einstein field equations</a></div> <p>The solutions of the Einstein field equations are <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metrics</a> of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as <a href="/wiki/Post-Newtonian_approximation" class="mw-redirect" title="Post-Newtonian approximation">post-Newtonian approximations</a>. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.<sup id="cite_ref-Stephani_et_al_9-1" class="reference"><a href="#cite_note-Stephani_et_al-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The study of exact solutions of Einstein's field equations is one of the activities of <a href="/wiki/Physical_cosmology" title="Physical cosmology">cosmology</a>. It leads to the prediction of <a href="/wiki/Black_hole" title="Black hole">black holes</a> and to different models of evolution of the <a href="/wiki/Universe" title="Universe">universe</a>. </p><p>One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,<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> self-similar solutions to the Einstein field equations are fixed points of the resulting <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>. New solutions have been discovered using these methods by LeBlanc<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> and Kohli and Haslam.<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> </p> <div class="mw-heading mw-heading2"><h2 id="The_linearized_EFE">The linearized EFE</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=12" title="Edit section: The linearized EFE"><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/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></div> <p>The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> is very weak and the <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of <a href="/wiki/Gravitational_radiation" class="mw-redirect" title="Gravitational radiation">gravitational radiation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Polynomial_form">Polynomial form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=13" title="Edit section: Polynomial form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written <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 \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mstyle> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mi>λ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>κ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>λ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>μ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775f0c22c59b27b89ecca924a22db339a493ff3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.521ex; height:3.509ex;" alt="{\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}"></span> using the <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>; and the inverse of the metric in 4 dimensions can be written as: <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 g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>κ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mi>λ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>λ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>μ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eee850494afee86e00e42b9a1b9aa2e5019dcfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.423ex; height:7.343ex;" alt="{\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.}"></span> </p><p>Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of <span class="texhtml">det(<i>g</i>)</span> to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The <a href="/wiki/Einstein-Hilbert_action" class="mw-redirect" title="Einstein-Hilbert action">Einstein-Hilbert action</a> from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<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=Einstein_field_equations&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Conformastatic_spacetimes" title="Conformastatic spacetimes">Conformastatic spacetimes</a></li> <li><a href="/wiki/Einstein%E2%80%93Hilbert_action" title="Einstein–Hilbert action">Einstein–Hilbert action</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Exact solutions in general relativity</a></li> <li><a href="/wiki/General_relativity_resources" class="mw-redirect" title="General relativity resources">General relativity resources</a></li> <li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History of general relativity</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/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=15" title="Edit section: Notes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-ein-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ein_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ein_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFEinstein1916" class="citation journal cs1">Einstein, Albert (1916). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120206225139/http://www.alberteinstein.info/gallery/gtext3.html">"The Foundation of the General Theory of Relativity"</a>. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i>. <b>354</b> (7): 769. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1916AnP...354..769E">1916AnP...354..769E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19163540702">10.1002/andp.19163540702</a>. Archived from <a rel="nofollow" class="external text" href="http://www.alberteinstein.info/gallery/science.html">the original</a> <span class="cs1-format">(<a href="/wiki/PDF" title="PDF">PDF</a>)</span> on 2012-02-06.</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=The+Foundation+of+the+General+Theory+of+Relativity&rft.volume=354&rft.issue=7&rft.pages=769&rft.date=1916&rft_id=info%3Adoi%2F10.1002%2Fandp.19163540702&rft_id=info%3Abibcode%2F1916AnP...354..769E&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=http%3A%2F%2Fwww.alberteinstein.info%2Fgallery%2Fscience.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-Ein1915-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ein1915_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1915" class="citation journal cs1"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (November 25, 1915). <a rel="nofollow" class="external text" href="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/6E3MAXK4/index.meta">"Die Feldgleichungen der Gravitation"</a>. <i>Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin</i>: 844–847<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-08-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+Preussischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=Die+Feldgleichungen+der+Gravitation&rft.pages=844-847&rft.date=1915-11-25&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=http%3A%2F%2Fecho.mpiwg-berlin.mpg.de%2FECHOdocuView%3Furl%3D%2Fpermanent%2Fecho%2Feinstein%2Fsitzungsberichte%2F6E3MAXK4%2Findex.meta&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMisnerThorneWheeler1973916_[ch._34]-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMisnerThorneWheeler1973916_[ch._34]_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler (1973)</a>, p. 916 [ch. 34].</span> </li> <li id="cite_note-Carroll-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Carroll_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll2004" class="citation book cs1"><a href="/wiki/Sean_M._Carroll" title="Sean M. Carroll">Carroll, Sean</a> (2004). <i>Spacetime and Geometry – An Introduction to General Relativity</i>. Addison Wesley. pp. 151–159. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-8732-3" title="Special:BookSources/0-8053-8732-3"><bdi>0-8053-8732-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+and+Geometry+%E2%80%93+An+Introduction+to+General+Relativity&rft.pages=151-159&rft.pub=Addison+Wesley&rft.date=2004&rft.isbn=0-8053-8732-3&rft.aulast=Carroll&rft.aufirst=Sean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></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="CITEREFGrønHervik2007" class="citation book cs1">Grøn, Øyvind; Hervik, Sigbjorn (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IyJhCHAryuUC&pg=PA180"><i>Einstein's General Theory of Relativity: With Modern Applications in Cosmology</i></a> (illustrated ed.). Springer Science & Business Media. p. 180. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-69200-5" title="Special:BookSources/978-0-387-69200-5"><bdi>978-0-387-69200-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein%27s+General+Theory+of+Relativity%3A+With+Modern+Applications+in+Cosmology&rft.pages=180&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-0-387-69200-5&rft.aulast=Gr%C3%B8n&rft.aufirst=%C3%98yvind&rft.au=Hervik%2C+Sigbjorn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIyJhCHAryuUC%26pg%3DPA180&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">With the choice of the Einstein gravitational constant as given here, <span class="texhtml"><i>κ</i> = 8<i>πG</i>/<i>c</i><span style="padding-left:0.12em;"><sup>4</sup></span></span>, the stress–energy tensor on the right side of the equation must be written with each component in units of energy density (i.e., energy per volume, equivalently pressure). In Einstein's original publication, the choice is <span class="texhtml"><i>κ</i> = 8<i>πG</i>/<i>c</i><span style="padding-left:0.12em;"><sup>2</sup></span></span>, in which case the stress–energy tensor components have units of mass density.</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="CITEREFAdlerBazinSchiffer1975" class="citation book cs1">Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/1046135"><i>Introduction to general relativity</i></a> (2d ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-000423-4" title="Special:BookSources/0-07-000423-4"><bdi>0-07-000423-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1046135">1046135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+general+relativity&rft.place=New+York&rft.edition=2d&rft.pub=McGraw-Hill&rft.date=1975&rft_id=info%3Aoclcnum%2F1046135&rft.isbn=0-07-000423-4&rft.aulast=Adler&rft.aufirst=Ronald&rft.au=Bazin%2C+Maurice&rft.au=Schiffer%2C+Menahem&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1046135&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-SW1993-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-SW1993_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1993" class="citation book cs1">Weinberg, Steven (1993). <i>Dreams of a Final Theory: the search for the fundamental laws of nature</i>. Vintage Press. pp. 107, 233. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-09-922391-0" title="Special:BookSources/0-09-922391-0"><bdi>0-09-922391-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dreams+of+a+Final+Theory%3A+the+search+for+the+fundamental+laws+of+nature&rft.pages=107%2C+233&rft.pub=Vintage+Press&rft.date=1993&rft.isbn=0-09-922391-0&rft.aulast=Weinberg&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-Stephani_et_al-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stephani_et_al_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stephani_et_al_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephaniKramerMacCallumHoenselaers2003" class="citation book cs1">Stephani, Hans; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003). <i>Exact Solutions of Einstein's Field Equations</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-46136-7" title="Special:BookSources/0-521-46136-7"><bdi>0-521-46136-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exact+Solutions+of+Einstein%27s+Field+Equations&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-46136-7&rft.aulast=Stephani&rft.aufirst=Hans&rft.au=Kramer%2C+D.&rft.au=MacCallum%2C+M.&rft.au=Hoenselaers%2C+C.&rft.au=Herlt%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRendall2005" class="citation journal cs1">Rendall, Alan D. (2005). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5256071">"Theorems on Existence and Global Dynamics for the Einstein Equations"</a>. <i>Living Rev. Relativ</i>. <b>8</b> (1). Article number: 6. <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/gr-qc/0505133">gr-qc/0505133</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/2005LRR.....8....6R">2005LRR.....8....6R</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-2005-6">10.12942/lrr-2005-6</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/PMC5256071">5256071</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/28179868">28179868</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Rev.+Relativ.&rft.atitle=Theorems+on+Existence+and+Global+Dynamics+for+the+Einstein+Equations&rft.volume=8&rft.issue=1&rft.pages=Article+number%3A+6&rft.date=2005&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256071%23id-name%3DPMC&rft_id=info%3Abibcode%2F2005LRR.....8....6R&rft_id=info%3Aarxiv%2Fgr-qc%2F0505133&rft_id=info%3Apmid%2F28179868&rft_id=info%3Adoi%2F10.12942%2Flrr-2005-6&rft.aulast=Rendall&rft.aufirst=Alan+D.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5256071&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMisnerThorneWheeler1973501ff-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMisnerThorneWheeler1973501ff_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler (1973)</a>, p. 501ff.</span> </li> <li id="cite_note-FOOTNOTEWeinberg1972-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeinberg1972_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg1972">Weinberg (1972)</a>.</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="CITEREFPeebles1980" class="citation book cs1">Peebles, Phillip James Edwin (1980). <i>The Large-scale Structure of the Universe</i>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08239-1" title="Special:BookSources/0-691-08239-1"><bdi>0-691-08239-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Large-scale+Structure+of+the+Universe&rft.pub=Princeton+University+Press&rft.date=1980&rft.isbn=0-691-08239-1&rft.aulast=Peebles&rft.aufirst=Phillip+James+Edwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" 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="CITEREFEfstathiouSutherlandMaddox1990" class="citation journal cs1">Efstathiou, G.; Sutherland, W. J.; Maddox, S. J. (1990). "The cosmological constant and cold dark matter". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>348</b> (6303): 705. <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/1990Natur.348..705E">1990Natur.348..705E</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%2F348705a0">10.1038/348705a0</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:12988317">12988317</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=The+cosmological+constant+and+cold+dark+matter&rft.volume=348&rft.issue=6303&rft.pages=705&rft.date=1990&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12988317%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F348705a0&rft_id=info%3Abibcode%2F1990Natur.348..705E&rft.aulast=Efstathiou&rft.aufirst=G.&rft.au=Sutherland%2C+W.+J.&rft.au=Maddox%2C+S.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" 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="CITEREFCollinsMartinSquires1989" class="citation book cs1">Collins, P. 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University of Toronto. Archived from <a rel="nofollow" class="external text" href="http://www.news.utoronto.ca/bin6/051122-1839.asp">the original</a> on 2007-03-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=News%40UofT&rft.atitle=Was+Einstein%27s+%27biggest+blunder%27+a+stellar+success%3F&rft.date=2005-11-22&rft.aulast=Wahl&rft.aufirst=Nicolle&rft_id=http%3A%2F%2Fwww.news.utoronto.ca%2Fbin6%2F051122-1839.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-turner-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-turner_19-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTurner2001" class="citation journal cs1">Turner, Michael S. (May 2001). "Making Sense of the New Cosmology". <i>Int. J. Mod. Phys. A</i>. <b>17</b> (S1): 180–196. <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/astro-ph/0202008">astro-ph/0202008</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/2002IJMPA..17S.180T">2002IJMPA..17S.180T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X02013113">10.1142/S0217751X02013113</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:16669258">16669258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Int.+J.+Mod.+Phys.+A&rft.atitle=Making+Sense+of+the+New+Cosmology&rft.volume=17&rft.issue=S1&rft.pages=180-196&rft.date=2001-05&rft_id=info%3Aarxiv%2Fastro-ph%2F0202008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16669258%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0217751X02013113&rft_id=info%3Abibcode%2F2002IJMPA..17S.180T&rft.aulast=Turner&rft.aufirst=Michael+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2005" class="citation book cs1">Brown, Harvey (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T6IVyWiPQksC&q=Maxwell+and+potential+and+%22generally+covariant%22&pg=PA164"><i>Physical Relativity</i></a>. 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Quantum Grav</i>. <b>3</b> (6): 1105–1124. <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/1986CQGra...3.1105H">1986CQGra...3.1105H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F3%2F6%2F011">10.1088/0264-9381/3/6/011</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:250907312">250907312</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Class.+Quantum+Grav.&rft.atitle=Self-similar+spatially+homogeneous+cosmologies%3A+orthogonal+perfect+fluid+and+vacuum+solutions&rft.volume=3&rft.issue=6&rft.pages=1105-1124&rft.date=1986&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250907312%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F3%2F6%2F011&rft_id=info%3Abibcode%2F1986CQGra...3.1105H&rft.aulast=Hsu&rft.aufirst=L.&rft.au=Wainwright%2C+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeBlanc1997" class="citation journal cs1">LeBlanc, V. G. (1997). "Asymptotic states of magnetic Bianchi I cosmologies". <i>Class. Quantum Grav</i>. <b>14</b> (8): 2281. <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/1997CQGra..14.2281L">1997CQGra..14.2281L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F14%2F8%2F025">10.1088/0264-9381/14/8/025</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:250876974">250876974</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Class.+Quantum+Grav.&rft.atitle=Asymptotic+states+of+magnetic+Bianchi+I+cosmologies&rft.volume=14&rft.issue=8&rft.pages=2281&rft.date=1997&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250876974%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F14%2F8%2F025&rft_id=info%3Abibcode%2F1997CQGra..14.2281L&rft.aulast=LeBlanc&rft.aufirst=V.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKohliHaslam2013" class="citation journal cs1">Kohli, Ikjyot Singh; Haslam, Michael C. (2013). "Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model". <i>Phys. Rev. D</i>. <b>88</b> (6): 063518. <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/1304.8042">1304.8042</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/2013PhRvD..88f3518K">2013PhRvD..88f3518K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.88.063518">10.1103/physrevd.88.063518</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:119178273">119178273</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+D&rft.atitle=Dynamical+systems+approach+to+a+Bianchi+type+I+viscous+magnetohydrodynamic+model&rft.volume=88&rft.issue=6&rft.pages=063518&rft.date=2013&rft_id=info%3Aarxiv%2F1304.8042&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119178273%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2Fphysrevd.88.063518&rft_id=info%3Abibcode%2F2013PhRvD..88f3518K&rft.aulast=Kohli&rft.aufirst=Ikjyot+Singh&rft.au=Haslam%2C+Michael+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatanaev2006" class="citation journal cs1">Katanaev, M. O. (2006). "Polynomial form of the Hilbert–Einstein action". <i>Gen. Rel. Grav</i>. <b>38</b> (8): 1233–1240. <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/gr-qc/0507026">gr-qc/0507026</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/2006GReGr..38.1233K">2006GReGr..38.1233K</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%2Fs10714-006-0310-5">10.1007/s10714-006-0310-5</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6263993">6263993</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Gen.+Rel.+Grav.&rft.atitle=Polynomial+form+of+the+Hilbert%E2%80%93Einstein+action&rft.volume=38&rft.issue=8&rft.pages=1233-1240&rft.date=2006&rft_id=info%3Aarxiv%2Fgr-qc%2F0507026&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6263993%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10714-006-0310-5&rft_id=info%3Abibcode%2F2006GReGr..38.1233K&rft.aulast=Katanaev&rft.aufirst=M.+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></span> </li> </ol></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=Einstein_field_equations&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>See <a href="/wiki/General_relativity_resources" class="mw-redirect" title="General relativity resources">General relativity resources</a>. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1"><a href="/wiki/Charles_Misner" class="mw-redirect" title="Charles Misner">Misner, Charles W.</a>; <a href="/wiki/Kip_S._Thorne" class="mw-redirect" title="Kip S. Thorne">Thorne, Kip S.</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (1973). <i><a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Gravitation</a></i>. San Francisco: <a href="/wiki/W._H._Freeman" class="mw-redirect" title="W. H. Freeman">W. H. Freeman</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0344-0" title="Special:BookSources/978-0-7167-0344-0"><bdi>978-0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.place=San+Francisco&rft.pub=W.+H.+Freeman&rft.date=1973&rft.isbn=978-0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles+W.&rft.au=Thorne%2C+Kip+S.&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1972" class="citation book cs1"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1972). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/gravitationcosmo00stev_0"><i>Gravitation and Cosmology</i></a></span>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-92567-5" title="Special:BookSources/0-471-92567-5"><bdi>0-471-92567-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Cosmology&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=0-471-92567-5&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgravitationcosmo00stev_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeacock1999" class="citation book cs1"><a href="/wiki/John_A._Peacock" title="John A. Peacock">Peacock, John A.</a> (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/cosmologicalphys0000peac"><i>Cosmological Physics</i></a></span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521410724" title="Special:BookSources/978-0521410724"><bdi>978-0521410724</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cosmological+Physics&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0521410724&rft.aulast=Peacock&rft.aufirst=John+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcosmologicalphys0000peac&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" 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=Einstein_field_equations&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output 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width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/General_Relativity" class="extiw" title="v:General Relativity"> General Relativity</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Einstein_equations">"Einstein equations"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Einstein+equations&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DEinstein_equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEinstein+field+equations" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110621005940/http://www.black-holes.org/relativity6.html">Caltech Tutorial on Relativity</a> — A simple introduction to Einstein's Field Equations.</li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/einstein/einstein.html">The Meaning of Einstein's Equation</a> — An explanation of Einstein's field equation, its derivation, and some of its consequences</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2">Video Lecture on Einstein's Field Equations</a> by <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a> Physics Professor Edmund Bertschinger.</li> <li><a rel="nofollow" class="external text" href="http://scitation.aip.org/content/aip/magazine/physicstoday/article/68/11/10.1063/PT.3.2979">Arch and scaffold: How Einstein found his field equations</a> Physics Today November 2015, History of the Development of the Field Equations</li></ul> <div class="mw-heading mw-heading3"><h3 id="External_images">External images</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Einstein_field_equations&action=edit&section=18" title="Edit section: External images"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180226091926/https://www.ilorentz.org/history/wallformulas/images/pages/page_2.html">The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden</a></li> <li><a href="/wiki/Suzanne_Imber" title="Suzanne Imber">Suzanne Imber</a>, <a rel="nofollow" class="external text" href="https://imaggeo.egu.eu/view/886/">"The impact of general relativity on the Atacama Desert"</a>, Einstein field equation on the side of a train in Bolivia.</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 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.navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Albert_Einstein" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Albert_Einstein" title="Template:Albert Einstein"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Albert_Einstein" title="Template talk:Albert Einstein"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Albert_Einstein" title="Special:EditPage/Template:Albert Einstein"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Albert_Einstein" style="font-size:114%;margin:0 4em"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li></ul></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a></li> <li><a href="/wiki/Photoelectric_effect" title="Photoelectric effect">Photoelectric effect</a></li> <li><a href="/wiki/Einstein_coefficients" title="Einstein coefficients">Einstein coefficients</a></li> <li><a href="/wiki/Einstein_solid" title="Einstein solid">Einstein solid</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a class="mw-selflink selflink">Einstein field equations</a></li> <li><a href="/wiki/Einstein_radius" title="Einstein radius">Einstein radius</a></li> <li><a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a></li> <li><a href="/wiki/Cosmological_constant" title="Cosmological constant">Cosmological constant</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_correlations" title="Bose–Einstein correlations">Bose–Einstein correlations</a></li> <li><a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a></li> <li><a href="/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations" title="Einstein–Infeld–Hoffmann equations">Einstein–Infeld–Hoffmann equations</a></li> <li><a href="/wiki/Einstein%E2%80%93de_Haas_effect" title="Einstein–de Haas effect">Einstein–de Haas effect</a></li> <li><a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a></li> <li><a href="/wiki/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein debates</a></li> <li><a href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</a></li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Thought experiments</a></li> <li><a href="/wiki/Einstein%27s_unsuccessful_investigations" title="Einstein's unsuccessful investigations">Unsuccessful investigations</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational wave</a></li> <li><a href="/wiki/Tea_leaf_paradox" title="Tea leaf paradox">Tea leaf paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_scientific_publications_by_Albert_Einstein" title="List of scientific publications by Albert Einstein">Works</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Annus_mirabilis_papers" title="Annus mirabilis papers"><i>Annus mirabilis</i> papers</a> (1905)</li> <li>"<a href="/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen" title="Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen">Investigations on the Theory of Brownian Movement</a>" (1905)</li> <li><i><a href="/wiki/Relativity:_The_Special_and_the_General_Theory" title="Relativity: The Special and the General Theory">Relativity: The Special and the General Theory</a></i> (1916)</li> <li><i><a href="/wiki/The_Meaning_of_Relativity" title="The Meaning of Relativity">The Meaning of Relativity</a></i> (1922)</li> <li><i><a href="/wiki/The_World_as_I_See_It_(book)" title="The World as I See It (book)">The World as I See It</a></i> (1934)</li> <li><i><a href="/wiki/The_Evolution_of_Physics" title="The Evolution of Physics">The Evolution of Physics</a></i> (1938)</li> <li>"<a href="/wiki/Why_Socialism%3F" title="Why Socialism?">Why Socialism?</a>" (1949)</li> <li><a href="/wiki/Russell%E2%80%93Einstein_Manifesto" title="Russell–Einstein Manifesto">Russell–Einstein Manifesto</a> (1955)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Albert_Einstein_in_popular_culture" title="Albert Einstein in popular culture">In popular<br />culture</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Die_Grundlagen_der_Einsteinschen_Relativit%C3%A4ts-Theorie" title="Die Grundlagen der Einsteinschen Relativitäts-Theorie">Die Grundlagen der Einsteinschen Relativitäts-Theorie</a></i> (1922 documentary)</li> <li><i><a href="/wiki/The_Einstein_Theory_of_Relativity" title="The Einstein Theory of Relativity">The Einstein Theory of Relativity</a></i> (1923 documentary)</li> <li><i><a href="/wiki/Relics:_Einstein%27s_Brain" title="Relics: Einstein's Brain">Relics: Einstein's Brain</a></i> (1994 documentary)</li> <li><i><a href="/wiki/Insignificance_(film)" title="Insignificance (film)">Insignificance</a></i> (1985 film)</li> <li><i><a href="/wiki/Young_Einstein" title="Young Einstein">Young Einstein</a></i> (1988 film)</li> <li><i><a href="/wiki/Picasso_at_the_Lapin_Agile" title="Picasso at the Lapin Agile">Picasso at the Lapin Agile</a></i> (1993 play)</li> <li><i><a href="/wiki/I.Q._(film)" title="I.Q. (film)">I.Q.</a></i> (1994 film)</li> <li><i><a href="/wiki/Einstein%27s_Gift" title="Einstein's Gift">Einstein's Gift</a></i> (2003 play)</li> <li><i><a href="/wiki/Einstein_and_Eddington" title="Einstein and Eddington">Einstein and Eddington</a></i> (2008 TV film)</li> <li><i><a href="/wiki/Genius_(American_TV_series)" title="Genius (American TV series)">Genius</a></i> (2017 series)</li> <li><i><a href="/wiki/Oppenheimer_(film)" title="Oppenheimer (film)">Oppenheimer</a></i> (2023 film)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Prizes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_Einstein_Award" title="Albert Einstein Award">Albert Einstein Award</a></li> <li><a href="/wiki/Albert_Einstein_Medal" title="Albert Einstein Medal">Albert Einstein Medal</a></li> <li><a href="/wiki/Kalinga_Prize" title="Kalinga Prize">Kalinga Prize</a></li> <li><a href="/wiki/Albert_Einstein_Peace_Prize" title="Albert Einstein Peace Prize">Albert Einstein Peace Prize</a></li> <li><a href="/wiki/Albert_Einstein_World_Award_of_Science" title="Albert Einstein World Award of Science">Albert Einstein World Award of Science</a></li> <li><a href="/wiki/Einstein_Prize_for_Laser_Science" title="Einstein Prize for Laser Science">Einstein Prize for Laser Science</a></li> <li><a href="/wiki/Einstein_Prize_(APS)" title="Einstein Prize (APS)">Einstein Prize (APS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books about<br />Einstein</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Albert_Einstein:_Creator_and_Rebel" title="Albert Einstein: Creator and Rebel">Albert Einstein: Creator and Rebel</a></i></li> <li><i><a href="/wiki/Einstein_and_Religion" title="Einstein and Religion">Einstein and Religion</a></i></li> <li><i><a href="/wiki/Einstein_for_Beginners" title="Einstein for Beginners">Einstein for Beginners</a></i></li> <li><i><a href="/wiki/Einstein:_His_Life_and_Universe" title="Einstein: His Life and Universe">Einstein: His Life and Universe</a></i></li> <li><i><a href="/wiki/Einstein%27s_Cosmos" title="Einstein's Cosmos">Einstein's Cosmos</a></i></li> <li><i><a href="/wiki/I_Am_Albert_Einstein" title="I Am Albert Einstein">I Am Albert Einstein</a></i></li> <li><i><a href="/wiki/Introducing_Relativity" title="Introducing Relativity">Introducing Relativity</a></i></li> <li><i><a href="/wiki/Subtle_is_the_Lord" title="Subtle is the Lord">Subtle is the Lord</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Einstein_family" title="Einstein family">Family</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mileva_Mari%C4%87" title="Mileva Marić">Mileva Marić</a> (first wife)</li> <li><a href="/wiki/Elsa_Einstein" title="Elsa Einstein">Elsa Einstein</a> (second wife; cousin)</li> <li><a href="/wiki/Lieserl_Einstein" class="mw-redirect" title="Lieserl Einstein">Lieserl Einstein</a> (daughter)</li> <li><a href="/wiki/Hans_Albert_Einstein" title="Hans Albert Einstein">Hans Albert Einstein</a> (son)</li> <li><a href="/wiki/Pauline_Koch" class="mw-redirect" title="Pauline Koch">Pauline Koch</a> (mother)</li> <li><a href="/wiki/Hermann_Einstein" class="mw-redirect" title="Hermann Einstein">Hermann Einstein</a> (father)</li> <li><a href="/wiki/Maja_Einstein" title="Maja Einstein">Maja Einstein</a> (sister)</li> <li><a href="/wiki/Einstein_family#Eduard_"Tete"_Einstein_(Albert's_second_son)" title="Einstein family">Eduard Einstein</a> (son)</li> <li><a href="/wiki/Murder_of_the_family_of_Robert_Einstein" title="Murder of the family of Robert Einstein">Robert Einstein</a> (cousin)</li> <li><a href="/wiki/Bernhard_Caesar_Einstein" title="Bernhard Caesar Einstein">Bernhard Caesar Einstein</a> (grandson)</li> <li><a href="/wiki/Evelyn_Einstein" title="Evelyn Einstein">Evelyn Einstein</a> (granddaughter)</li> <li><a href="/wiki/Thomas_Martin_Einstein" class="mw-redirect" title="Thomas Martin Einstein">Thomas Martin Einstein</a> (great-grandson)</li> <li><a href="/wiki/Siegbert_Einstein" title="Siegbert Einstein">Siegbert Einstein</a> (distant cousin)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_awards_and_honors_received_by_Albert_Einstein" title="List of awards and honors received by Albert Einstein">Awards and honors</a></li> <li><a href="/wiki/Brain_of_Albert_Einstein" title="Brain of Albert Einstein">Brain</a></li> <li><a href="/wiki/Albert_Einstein_House" title="Albert Einstein House">House</a></li> <li><a href="/wiki/Albert_Einstein_Memorial" title="Albert Einstein Memorial">Memorial</a></li> <li><a href="/wiki/Political_views_of_Albert_Einstein" title="Political views of Albert Einstein">Political views</a></li> <li><a href="/wiki/Religious_and_philosophical_views_of_Albert_Einstein" title="Religious and philosophical views of Albert Einstein">Religious views</a></li> <li><a href="/wiki/List_of_things_named_after_Albert_Einstein" title="List of things named after Albert Einstein">Things named after</a></li> <li><a href="/wiki/Einstein%E2%80%93Oppenheimer_relationship" title="Einstein–Oppenheimer relationship">Einstein–Oppenheimer relationship</a></li> <li><a href="/wiki/Albert_Einstein_Archives" title="Albert Einstein Archives">Albert Einstein Archives</a></li> <li><a href="/wiki/Einstein%27s_Blackboard" title="Einstein's Blackboard">Einstein's Blackboard</a></li> <li><a href="/wiki/Einstein_Papers_Project" title="Einstein Papers Project">Einstein Papers Project</a></li> <li><a href="/wiki/Einstein_refrigerator" title="Einstein refrigerator">Einstein refrigerator</a></li> <li><a href="/wiki/Einsteinhaus" title="Einsteinhaus">Einsteinhaus</a></li> <li><a href="/wiki/Einsteinium" title="Einsteinium">Einsteinium</a></li> <li><a href="/wiki/Max_Talmey" title="Max Talmey">Max Talmey</a></li> <li><a href="/wiki/Emergency_Committee_of_Atomic_Scientists" title="Emergency Committee of Atomic Scientists">Emergency Committee of Atomic Scientists</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><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> <b><a href="/wiki/Category:Albert_Einstein" title="Category:Albert Einstein">Category</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Relativity" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a class="mw-selflink selflink">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. 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Einstein field equations - Wikipedia