Schwarzschild metric

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<span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singularities_and_black_holes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Singularities_and_black_holes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Singularities and black holes</span> </div> </a> <ul id="toc-Singularities_and_black_holes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_coordinates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Alternative coordinates</span> </div> </a> <ul id="toc-Alternative_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flamm's_paraboloid" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flamm's_paraboloid"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Flamm's paraboloid</span> </div> </a> <ul id="toc-Flamm's_paraboloid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_motion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbital_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Orbital motion</span> </div> </a> <ul id="toc-Orbital_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Symmetries</span> </div> </a> <ul id="toc-Symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curvatures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Curvatures"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Curvatures</span> </div> </a> <ul id="toc-Curvatures-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> </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 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Available in 38 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-38" 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">38 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%B5%D9%81%D9%88%D9%81%D8%A9_%D8%B4%D9%88%D8%A7%D8%B1%D8%B2%D8%B4%D9%8A%D9%84%D8%AF" title="مصفوفة شوارزشيلد – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة شوارزشيلد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D0%B4" title="Метрика на Шварцшилд – Bulgarian" lang="bg" hreflang="bg" data-title="Метрика на Шварцшилд" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A8trica_de_Schwarzschild" title="Mètrica de Schwarzschild – Catalan" lang="ca" hreflang="ca" data-title="Mètrica de Schwarzschild" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Schwarzschildova_metrika" title="Schwarzschildova metrika – Czech" lang="cs" hreflang="cs" data-title="Schwarzschildova metrika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Schwarzschild-metrik" title="Schwarzschild-metrik – Danish" lang="da" hreflang="da" data-title="Schwarzschild-metrik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schwarzschild-Metrik" title="Schwarzschild-Metrik – German" lang="de" hreflang="de" data-title="Schwarzschild-Metrik" 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/Schwarzschildi_meetrika" title="Schwarzschildi meetrika – Estonian" lang="et" hreflang="et" data-title="Schwarzschildi meetrika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CE%A3%CE%B2%CE%AC%CF%81%CF%84%CF%83%CE%B9%CE%BB%CE%BD%CF%84" title="Μετρική Σβάρτσιλντ – Greek" lang="el" hreflang="el" data-title="Μετρική Σβάρτσιλντ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A9trica_de_Schwarzschild" title="Métrica de Schwarzschild – Spanish" lang="es" hreflang="es" data-title="Métrica de Schwarzschild" 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/Schwarzschilden_metrika" title="Schwarzschilden metrika – Basque" lang="eu" hreflang="eu" data-title="Schwarzschilden metrika" 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%AA%D8%B1%DB%8C%DA%A9_%D8%B4%D9%88%D8%A7%D8%B1%D8%AA%D8%B3%E2%80%8C%D8%B4%DB%8C%D9%84%D8%AF" title="متریک شوارتس‌شیلد – Persian" lang="fa" hreflang="fa" data-title="متریک شوارتس‌شیلد" 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/M%C3%A9trique_de_Schwarzschild" title="Métrique de Schwarzschild – French" lang="fr" hreflang="fr" data-title="Métrique de Schwarzschild" 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%8A%88%EB%B0%94%EB%A5%B4%EC%B8%A0%EC%8B%A4%ED%8A%B8_%EA%B3%84%EB%9F%89" title="슈바르츠실트 계량 – Korean" lang="ko" hreflang="ko" data-title="슈바르츠실트 계량" 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/%D5%87%D5%BE%D5%A1%D6%80%D6%81%D5%B7%D5%AB%D5%AC%D5%A4%D5%AB_%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Շվարցշիլդի չափականություն – Armenian" lang="hy" hreflang="hy" data-title="Շվարցշիլդի չափականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Schwarzschildova_metrika" title="Schwarzschildova metrika – Croatian" lang="hr" hreflang="hr" data-title="Schwarzschildova metrika" 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/Metrik_Schwarzschild" title="Metrik Schwarzschild – Indonesian" lang="id" hreflang="id" data-title="Metrik Schwarzschild" 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/Spaziotempo_di_Schwarzschild" title="Spaziotempo di Schwarzschild – Italian" lang="it" hreflang="it" data-title="Spaziotempo di Schwarzschild" 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%98%D7%A8%D7%99%D7%A7%D7%AA_%D7%A9%D7%95%D7%95%D7%A8%D7%A6%D7%A9%D7%99%D7%9C%D7%93" title="מטריקת שוורצשילד – Hebrew" lang="he" hreflang="he" data-title="מטריקת שוורצשילד" 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/Schwarzschild-metrika" title="Schwarzschild-metrika – Hungarian" lang="hu" hreflang="hu" data-title="Schwarzschild-metrika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Schwarzschildmetriek" title="Schwarzschildmetriek – Dutch" lang="nl" hreflang="nl" data-title="Schwarzschildmetriek" 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%B7%E3%83%A5%E3%83%AF%E3%83%AB%E3%83%84%E3%82%B7%E3%83%AB%E3%83%88%E8%A7%A3" title="シュワルツシルト解 – Japanese" lang="ja" hreflang="ja" data-title="シュワルツシルト解" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Schwarzschilds_l%C3%B8sning" title="Schwarzschilds løsning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Schwarzschilds løsning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BC%E0%A8%B5%E0%A8%BE%E0%A8%B0%E0%A8%9C%E0%A8%BC%E0%A8%9A%E0%A8%BF%E0%A8%B2%E0%A8%A1_%E0%A8%AE%E0%A9%80%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A8%95" title="ਸ਼ਵਾਰਜ਼ਚਿਲਡ ਮੀਟ੍ਰਿਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਸ਼ਵਾਰਜ਼ਚਿਲਡ ਮੀਟ੍ਰਿਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metryka_Schwarzschilda" title="Metryka Schwarzschilda – Polish" lang="pl" hreflang="pl" data-title="Metryka Schwarzschilda" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A9trica_de_Schwarzschild" title="Métrica de Schwarzschild – Portuguese" lang="pt" hreflang="pt" data-title="Métrica de Schwarzschild" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Solu%C8%9Bia_Schwarzschild" title="Soluția Schwarzschild – Romanian" lang="ro" hreflang="ro" data-title="Soluția Schwarzschild" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D1%8C%D0%B4%D0%B0" title="Метрика Шварцшильда – Russian" lang="ru" hreflang="ru" data-title="Метрика Шварцшильда" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Schwarzschild_metric" title="Schwarzschild metric – Simple English" lang="en-simple" hreflang="en-simple" data-title="Schwarzschild metric" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Schwarzschildova_metrika" title="Schwarzschildova metrika – Slovak" lang="sk" hreflang="sk" data-title="Schwarzschildova metrika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Schwarzschildova_metrika" title="Schwarzschildova metrika – Slovenian" lang="sl" hreflang="sl" data-title="Schwarzschildova metrika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Schwarzschildova_metrika" title="Schwarzschildova metrika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Schwarzschildova metrika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Schwarzschildmetrik" title="Schwarzschildmetrik – Swedish" lang="sv" hreflang="sv" data-title="Schwarzschildmetrik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Solusyong_Schwarzschild" title="Solusyong Schwarzschild – Tagalog" lang="tl" hreflang="tl" data-title="Solusyong Schwarzschild" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Schwarzschild_metri%C4%9Fi" title="Schwarzschild metriği – Turkish" lang="tr" hreflang="tr" data-title="Schwarzschild metriği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D1%8C%D0%B4%D0%B0" title="Метрика Шварцшильда – Ukrainian" lang="uk" hreflang="uk" data-title="Метрика Шварцшильда" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B4%D9%88%D8%A7%D8%B1%D8%B2%DA%86%D8%A7%D8%A6%D9%84%DA%88_%D9%85%DB%8C%D9%B9%D8%B1%DA%A9" title="شوارزچائلڈ میٹرک – Urdu" lang="ur" hreflang="ur" data-title="شوارزچائلڈ میٹرک" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%AAtric_Schwarzschild" title="Mêtric Schwarzschild – Vietnamese" lang="vi" hreflang="vi" data-title="Mêtric Schwarzschild" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%B2%E7%93%A6%E8%A5%BF%E5%BA%A6%E8%A6%8F" title="史瓦西度規 – Chinese" lang="zh" 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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 mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/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 <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, the <b>Schwarzschild metric</b> (also known as the <b>Schwarzschild solution</b>) is an exact solution to the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> that describes the <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> outside a spherical mass, on the assumption that the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> of the mass, <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of the mass, and universal <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many <a href="/wiki/Star" title="Star">stars</a> and <a href="/wiki/Planet" title="Planet">planets</a>, including Earth and the Sun. It was found by <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> in 1916. </p><p>According to <a href="/wiki/Birkhoff%27s_theorem_(relativity)" title="Birkhoff's theorem (relativity)">Birkhoff's theorem</a>, the Schwarzschild metric is the most general <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">spherically symmetric</a> <a href="/wiki/Vacuum_solution_(general_relativity)" title="Vacuum solution (general relativity)">vacuum solution</a> of the Einstein field equations. A <b>Schwarzschild black hole</b> or <b>static black hole</b> is a <a href="/wiki/Black_hole" title="Black hole">black hole</a> that has neither electric charge nor angular momentum (non-rotating). A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass. </p><p>The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a>, which is situated at the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</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 r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc67a392f63faf10fdb2eee39b5f382dbb5a936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.929ex; height:2.009ex;" alt="{\displaystyle r_{\text{s}}}"></span>), often called the radius of a black hole. The boundary is not a physical surface, and a person who fell through the event horizon (before being torn apart by tidal forces) would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass <span class="texhtml mvar" style="font-style:italic;">M</span>, so in principle (within the theory of general relativity) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation. </p><p>In the vicinity of a Schwarzschild black hole, space curves so much that even light rays are deflected, and very nearby light can be deflected so much that it travels several times around the black hole.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=1" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Derivation_of_the_Schwarzschild_solution" title="Derivation of the Schwarzschild solution">Derivation of the Schwarzschild solution</a></div> <p>The Schwarzschild metric is a <a href="/wiki/Spherically_symmetric_spacetime" title="Spherically symmetric spacetime">spherically symmetric Lorentzian metric</a> (here, with signature convention <span class="texhtml">(+, -, -, -)</span>), defined on (a subset of) <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 \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>O</mi> </mrow> <mo>)</mo> </mrow> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c2d171576bc87d67329a1b4c8954036e49479a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.473ex; height:3.343ex;" alt="{\displaystyle \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}}"></span> 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 E^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab83515bfb94cf95187779f517841793ac61ee5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.848ex; height:2.676ex;" alt="{\displaystyle E^{3}}"></span> is 3 dimensional Euclidean space, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\subset E^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⊂</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\subset E^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392ee3032388b96ebe6329e47cc263fc2fad2050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.522ex; height:2.676ex;" alt="{\displaystyle S^{2}\subset E^{3}}"></span> is the two sphere. The rotation group <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 \mathrm {SO} (3)=\mathrm {SO} (E^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)=\mathrm {SO} (E^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee665f54ba7e55593d865c60c35b484b91451e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.929ex; height:3.176ex;" alt="{\displaystyle \mathrm {SO} (3)=\mathrm {SO} (E^{3})}"></span> acts on the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{3}-O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{3}-O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7e41178a267c4566940c7b98efe15bc431ed51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.462ex; height:2.843ex;" alt="{\displaystyle E^{3}-O}"></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 S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}"></span> factor as rotations around the center <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>, while leaving the first <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> factor unchanged. The Schwarzschild metric is a solution of <a href="/wiki/Einstein%27s_field_equation#Vacuum_field_equations" class="mw-redirect" title="Einstein's field equation">Einstein's field equations</a> in empty space, meaning that it is valid only <i>outside</i> the gravitating body. That is, for a spherical body of radius <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> the solution is valid for <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>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8971c9610113faec012a76ec2d47fa6235e16d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r>R}"></span>. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a088bb7b6b4be9ff6b0fdb33fc1dd53af91e356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r=R}"></span>⁠</span>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> such as the <a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior Schwarzschild metric</a>. </p><p>In <a href="/wiki/Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</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 (t,r,\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,r,\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa61ce2c0a7b5e0dd50e2627d92d4af74500d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.275ex; height:2.843ex;" alt="{\displaystyle (t,r,\theta ,\phi )}"></span> the Schwarzschild metric (or equivalently, the <a href="/wiki/Line_element" title="Line element">line element</a> for <a href="/wiki/Proper_time" title="Proper time">proper time</a>) has the 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 {ds}^{2}=c^{2}\,{d\tau }^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}-r^{2}{d\Omega }^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {ds}^{2}=c^{2}\,{d\tau }^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}-r^{2}{d\Omega }^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdeed9926914c112e63988880392aea456715766" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.983ex; height:5.176ex;" alt="{\displaystyle {ds}^{2}=c^{2}\,{d\tau }^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}-r^{2}{d\Omega }^{2},}"></span> 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 {d\Omega }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\Omega }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e780fff4613e5fe792434de0be17fdfb6d40c1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.948ex; height:2.676ex;" alt="{\displaystyle {d\Omega }^{2}}"></span> is the metric on the two sphere, i.e. <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d\Omega }^{2}=\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\Omega }^{2}=\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19ff7d8c3e3287a1830d7c126f989a3ce707a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.807ex; height:3.343ex;" alt="{\displaystyle {d\Omega }^{2}=\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}"></span>⁠</span>. Furthermore, </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1f64087a9799c1521cfd4cfbf7473fd887070a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.528ex; height:2.676ex;" alt="{\displaystyle d\tau ^{2}}"></span> is positive for timelike curves, in which case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is the <a href="/wiki/Proper_time" title="Proper time">proper time</a> (time measured by a clock moving along the same <a href="/wiki/World_line" title="World line">world line</a> with a <a href="/wiki/Test_particle" title="Test particle">test particle</a>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is, for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fc3420fc0caf8350f7083bf5c5906f12a4bab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:2.176ex;" alt="{\displaystyle r>r_{\text{s}}}"></span>⁠</span>, the time coordinate (measured by a clock located infinitely far from the massive body and stationary with respect to it),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is, for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fc3420fc0caf8350f7083bf5c5906f12a4bab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:2.176ex;" alt="{\displaystyle r>r_{\text{s}}}"></span>⁠</span>, the radial coordinate (measured as the circumference, divided by 2<span class="texhtml mvar" style="font-style:italic;">π</span>, of a sphere centered around the massive body),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is a point on the two sphere <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}"></span>⁠</span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> (angle from north, in units of <a href="/wiki/Radian" title="Radian">radians</a>) defined after arbitrarily choosing a <i>z</i>-axis,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is the <a href="/wiki/Longitude" title="Longitude">longitude</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> (also in radians) around the chosen <i>z</i>-axis, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc67a392f63faf10fdb2eee39b5f382dbb5a936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.929ex; height:2.009ex;" alt="{\displaystyle r_{\text{s}}}"></span> is the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a> of the massive body, a <a href="/wiki/Scale_factor" class="mw-redirect" title="Scale factor">scale factor</a> which is related to its mass <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> by <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}={2GM}/{c^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>G</mi> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}={2GM}/{c^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723c3432f2e0fc079f6b3c1ec19501c13dca29fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.682ex; height:3.176ex;" alt="{\displaystyle r_{\text{s}}={2GM}/{c^{2}}}"></span>⁠</span>, 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>.<sup id="cite_ref-landau_1975_5-0" class="reference"><a href="#cite_note-landau_1975-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <p>The Schwarzschild metric has a singularity for <span class="texhtml"><i>r</i> = 0</span>, which is an intrinsic curvature singularity. It also seems to have a singularity on the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span>. Depending on the point of view, the metric is therefore defined only on the exterior region <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>r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fc3420fc0caf8350f7083bf5c5906f12a4bab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:2.176ex;" alt="{\displaystyle r>r_{\text{s}}}"></span>, only on the interior region <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<r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbf1ebaf3a6b2c0d7e715280ce775340c5e93e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:2.176ex;" alt="{\displaystyle r<r_{\text{s}}}"></span> or their disjoint union. However, the metric is actually non-singular across the event horizon, as one sees in suitable coordinates (see below). For <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\gg r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≫</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\gg r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6543b1c35334854ed5d35e805969bcb1a6c549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.592ex; height:2.176ex;" alt="{\displaystyle r\gg r_{\text{s}}}"></span>⁠</span>, the Schwarzschild metric is asymptotic to the standard Lorentz metric on Minkowski space. For almost all astrophysical objects, the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{\text{s}}}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r_{\text{s}}}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4096bc5ff60bcab972192b08a73e45ade4bd812b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.765ex; height:4.843ex;" alt="{\displaystyle {\frac {r_{\text{s}}}{R}}}"></span> is extremely small. For example, the Schwarzschild radius <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_{\text{s}}^{({\text{Earth}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Earth</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}^{({\text{Earth}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c5f5a44560547a1dd9b26b5971bb0bcf976391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.7ex; height:3.343ex;" alt="{\displaystyle r_{\text{s}}^{({\text{Earth}})}}"></span> of the Earth is roughly <span class="nowrap"><span data-sort-value="6997890000000000000♠"></span>8.9 mm</span>, while the Sun, which is <span class="nowrap"><span data-sort-value="7005330000000000000♠"></span>3.3<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>5</sup></span> times as massive<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> has a Schwarzschild radius <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_{\text{s}}^{({\text{Sun}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sun</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}^{({\text{Sun}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00581f533a8c0131aae3949ec03f45ceda452113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:3.343ex;" alt="{\displaystyle r_{\text{s}}^{({\text{Sun}})}}"></span> of approximately 3.0 km. The ratio becomes large only in close proximity to <a href="/wiki/Black_hole" title="Black hole">black holes</a> and other ultra-dense objects such as <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a>. </p><p>The radial coordinate turns out to have physical significance as the "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line".<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><p>The Schwarzschild solution is analogous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schwarzschild solution is named in honour of <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a>, who found the exact solution in 1915 and published it in January 1916,<sup id="cite_ref-Schwarzschild1916_9-0" class="reference"><a href="#cite_note-Schwarzschild1916-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> a little more than a month after the publication of Einstein's theory of general relativity. It was the first <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solution</a> of the Einstein field equations other than the trivial <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">flat space solution</a>. Schwarzschild died shortly after his paper was published, as a result of a disease (thought to be <a href="/wiki/Pemphigus" title="Pemphigus">pemphigus</a>) he developed while serving in the <a href="/wiki/Reichswehr" title="Reichswehr">German army</a> during <a href="/wiki/World_War_I" title="World War I">World War I</a>.<sup id="cite_ref-MacTutorBio_10-0" class="reference"><a href="#cite_note-MacTutorBio-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Johannes_Droste" title="Johannes Droste">Johannes Droste</a> in 1916<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system. In this paper he also introduced what is now known as the Schwarzschild radial coordinate (<span class="texhtml mvar" style="font-style:italic;">r</span> in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius. </p><p>A more complete analysis of the singularity structure was given by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a><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> in the following year, identifying the singularities both at <span class="texhtml"><i>r</i> = 0</span> and <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span>. Although there was general consensus that the singularity at <span class="texhtml"><i>r</i> = 0</span> was a 'genuine' physical singularity, the nature of the singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> remained unclear.<sup id="cite_ref-earman_14-0" class="reference"><a href="#cite_note-earman-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>In 1921, <a href="/wiki/Paul_Painlev%C3%A9" title="Paul Painlevé">Paul Painlevé</a> and in 1922 <a href="/wiki/Allvar_Gullstrand" title="Allvar Gullstrand">Allvar Gullstrand</a> independently produced a metric, a spherically symmetric solution of Einstein's equations, which we now know is coordinate transformation of the Schwarzschild metric, <a href="/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" title="Gullstrand–Painlevé coordinates">Gullstrand–Painlevé coordinates</a>, in which there was no singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span>. They, however, did not recognize that their solutions were just coordinate transforms, and in fact used their solution to argue that Einstein's theory was wrong. In 1924 <a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Arthur Eddington</a> produced the first coordinate transformation (<a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a>) that showed that the singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> was a coordinate artifact, although he also seems to have been unaware of the significance of this discovery. Later, in 1932, <a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Georges Lemaître</a> gave a different coordinate transformation (<a href="/wiki/Lema%C3%AEtre_coordinates" title="Lemaître coordinates">Lemaître coordinates</a>) to the same effect and was the first to recognize that this implied that the singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> was not physical. In 1939 <a href="/wiki/Howard_Percy_Robertson" class="mw-redirect" title="Howard Percy Robertson">Howard Robertson</a> showed that a free falling observer descending in the Schwarzschild metric would cross the <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> singularity in a finite amount of <a href="/wiki/Proper_time" title="Proper time">proper time</a> even though this would take an infinite amount of time in terms of coordinate time <span class="texhtml mvar" style="font-style:italic;">t</span>.<sup id="cite_ref-earman_14-1" class="reference"><a href="#cite_note-earman-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>In 1950, <a href="/wiki/John_Lighton_Synge" title="John Lighton Synge">John Synge</a> produced a paper<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> that showed the maximal <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic extension</a> of the Schwarzschild metric, again showing that the singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> was a coordinate artifact and that it represented two horizons. A similar result was later rediscovered by <a href="/wiki/George_Szekeres" title="George Szekeres">George Szekeres</a>,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and independently <a href="/wiki/Martin_Kruskal" class="mw-redirect" title="Martin Kruskal">Martin Kruskal</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The new coordinates nowadays known as <a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a> were much simpler than Synge's but both provided a single set of coordinates that covered the entire spacetime. However, perhaps due to the obscurity of the journals in which the papers of Lemaître and Synge were published their conclusions went unnoticed, with many of the major players in the field including Einstein believing that the singularity at the Schwarzschild radius was physical.<sup id="cite_ref-earman_14-2" class="reference"><a href="#cite_note-earman-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Synge's later derivation of the Kruskal–Szekeres metric solution,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> which was motivated by a desire to avoid "using 'bad' [Schwarzschild] coordinates to obtain 'good' [Kruskal–Szekeres] coordinates", has been generally under-appreciated in the literature, but was adopted by Chandrasekhar in his black hole monograph.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Real progress was made in the 1960s when the mathematically rigorous formulation cast in terms of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> entered the field of general relativity, allowing more exact definitions of what it means for a <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifold</a> to be singular. This led to definitive identification of the <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> singularity in the Schwarzschild metric as an <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a>, i.e., a hypersurface in spacetime that can be crossed in only one direction.<sup id="cite_ref-earman_14-3" class="reference"><a href="#cite_note-earman-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Singularities_and_black_holes">Singularities and black holes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=3" title="Edit section: Singularities and black holes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schwarzschild solution appears to have <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a> at <span class="texhtml"><i>r</i> = 0</span> and <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span>; some of the metric components "blow up" (entail division by zero or multiplication by infinity) at these radii. Since the Schwarzschild metric is expected to be valid only for those radii larger than the radius <span class="texhtml mvar" style="font-style:italic;">R</span> of the gravitating body, there is no problem as long as <span class="texhtml"><i>R</i> > <i>r</i><sub>s</sub></span>. For ordinary stars and planets this is always the case. For example, the radius of the <a href="/wiki/Sun" title="Sun">Sun</a> is approximately <span class="nowrap"><span data-sort-value="7008700000000000000♠"></span>700<span style="margin-left:.25em;">000</span> km</span>, while its Schwarzschild radius is only <span class="nowrap"><span data-sort-value="7003300000000000000♠"></span>3 km</span>. </p><p>The singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> divides the Schwarzschild coordinates in two <a href="/wiki/Connectedness" title="Connectedness">disconnected</a> <a href="/wiki/Coordinate_patch" class="mw-redirect" title="Coordinate patch">patches</a>. The <i>exterior Schwarzschild solution</i> with <span class="texhtml"><i>r</i> > <i>r</i><sub>s</sub></span> is the one that is related to the gravitational fields of stars and planets. The <i>interior Schwarzschild solution</i> with <span class="texhtml">0 ≤ <i>r</i> < <i>r</i><sub>s</sub></span>, which contains the singularity at <span class="texhtml"><i>r</i> = 0</span>, is completely separated from the outer patch by the singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span>. The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> is an illusion however; it is an instance of what is called a <i><a href="/wiki/Coordinate_singularity" title="Coordinate singularity">coordinate singularity</a></i>. As the name implies, the singularity arises from a bad choice of coordinates or <a href="/wiki/Coordinate_conditions" title="Coordinate conditions">coordinate conditions</a>. When changing to a different coordinate system (for example <a href="/wiki/Lema%C3%AEtre_coordinates" title="Lemaître coordinates">Lemaître coordinates</a>, <a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a>, <a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a>, Novikov coordinates, or <a href="/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" title="Gullstrand–Painlevé coordinates">Gullstrand–Painlevé coordinates</a>) the metric becomes regular at <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> and can extend the external patch to values of <span class="texhtml mvar" style="font-style:italic;">r</span> smaller than <span class="texhtml"><i>r</i><sub>s</sub></span>. Using a different coordinate transformation one can then relate the extended external patch to the inner patch.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The case <span class="texhtml"><i>r</i> = 0</span> is different, however. If one asks that the solution be valid for all <span class="texhtml mvar" style="font-style:italic;">r</span> one runs into a true physical singularity, or <i><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">gravitational singularity</a></i>, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the <a href="/wiki/Kretschmann_invariant" class="mw-redirect" title="Kretschmann invariant">Kretschmann invariant</a>, which is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }={\frac {12r_{\mathrm {s} }^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>48</mn> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }={\frac {12r_{\mathrm {s} }^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cca4ce179c4155fb44a4483525f8a0c3a932633a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.338ex; height:6.009ex;" alt="{\displaystyle R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }={\frac {12r_{\mathrm {s} }^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}\,.}"></span></dd></dl> <p>At <span class="texhtml"><i>r</i> = 0</span> the curvature becomes infinite, indicating the presence of a singularity. At this point the metric cannot be extended in a smooth manner (the Kretschmann invariant involves second derivatives of the metric), spacetime itself is then no longer well-defined. Furthermore, Sbierski<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> showed the metric cannot be extended even in a continuous manner. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case. </p><p>The Schwarzschild solution, taken to be valid for all <span class="texhtml"><i>r</i> > 0</span>, is called a <b>Schwarzschild black hole</b>. It is a perfectly valid solution of the Einstein field equations, although (like other black holes) it has rather bizarre properties. For <span class="texhtml"><i>r</i> < <i>r</i><sub>s</sub></span> the Schwarzschild radial coordinate <span class="texhtml mvar" style="font-style:italic;">r</span> becomes <a href="/wiki/Spacetime#Time-like_interval" title="Spacetime">timelike</a> and the time coordinate <span class="texhtml mvar" style="font-style:italic;">t</span> becomes <a href="/wiki/Spacetime#Space-like_interval" title="Spacetime">spacelike</a>.<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> A curve at constant <span class="texhtml mvar" style="font-style:italic;">r</span> is no longer a possible <a href="/wiki/Worldline" class="mw-redirect" title="Worldline">worldline</a> of a particle or observer, not even if a force is exerted to try to keep it there; this occurs because spacetime has been curved so much that the direction of cause and effect (the particle's future <a href="/wiki/Light_cone" title="Light cone">light cone</a>) points into the singularity.<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. (November 2010)">citation needed</span></a></i>]</sup> The surface <span class="texhtml"><i>r</i> = <i>r</i><sub>s</sub></span> demarcates what is called the <i><a href="/wiki/Event_horizon" title="Event horizon">event horizon</a></i> of the black hole. It represents the point past which light can no longer escape the gravitational field. Any physical object whose radius <span class="texhtml mvar" style="font-style:italic;">R</span> becomes less than or equal to the Schwarzschild radius has undergone <a href="/wiki/Gravitational_collapse" title="Gravitational collapse">gravitational collapse</a> and become a black hole. </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_coordinates">Alternative coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=4" title="Edit section: Alternative coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schwarzschild solution can be expressed in a range of different choices of coordinates besides the Schwarzschild coordinates used above. Different choices tend to highlight different features of the solution. The table below shows some popular choices. </p> <table class="wikitable" style="margin: 1em auto 1em auto;"> <caption>Alternative coordinates<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> </caption> <tbody><tr> <th>Coordinates </th> <th>Line element </th> <th>Notes </th> <th>Features </th></tr> <tr> <td><a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a><br />(ingoing) </td> <td><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 \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dv^{2}-2\,dv\,dr-r^{2}\,g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dv^{2}-2\,dv\,dr-r^{2}\,g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89b34c580ab45c19757fad30f184663ff660995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.959ex; height:4.843ex;" alt="{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dv^{2}-2\,dv\,dr-r^{2}\,g_{\Omega }}"></span> </td> <td> </td> <td>regular at future horizon<br /> past horizon is at <span class="texhtml"><i>v</i> = −∞</span> </td></tr> <tr> <td><a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a><br />(outgoing) </td> <td><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 \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,du^{2}+2\,du\,dr-r^{2}g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,du^{2}+2\,du\,dr-r^{2}g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a4abf757ddacfdebe4a7e69da007c0169b7a5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.976ex; height:4.843ex;" alt="{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,du^{2}+2\,du\,dr-r^{2}g_{\Omega }}"></span> </td> <td> </td> <td>regular at past horizon<br />extends across past horizon<br />future horizon at <span class="texhtml"><i>u</i> = ∞</span> </td></tr> <tr> <td><a href="/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" title="Gullstrand–Painlevé coordinates">Gullstrand–Painlevé coordinates</a> </td> <td><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 \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dT^{2}\pm 2{\sqrt {\frac {r_{\mathrm {s} }}{r}}}\,dT\,dr-dr^{2}-r^{2}\,g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>±</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>−</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dT^{2}\pm 2{\sqrt {\frac {r_{\mathrm {s} }}{r}}}\,dT\,dr-dr^{2}-r^{2}\,g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3097599383817428e46e69f9d089e22624d1f6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.307ex; height:6.343ex;" alt="{\displaystyle \left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dT^{2}\pm 2{\sqrt {\frac {r_{\mathrm {s} }}{r}}}\,dT\,dr-dr^{2}-r^{2}\,g_{\Omega }}"></span> </td> <td> </td> <td>regular at past and future horizons </td></tr> <tr> <td><a href="/wiki/Isotropic_coordinates" title="Isotropic coordinates">Isotropic coordinates</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(1-{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}{\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}}\,{dt}^{2}-\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{4}\,\left(dx^{2}+dy^{2}+dz^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow> <mn>4</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow> <mn>4</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow> <mn>4</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(1-{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}{\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}}\,{dt}^{2}-\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{4}\,\left(dx^{2}+dy^{2}+dz^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f1c0f22854d0576c1b6648ef69f93939f76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:48.482ex; height:11.176ex;" alt="{\displaystyle {\frac {\left(1-{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}{\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}}\,{dt}^{2}-\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{4}\,\left(dx^{2}+dy^{2}+dz^{2}\right)}"></span> </td> <td><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={\sqrt {x^{2}+y^{2}+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\sqrt {x^{2}+y^{2}+z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5252401065014a7a4fe74a77e1e1550bf9583e47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.61ex; height:4.843ex;" alt="{\displaystyle R={\sqrt {x^{2}+y^{2}+z^{2}}}}"></span><sup id="cite_ref-eddntn1923_24-0" class="reference"><a href="#cite_note-eddntn1923-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><br /> Valid only outside the event horizon: <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>r_{\text{s}}/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>r_{\text{s}}/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f260394d42c6229e50cc7d9a46dfce6e0d62fc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.116ex; height:2.843ex;" alt="{\displaystyle R>r_{\text{s}}/4}"></span> </td> <td>isotropic lightcones on constant time slices </td></tr> <tr> <td><a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4r_{\mathrm {s} }^{3}}{r}}e^{-{\frac {r}{r_{\mathrm {s} }}}}\,\left(dT^{2}-dR^{2}\right)-r^{2}\,g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mi>r</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4r_{\mathrm {s} }^{3}}{r}}e^{-{\frac {r}{r_{\mathrm {s} }}}}\,\left(dT^{2}-dR^{2}\right)-r^{2}\,g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a04e5a783c6923429161a400b63d5b3e9faa47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.42ex; height:5.676ex;" alt="{\displaystyle {\frac {4r_{\mathrm {s} }^{3}}{r}}e^{-{\frac {r}{r_{\mathrm {s} }}}}\,\left(dT^{2}-dR^{2}\right)-r^{2}\,g_{\Omega }}"></span> </td> <td><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^{2}-R^{2}=\left(1-{\frac {r}{r_{\mathrm {s} }}}\right)e^{\frac {r}{r_{\mathrm {s} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}-R^{2}=\left(1-{\frac {r}{r_{\mathrm {s} }}}\right)e^{\frac {r}{r_{\mathrm {s} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f93bff8eb47fed6372a1002f2bc63e547c6b3fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.521ex; height:6.176ex;" alt="{\displaystyle T^{2}-R^{2}=\left(1-{\frac {r}{r_{\mathrm {s} }}}\right)e^{\frac {r}{r_{\mathrm {s} }}}}"></span> </td> <td>regular at horizon; maximally extends to full spacetime </td></tr> <tr> <td><a href="/wiki/Lema%C3%AEtre_coordinates" title="Lemaître coordinates">Lemaître coordinates</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dT^{2}-{\frac {r_{\mathrm {s} }}{r}}\,dR^{2}-r^{2}\,g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dT^{2}-{\frac {r_{\mathrm {s} }}{r}}\,dR^{2}-r^{2}\,g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa817e363027082d71e05d203ba05bb002df2d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.875ex; height:4.676ex;" alt="{\displaystyle dT^{2}-{\frac {r_{\mathrm {s} }}{r}}\,dR^{2}-r^{2}\,g_{\Omega }}"></span> </td> <td><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=\left({\tfrac {3}{2}}(R\pm T)\right)^{\frac {2}{3}}r_{\mathrm {s} }^{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>±</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\left({\tfrac {3}{2}}(R\pm T)\right)^{\frac {2}{3}}r_{\mathrm {s} }^{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a976e49f8e568bf311002bc2594854c8298b6b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.151ex; height:5.843ex;" alt="{\displaystyle r=\left({\tfrac {3}{2}}(R\pm T)\right)^{\frac {2}{3}}r_{\mathrm {s} }^{\frac {1}{3}}}"></span> </td> <td>regular at either past or future horizon </td></tr> <tr> <td><a href="/wiki/Harmonic_coordinates" title="Harmonic coordinates">Harmonic coordinates</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\rho -r_{\mathrm {s} }/2}{\rho +r_{\mathrm {s} }/2}}dt^{2}-{\frac {\rho +r_{\mathrm {s} }/2}{\rho -r_{\mathrm {s} }/2}}d\rho ^{2}-(\rho +r_{\mathrm {s} }/2)^{2}g_{\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <mi>ρ</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <mi>ρ</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\rho -r_{\mathrm {s} }/2}{\rho +r_{\mathrm {s} }/2}}dt^{2}-{\frac {\rho +r_{\mathrm {s} }/2}{\rho -r_{\mathrm {s} }/2}}d\rho ^{2}-(\rho +r_{\mathrm {s} }/2)^{2}g_{\Omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bb81ddfb3a6ec57a50ce208a8afda38c1b117e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.214ex; height:6.509ex;" alt="{\displaystyle {\frac {\rho -r_{\mathrm {s} }/2}{\rho +r_{\mathrm {s} }/2}}dt^{2}-{\frac {\rho +r_{\mathrm {s} }/2}{\rho -r_{\mathrm {s} }/2}}d\rho ^{2}-(\rho +r_{\mathrm {s} }/2)^{2}g_{\Omega }}"></span> </td> <td><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 \rho =r-r_{\mathrm {s} }/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =r-r_{\mathrm {s} }/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd444d1b41f818d7852de9238076aeb231576cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.443ex; height:2.843ex;" alt="{\displaystyle \rho =r-r_{\mathrm {s} }/2}"></span> </td> <td> </td></tr></tbody></table> <p>In table above, some shorthand has been introduced for brevity. The speed of light <span class="texhtml mvar" style="font-style:italic;">c</span> <a href="/wiki/Natural_units" title="Natural units">has been set to one</a>. The notation </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_{\Omega }=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\Omega }=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd90f9f4a9548e15e659f4ed23e509f265d6d56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.392ex; height:3.176ex;" alt="{\displaystyle g_{\Omega }=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}}"></span></dd></dl> <p>is used for the metric of a unit radius 2-dimensional sphere. Moreover, in each entry <span class="texhtml"><i>R</i></span> and <span class="texhtml"><i>T</i></span> denote alternative choices of radial and time coordinate for the particular coordinates. Note, the <span class="texhtml"><i>R</i></span> or <span class="texhtml"><i>T</i></span> may vary from entry to entry. </p><p>The Kruskal–Szekeres coordinates have the form to which the <a href="/wiki/Belinski%E2%80%93Zakharov_transform" title="Belinski–Zakharov transform">Belinski–Zakharov transform</a> can be applied. This implies that the Schwarzschild black hole is a form of <a href="/wiki/Gravitational_soliton" title="Gravitational soliton">gravitational soliton</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Flamm's_paraboloid"><span id="Flamm.27s_paraboloid"></span>Flamm's paraboloid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=5" title="Edit section: Flamm's paraboloid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Flamm.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Flamm.jpg/220px-Flamm.jpg" decoding="async" width="220" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Flamm.jpg/330px-Flamm.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Flamm.jpg/440px-Flamm.jpg 2x" data-file-width="830" data-file-height="445" /></a><figcaption>A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravity well</a>.</figcaption></figure> <p>The spatial curvature of the Schwarzschild solution for <span class="texhtml"><i>r</i> > <i>r</i><sub>s</sub></span> can be visualized as the graphic shows. Consider a constant time equatorial slice <span class="texhtml"> <i>H</i></span> through the Schwarzschild solution by fixing <span class="texhtml"><i>θ</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, <span class="texhtml mvar" style="font-style:italic;">t</span> = constant, and letting the remaining Schwarzschild coordinates <span class="texhtml">(<i>r</i>, <i>φ</i>)</span> vary. Imagine now that there is an additional Euclidean dimension <span class="texhtml mvar" style="font-style:italic;">w</span>, which has no physical reality (it is not part of spacetime). Then replace the <span class="texhtml">(<i>r</i>, <i>φ</i>)</span> plane with a surface dimpled in the <span class="texhtml mvar" style="font-style:italic;">w</span> direction according to the equation (<i>Flamm's paraboloid</i>) </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 w=2{\sqrt {r_{\mathrm {s} }\left(r-r_{\mathrm {s} }\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=2{\sqrt {r_{\mathrm {s} }\left(r-r_{\mathrm {s} }\right)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1767eba7270e0fd70f1bc818995a35563625af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.839ex; height:4.843ex;" alt="{\displaystyle w=2{\sqrt {r_{\mathrm {s} }\left(r-r_{\mathrm {s} }\right)}}.}"></span></dd></dl> <p>This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of <i>w</i> above, </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 dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}={\frac {dr^{2}}{1-{\frac {r_{\mathrm {s} }}{r}}}}+r^{2}\,d\varphi ^{2}=-ds^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}={\frac {dr^{2}}{1-{\frac {r_{\mathrm {s} }}{r}}}}+r^{2}\,d\varphi ^{2}=-ds^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baae470604b4106cc1b685aa3ff70ebf3a9d8a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.807ex; height:7.009ex;" alt="{\displaystyle dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}={\frac {dr^{2}}{1-{\frac {r_{\mathrm {s} }}{r}}}}+r^{2}\,d\varphi ^{2}=-ds^{2}}"></span></dd></dl> <p>Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. It should not, however, be confused with a <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravity well</a>. No ordinary (massive or massless) particle can have a worldline lying on the paraboloid, since all distances on it are <a href="/wiki/Spacelike" class="mw-redirect" title="Spacelike">spacelike</a> (this is a cross-section at one moment of time, so any particle moving on it would have an infinite <a href="/wiki/Velocity" title="Velocity">velocity</a>). A <a href="/wiki/Tachyon" title="Tachyon">tachyon</a> could have a spacelike worldline that lies entirely on a single paraboloid. However, even in that case its <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> path is not the trajectory one gets through a "rubber sheet" analogy of gravitational well: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's geodesic path still curves toward the central mass, not away. See the <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravity well</a> article for more information. </p><p>Flamm's paraboloid may be derived as follows. The Euclidean metric in the <a href="/wiki/Cylindrical_coordinates" class="mw-redirect" title="Cylindrical coordinates">cylindrical coordinates</a> <span class="texhtml">(<i>r</i>, <i>φ</i>, <i>w</i>)</span> is written </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 -ds^{2}=dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -ds^{2}=dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2418ffb4b3887430bf7861d29f56f13eea07f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.515ex; height:3.176ex;" alt="{\displaystyle -ds^{2}=dw^{2}+dr^{2}+r^{2}\,d\varphi ^{2}\,.}"></span></dd></dl> <p>Letting the surface be described by the function <span class="texhtml"><i>w</i> = <i>w</i>(<i>r</i>)</span>, the Euclidean metric can 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 -ds^{2}=\left(1+\left({\frac {dw}{dr}}\right)^{2}\right)\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>w</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -ds^{2}=\left(1+\left({\frac {dw}{dr}}\right)^{2}\right)\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4737d65d20fda1b9906d3eff1a14c33e9689f993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.39ex; height:7.509ex;" alt="{\displaystyle -ds^{2}=\left(1+\left({\frac {dw}{dr}}\right)^{2}\right)\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}"></span></dd></dl> <p>Comparing this with the Schwarzschild metric in the equatorial plane (<span class="texhtml"><i>θ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>) at a fixed time (<span class="texhtml mvar" style="font-style:italic;">t</span> = constant, <span class="texhtml"><i>dt</i> = 0</span>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -ds^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -ds^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21630e7774856f745f82cd3b4447b4b32347efc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.004ex; height:5.176ex;" alt="{\displaystyle -ds^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}\,d\varphi ^{2}\,,}"></span></dd></dl> <p>yields an integral expression for <span class="texhtml"><i>w</i>(<i>r</i>)</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(r)=\int {\frac {dr}{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}}=2r_{\mathrm {s} }{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}+{\mbox{constant}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>constant</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(r)=\int {\frac {dr}{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}}=2r_{\mathrm {s} }{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}+{\mbox{constant}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/439f8e2e7c4a8f482489f179de809075d46142e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:46.421ex; height:8.176ex;" alt="{\displaystyle w(r)=\int {\frac {dr}{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}}=2r_{\mathrm {s} }{\sqrt {{\frac {r}{r_{\mathrm {s} }}}-1}}+{\mbox{constant}}}"></span></dd></dl> <p>whose solution is Flamm's paraboloid. </p> <div class="mw-heading mw-heading2"><h2 id="Orbital_motion">Orbital motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=6" title="Edit section: Orbital motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Newton.vs.Schwarzschild.thumbnail.250px.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Newton.vs.Schwarzschild.thumbnail.250px.png/220px-Newton.vs.Schwarzschild.thumbnail.250px.png" decoding="async" width="220" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7d/Newton.vs.Schwarzschild.thumbnail.250px.png 1.5x" data-file-width="250" data-file-height="120" /></a><figcaption>Comparison between the orbit of a test particle in Newtonian (left) and Schwarzschild (right) spacetime; note the <a href="/wiki/Apsidal_precession" title="Apsidal precession">apsidal precession</a> on the right.</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Schwarzschild_geodesics" title="Schwarzschild geodesics">Schwarzschild geodesics</a></div> <p>A particle orbiting in the Schwarzschild metric can have a stable circular orbit with <span class="texhtml"><i>r</i> > 3<i>r</i><sub>s</sub></span>. Circular orbits with <span class="texhtml mvar" style="font-style:italic;">r</span> between <span class="texhtml">1.5<i>r</i><sub>s</sub></span> and <span class="texhtml">3<i>r</i><sub>s</sub></span> are unstable, and no circular orbits exist for <span class="texhtml"><i>r</i> < 1.5<i>r</i><sub>s</sub></span>. The circular orbit of minimum radius <span class="texhtml">1.5<i>r</i><sub>s</sub></span> corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of <span class="texhtml mvar" style="font-style:italic;">r</span> between <span class="texhtml"><i>r</i><sub>s</sub></span> and <span class="texhtml">1.5<i>r</i><sub>s</sub></span>, but only if some force acts to keep it there. </p><p>Noncircular orbits, such as <a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a>'s, dwell longer at small radii than would be expected in <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newtonian gravity</a>. This can be seen as a less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever. Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as knife-edge orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward. </p> <div class="mw-heading mw-heading2"><h2 id="Symmetries">Symmetries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=7" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The isometry group of the Schwarzchild metric is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathrm {O} (3)\times \{\pm 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathrm {O} (3)\times \{\pm 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa88db60644b0686a4dd8b81f375f8012d2d6196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.434ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \times \mathrm {O} (3)\times \{\pm 1\}}"></span>⁠</span>, 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 \mathrm {O} (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e8b4adbc1c697bfe6aac6d78ecf14e1aae8efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (3)}"></span> is the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> of rotations and reflections in three dimensions, <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> comprises the time translations, 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 \{\pm 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\pm 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeb8470fc14dbbc80503092df4c0c36d56e66ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.296ex; height:2.843ex;" alt="{\displaystyle \{\pm 1\}}"></span> is the group generated by time reversal. </p><p>This is thus the subgroup of the ten-dimensional <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> which takes the time axis (trajectory of the star) to itself. It omits the spatial translations (three dimensions) and boosts (three dimensions). It retains the time translations (one dimension) and rotations (three dimensions). Thus it has four dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted. </p> <div class="mw-heading mw-heading2"><h2 id="Curvatures">Curvatures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarzschild_metric&action=edit&section=8" title="Edit section: Curvatures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Ricci curvature scalar and the <a href="/wiki/Ricci_curvature_tensor" class="mw-redirect" title="Ricci curvature tensor">Ricci curvature tensor</a> are both zero. Non-zero components of the <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a> are given by<sup id="cite_ref-ReferenceA_25-0" class="reference"><a href="#cite_note-ReferenceA-25"><span class="cite-bracket">[</span>25<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 -R^{t}{}_{rtr}=2R^{\theta }{}_{r\theta r}=2R^{\phi }{}_{r\phi r}={\frac {r_{\text{s}}}{r^{2}(r_{\text{s}}-r)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>θ</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>ϕ</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -R^{t}{}_{rtr}=2R^{\theta }{}_{r\theta r}=2R^{\phi }{}_{r\phi r}={\frac {r_{\text{s}}}{r^{2}(r_{\text{s}}-r)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81243401ab24cfe34bbedb36cde53aeea9b29d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.465ex; height:5.509ex;" alt="{\displaystyle -R^{t}{}_{rtr}=2R^{\theta }{}_{r\theta r}=2R^{\phi }{}_{r\phi r}={\frac {r_{\text{s}}}{r^{2}(r_{\text{s}}-r)}},}"></span></dd> <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 2R^{t}{}_{\theta t\theta }=2R^{r}{}_{\theta r\theta }=-R^{\phi }{}_{\theta \phi \theta }=-{\frac {r_{\text{s}}}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mi>t</mi> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mi>r</mi> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mi>ϕ</mi> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2R^{t}{}_{\theta t\theta }=2R^{r}{}_{\theta r\theta }=-R^{\phi }{}_{\theta \phi \theta }=-{\frac {r_{\text{s}}}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136110d2342c468100956339db96cd05bc92aa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.59ex; height:4.676ex;" alt="{\displaystyle 2R^{t}{}_{\theta t\theta }=2R^{r}{}_{\theta r\theta }=-R^{\phi }{}_{\theta \phi \theta }=-{\frac {r_{\text{s}}}{r}},}"></span></dd> <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 2R^{t}{}_{\phi t\phi }=2R^{r}{}_{\phi r\phi }=-R^{\theta }{}_{\phi \theta \phi }=-{\frac {r_{\text{s}}\sin ^{2}(\theta )}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>t</mi> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>r</mi> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>θ</mi> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2R^{t}{}_{\phi t\phi }=2R^{r}{}_{\phi r\phi }=-R^{\theta }{}_{\phi \theta \phi }=-{\frac {r_{\text{s}}\sin ^{2}(\theta )}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/467468d24bfb0d1a16c036535b4f2e0320f0bae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.621ex; height:5.843ex;" alt="{\displaystyle 2R^{t}{}_{\phi t\phi }=2R^{r}{}_{\phi r\phi }=-R^{\theta }{}_{\phi \theta \phi }=-{\frac {r_{\text{s}}\sin ^{2}(\theta )}{r}},}"></span></dd> <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^{r}{}_{trt}=-2R^{\theta }{}_{t\theta t}=-2R^{\phi }{}_{t\phi t}=c^{2}{\frac {r_{\text{s}}(r_{\text{s}}-r)}{r^{4}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>r</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>θ</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>ϕ</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{r}{}_{trt}=-2R^{\theta }{}_{t\theta t}=-2R^{\phi }{}_{t\phi t}=c^{2}{\frac {r_{\text{s}}(r_{\text{s}}-r)}{r^{4}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d568bb7c1460230e97895855730b6cdd57cbc1b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:43.569ex; height:6.009ex;" alt="{\displaystyle R^{r}{}_{trt}=-2R^{\theta }{}_{t\theta t}=-2R^{\phi }{}_{t\phi t}=c^{2}{\frac {r_{\text{s}}(r_{\text{s}}-r)}{r^{4}}},}"></span></dd></dl> <p>from which one can see that <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^{\gamma }{}_{\alpha \gamma \beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>γ</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\gamma }{}_{\alpha \gamma \beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f9e404fc449c5b5afd26b94fdd71b3a1101bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.268ex; height:3.009ex;" alt="{\displaystyle R^{\gamma }{}_{\alpha \gamma \beta }=0}"></span>. Six of these formulas are Eq. 5.13 in Carroll<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> and imply the other 6 by <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^{\alpha }{}_{\beta \gamma \delta }=g^{\alpha \kappa }g_{\beta \lambda }R^{\lambda }{}_{\kappa \delta \gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>κ</mi> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>λ</mi> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mi>δ</mi> <mi>γ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\alpha }{}_{\beta \gamma \delta }=g^{\alpha \kappa }g_{\beta \lambda }R^{\lambda }{}_{\kappa \delta \gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45049fd2a241de6f934d5babe6fac31d573c7218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.313ex; height:3.343ex;" alt="{\displaystyle R^{\alpha }{}_{\beta \gamma \delta }=g^{\alpha \kappa }g_{\beta \lambda }R^{\lambda }{}_{\kappa \delta \gamma }}"></span>. Components which are obtainable by other symmetries of the Riemann tensor are not displayed. </p><p>To understand the physical meaning of these quantities, it is useful to express the curvature tensor in an orthonormal basis. In an orthonormal basis of an observer the non-zero components in <a href="/wiki/Geometric_units" class="mw-redirect" title="Geometric units">geometric units</a> are<sup id="cite_ref-ReferenceA_25-1" class="reference"><a href="#cite_note-ReferenceA-25"><span class="cite-bracket">[</span>25<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 R^{\hat {r}}{}_{{\hat {t}}{\hat {r}}{\hat {t}}}=-R^{\hat {\theta }}{}_{{\hat {\phi }}{\hat {\theta }}{\hat {\phi }}}=-{\frac {r_{\text{s}}}{r^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\hat {r}}{}_{{\hat {t}}{\hat {r}}{\hat {t}}}=-R^{\hat {\theta }}{}_{{\hat {\phi }}{\hat {\theta }}{\hat {\phi }}}=-{\frac {r_{\text{s}}}{r^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d29a2b8ec8aaf14324f7907e1aa3f7542f8c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.009ex; height:5.009ex;" alt="{\displaystyle R^{\hat {r}}{}_{{\hat {t}}{\hat {r}}{\hat {t}}}=-R^{\hat {\theta }}{}_{{\hat {\phi }}{\hat {\theta }}{\hat {\phi }}}=-{\frac {r_{\text{s}}}{r^{3}}},}"></span></dd> <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^{\hat {\theta }}{}_{{\hat {t}}{\hat {\theta }}{\hat {t}}}=R^{\hat {\phi }}{}_{{\hat {t}}{\hat {\phi }}{\hat {t}}}=-R^{\hat {r}}{}_{{\hat {\theta }}{\hat {r}}{\hat {\theta }}}=-R^{\hat {r}}{}_{{\hat {\phi }}{\hat {r}}{\hat {\phi }}}={\frac {r_{\text{s}}}{2r^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\hat {\theta }}{}_{{\hat {t}}{\hat {\theta }}{\hat {t}}}=R^{\hat {\phi }}{}_{{\hat {t}}{\hat {\phi }}{\hat {t}}}=-R^{\hat {r}}{}_{{\hat {\theta }}{\hat {r}}{\hat {\theta }}}=-R^{\hat {r}}{}_{{\hat {\phi }}{\hat {r}}{\hat {\phi }}}={\frac {r_{\text{s}}}{2r^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3374027c12deaab9980944e4a0a3e32e5cc6bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:45.978ex; height:5.009ex;" alt="{\displaystyle R^{\hat {\theta }}{}_{{\hat {t}}{\hat {\theta }}{\hat {t}}}=R^{\hat {\phi }}{}_{{\hat {t}}{\hat {\phi }}{\hat {t}}}=-R^{\hat {r}}{}_{{\hat {\theta }}{\hat {r}}{\hat {\theta }}}=-R^{\hat {r}}{}_{{\hat {\phi }}{\hat {r}}{\hat {\phi }}}={\frac {r_{\text{s}}}{2r^{3}}}.}"></span></dd></dl> <p>Again, components which are obtainable by the symmetries of the Riemann tensor are not displayed. These results are invariant to any Lorentz boost, thus the components do not change for non-static observers. The <a href="/wiki/Geodesic_deviation" title="Geodesic deviation">geodesic deviation</a> equation shows that the tidal acceleration between two observers separated by <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 \xi ^{\hat {j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ^{\hat {j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a14c337239712793822b9fe9f6dfdaa7f63ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.088ex; height:3.509ex;" alt="{\displaystyle \xi ^{\hat {j}}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}\xi ^{\hat {j}}/D\tau ^{2}=-R^{\hat {j}}{}_{{\hat {t}}{\hat {k}}{\hat {t}}}\xi ^{\hat {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msub> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{2}\xi ^{\hat {j}}/D\tau ^{2}=-R^{\hat {j}}{}_{{\hat {t}}{\hat {k}}{\hat {t}}}\xi ^{\hat {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f945b00a312bd555a45bfab8c41540f1ada263e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.432ex; height:4.009ex;" alt="{\displaystyle D^{2}\xi ^{\hat {j}}/D\tau ^{2}=-R^{\hat {j}}{}_{{\hat {t}}{\hat {k}}{\hat {t}}}\xi ^{\hat {k}}}"></span>, so a body of length <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is stretched in the radial direction by an apparent acceleration <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_{\text{s}}/r^{3})c^{2}L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{\text{s}}/r^{3})c^{2}L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec10df002d74a1575d6ff29f13b258de23ef9051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.647ex; height:3.176ex;" alt="{\displaystyle (r_{\text{s}}/r^{3})c^{2}L}"></span> and squeezed in the perpendicular directions by <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_{\text{s}}/(2r^{3}))c^{2}L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(r_{\text{s}}/(2r^{3}))c^{2}L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef069677a025b8f958ecc721a48e9c16d5b36c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.427ex; height:3.176ex;" alt="{\displaystyle -(r_{\text{s}}/(2r^{3}))c^{2}L}"></span>. </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=Schwarzschild_metric&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Derivation_of_the_Schwarzschild_solution" title="Derivation of the Schwarzschild solution">Derivation of the Schwarzschild solution</a></li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström metric</a> (charged, non-rotating solution)</li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metric</a> (uncharged, rotating solution)</li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman metric</a> (charged, rotating solution)</li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a>, a general review</li> <li><a href="/wiki/Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</a></li> <li><a href="/wiki/Kruskal%E2%80%93Szekeres_coordinates" title="Kruskal–Szekeres coordinates">Kruskal–Szekeres coordinates</a></li> <li><a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein coordinates</a></li> <li><a href="/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" title="Gullstrand–Painlevé coordinates">Gullstrand–Painlevé coordinates</a></li> <li><a href="/wiki/Lema%C3%AEtre_coordinates" title="Lemaître coordinates">Lemaître coordinates</a> (Schwarzschild solution in <a href="/wiki/Synchronous_coordinates" class="mw-redirect" title="Synchronous coordinates">synchronous coordinates</a>)</li> <li><a href="/wiki/Frame_fields_in_general_relativity" title="Frame fields in general relativity">Frame fields in general relativity</a> (Lemaître observers in the Schwarzschild vacuum)</li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a> (metric and pressure equations of a static and spherically symmetric body of isotropic material)</li> <li><a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a></li></ul> <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=Schwarzschild_metric&action=edit&section=10" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLuminet1979" class="citation journal cs1">Luminet, J.-P. (1979-05-01). <a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979A&A....75..228L">"Image of a spherical black hole with thin accretion disk"</a>. <i>Astronomy and Astrophysics</i>. <b>75</b>: 228–235. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979A&A....75..228L">1979A&A....75..228L</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0004-6361">0004-6361</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomy+and+Astrophysics&rft.atitle=Image+of+a+spherical+black+hole+with+thin+accretion+disk.&rft.volume=75&rft.pages=228-235&rft.date=1979-05-01&rft.issn=0004-6361&rft_id=info%3Abibcode%2F1979A%26A....75..228L&rft.aulast=Luminet&rft.aufirst=J.-P.&rft_id=https%3A%2F%2Fui.adsabs.harvard.edu%2Fabs%2F1979A%26A....75..228L&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBozza2002" class="citation journal cs1">Bozza, V. (2002-11-22). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevD.66.103001">"Gravitational lensing in the strong field limit"</a>. <i>Physical Review D</i>. <b>66</b> (10): 103001. <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/0208075">gr-qc/0208075</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/2002PhRvD..66j3001B">2002PhRvD..66j3001B</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.66.103001">10.1103/PhysRevD.66.103001</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:119476658">119476658</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Gravitational+lensing+in+the+strong+field+limit&rft.volume=66&rft.issue=10&rft.pages=103001&rft.date=2002-11-22&rft_id=info%3Aarxiv%2Fgr-qc%2F0208075&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119476658%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.66.103001&rft_id=info%3Abibcode%2F2002PhRvD..66j3001B&rft.aulast=Bozza&rft.aufirst=V.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevD.66.103001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSneppen2021" class="citation journal cs1">Sneppen, Albert (2021-07-09). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270963">"Divergent reflections around the photon sphere of a black hole"</a>. <i>Scientific Reports</i>. <b>11</b> (1): 14247. <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/2021NatSR..1114247S">2021NatSR..1114247S</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%2Fs41598-021-93595-w">10.1038/s41598-021-93595-w</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2045-2322">2045-2322</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270963">8270963</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/34244573">34244573</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Divergent+reflections+around+the+photon+sphere+of+a+black+hole&rft.volume=11&rft.issue=1&rft.pages=14247&rft.date=2021-07-09&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8270963%23id-name%3DPMC&rft_id=info%3Abibcode%2F2021NatSR..1114247S&rft_id=info%3Apmid%2F34244573&rft_id=info%3Adoi%2F10.1038%2Fs41598-021-93595-w&rft.issn=2045-2322&rft.aulast=Sneppen&rft.aufirst=Albert&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8270963&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrolovZelnikov2011" class="citation book cs1">Frolov, Valeri; Zelnikov, Andrei (2011). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontobl00frol_255"><i>Introduction to Black Hole Physics</i></a></span>. Oxford. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontobl00frol_255/page/n188">168</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-969229-3" title="Special:BookSources/978-0-19-969229-3"><bdi>978-0-19-969229-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Black+Hole+Physics&rft.pages=168&rft.pub=Oxford&rft.date=2011&rft.isbn=978-0-19-969229-3&rft.aulast=Frolov&rft.aufirst=Valeri&rft.au=Zelnikov%2C+Andrei&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontobl00frol_255&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-landau_1975-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-landau_1975_5-0">^</a></b></span> <span class="reference-text">(<a href="#CITEREFLandauLiftshitz1975">Landau & Liftshitz 1975</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: no target: CITEREFLandauLiftshitz1975 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTennent1971" class="citation book cs1">Tennent, R.M., ed. (1971). <i>Science Data Book</i>. <a href="/wiki/Oliver_%26_Boyd" class="mw-redirect" title="Oliver & Boyd">Oliver & Boyd</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-05-002487-6" title="Special:BookSources/0-05-002487-6"><bdi>0-05-002487-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+Data+Book&rft.pub=Oliver+%26+Boyd&rft.date=1971&rft.isbn=0-05-002487-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGautreauHoffmann1978" class="citation journal cs1">Gautreau, Ronald; Hoffmann, Banesh (1978-05-15). "The Schwarzschild radial coordinate as a measure of proper distance". <i>Physical Review D</i>. <b>17</b> (10): 2552–2555. <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/1978PhRvD..17.2552G">1978PhRvD..17.2552G</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.17.2552">10.1103/PhysRevD.17.2552</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0556-2821">0556-2821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=The+Schwarzschild+radial+coordinate+as+a+measure+of+proper+distance&rft.volume=17&rft.issue=10&rft.pages=2552-2555&rft.date=1978-05-15&rft.issn=0556-2821&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.17.2552&rft_id=info%3Abibcode%2F1978PhRvD..17.2552G&rft.aulast=Gautreau&rft.aufirst=Ronald&rft.au=Hoffmann%2C+Banesh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhlers1997" class="citation journal cs1"><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers, Jürgen</a> (January 1997). <a rel="nofollow" class="external text" href="http://pubman.mpdl.mpg.de/pubman/item/escidoc:153004:1/component/escidoc:153003/328699.pdf">"Examples of Newtonian limits of relativistic spacetimes"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Classical_and_Quantum_Gravity" title="Classical and Quantum Gravity">Classical and Quantum Gravity</a></i>. <b>14</b> (1A): A119–A126. <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..14A.119E">1997CQGra..14A.119E</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%2F1A%2F010">10.1088/0264-9381/14/1A/010</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11858%2F00-001M-0000-0013-5AC5-F">11858/00-001M-0000-0013-5AC5-F</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250804865">250804865</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Examples+of+Newtonian+limits+of+relativistic+spacetimes&rft.volume=14&rft.issue=1A&rft.pages=A119-A126&rft.date=1997-01&rft_id=info%3Ahdl%2F11858%2F00-001M-0000-0013-5AC5-F&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250804865%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F14%2F1A%2F010&rft_id=info%3Abibcode%2F1997CQGra..14A.119E&rft.aulast=Ehlers&rft.aufirst=J%C3%BCrgen&rft_id=http%3A%2F%2Fpubman.mpdl.mpg.de%2Fpubman%2Fitem%2Fescidoc%3A153004%3A1%2Fcomponent%2Fescidoc%3A153003%2F328699.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-Schwarzschild1916-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schwarzschild1916_9-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarzschild1916" class="citation journal cs1">Schwarzschild, K. (1916). <a rel="nofollow" class="external text" href="https://archive.org/stream/sitzungsberichte1916deutsch#page/188/mode/2up">"Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie"</a>. <i><a href="/w/index.php?title=Sitzungsberichte_der_K%C3%B6niglich_Preussischen_Akademie_der_Wissenschaften&action=edit&redlink=1" class="new" title="Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (page does not exist)">Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften</a></i>. <b>7</b>: 189–196. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1916SPAW.......189S">1916SPAW.......189S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften&rft.atitle=%C3%9Cber+das+Gravitationsfeld+eines+Massenpunktes+nach+der+Einsteinschen+Theorie&rft.volume=7&rft.pages=189-196&rft.date=1916&rft_id=info%3Abibcode%2F1916SPAW.......189S&rft.aulast=Schwarzschild&rft.aufirst=K.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fsitzungsberichte1916deutsch%23page%2F188%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span> For a translation, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntociLoinger1999" class="citation arxiv cs1">Antoci, S.; Loinger, A. (1999). "On the gravitational field of a mass point according to Einstein's theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9905030">physics/9905030</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+the+gravitational+field+of+a+mass+point+according+to+Einstein%27s+theory&rft.date=1999&rft_id=info%3Aarxiv%2Fphysics%2F9905030&rft.aulast=Antoci&rft.aufirst=S.&rft.au=Loinger%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-MacTutorBio-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-MacTutorBio_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Schwarzschild.html">"Karl Schwarzschild"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Karl+Schwarzschild&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FSchwarzschild.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDroste1917" class="citation journal cs1">Droste, J. 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Misner">Misner, Charles W.</a>; <a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip S.</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (1973). <i>Gravitation</i>. New York: W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0334-1" title="Special:BookSources/978-0-7167-0334-1"><bdi>978-0-7167-0334-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.place=New+York&rft.pub=W.+H.+Freeman&rft.date=1973&rft.isbn=978-0-7167-0334-1&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%3ASchwarzschild+metric" 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="CITEREFCarroll2004" class="citation book cs1">Carroll, Sean (2004). <i>Spacetime and Geometry: An Introduction to General Relativity</i>. Addison Wesley. <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%3A+An+Introduction+to+General+Relativity&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%3ASchwarzschild+metric" 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=Schwarzschild_metric&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarzschild1916" class="citation journal cs1">Schwarzschild, K. (1916). <a rel="nofollow" class="external text" href="https://archive.org/stream/sitzungsberichte1916deutsch#page/188/mode/2up">"Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie"</a>. <i><a href="/w/index.php?title=Sitzungsberichte_der_K%C3%B6niglich_Preussischen_Akademie_der_Wissenschaften&action=edit&redlink=1" class="new" title="Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (page does not exist)">Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften</a></i>. <b>7</b>: 189–196. <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/1916AbhKP1916..189S">1916AbhKP1916..189S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften&rft.atitle=%C3%9Cber+das+Gravitationsfeld+eines+Massenpunktes+nach+der+Einsteinschen+Theorie&rft.volume=7&rft.pages=189-196&rft.date=1916&rft_id=info%3Abibcode%2F1916AbhKP1916..189S&rft.aulast=Schwarzschild&rft.aufirst=K.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fsitzungsberichte1916deutsch%23page%2F188%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li></ul> <dl><dd><ul><li><a class="external text" href="https://de.wikisource.org/wiki/%C3%9Cber_das_Gravitationsfeld_eines_Massenpunktes_nach_der_Einsteinschen_Theorie">Text of the original paper, in Wikisource</a></li> <li>Translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntociLoinger1999" class="citation arxiv cs1">Antoci, S.; Loinger, A. (1999). "On the gravitational field of a mass point according to Einstein's theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9905030">physics/9905030</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+the+gravitational+field+of+a+mass+point+according+to+Einstein%27s+theory&rft.date=1999&rft_id=info%3Aarxiv%2Fphysics%2F9905030&rft.aulast=Antoci&rft.aufirst=S.&rft.au=Loinger%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li>A commentary on the paper, giving a simpler derivation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBel2007" class="citation arxiv cs1">Bel, L. (2007). "Über das Gravitationsfeld eines Massenpunktesnach der Einsteinschen Theorie". <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/0709.2257">0709.2257</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/gr-qc">gr-qc</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=%C3%9Cber+das+Gravitationsfeld+eines+Massenpunktesnach+der+Einsteinschen+Theorie&rft.date=2007&rft_id=info%3Aarxiv%2F0709.2257&rft.aulast=Bel&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li></ul></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarzschild1916" class="citation journal cs1">Schwarzschild, K. (1916). <a rel="nofollow" class="external text" href="https://archive.org/stream/sitzungsberichte1916deutsch#page/424/mode/2up">"Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit"</a>. <i><a href="/w/index.php?title=Sitzungsberichte_der_K%C3%B6niglich_Preussischen_Akademie_der_Wissenschaften&action=edit&redlink=1" class="new" title="Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (page does not exist)">Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften</a></i>. <b>1</b>: 424.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften&rft.atitle=%C3%9Cber+das+Gravitationsfeld+einer+Kugel+aus+inkompressibler+Fl%C3%BCssigkeit&rft.volume=1&rft.pages=424&rft.date=1916&rft.aulast=Schwarzschild&rft.aufirst=K.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fsitzungsberichte1916deutsch%23page%2F424%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li></ul> <dl><dd><ul><li><a class="external text" href="https://de.wikisource.org/wiki/Gravitationsfeld_einer_Kugel_aus_inkompressibler_Fl%C3%BCssigkeit">Text of the original paper, in Wikisource</a></li> <li>Translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntoci1999" class="citation arxiv cs1">Antoci, S. (1999). "On the gravitational field of a sphere of incompressible fluid according to Einstein's theory". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9912033">physics/9912033</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+the+gravitational+field+of+a+sphere+of+incompressible+fluid+according+to+Einstein%27s+theory&rft.date=1999&rft_id=info%3Aarxiv%2Fphysics%2F9912033&rft.aulast=Antoci&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li></ul></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlamm1916" class="citation journal cs1">Flamm, L. (1916). "Beiträge zur Einstein'schen Gravitationstheorie". <i><a href="/wiki/Physikalische_Zeitschrift" title="Physikalische Zeitschrift">Physikalische Zeitschrift</a></i>. <b>17</b>: 448.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Beitr%C3%A4ge+zur+Einstein%27schen+Gravitationstheorie&rft.volume=17&rft.pages=448&rft.date=1916&rft.aulast=Flamm&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdlerBazinSchiffer1975" class="citation book cs1">Adler, R.; Bazin, M.; Schiffer, M. (1975). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoge0000adle"><i>Introduction to General Relativity</i></a></span> (2nd ed.). <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. Chapter 6. <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>.</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.pages=Chapter+6&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1975&rft.isbn=0-07-000423-4&rft.aulast=Adler&rft.aufirst=R.&rft.au=Bazin%2C+M.&rft.au=Schiffer%2C+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoge0000adle&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1951" class="citation book cs1">Landau, L. D.; Lifshitz, E. M. (1951). <i>The Classical Theory of Fields</i>. <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a>. Vol. 2 (4th Revised English ed.). <a href="/wiki/Pergamon_Press" title="Pergamon Press">Pergamon Press</a>. Chapter 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-08-025072-6" title="Special:BookSources/0-08-025072-6"><bdi>0-08-025072-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classical+Theory+of+Fields&rft.series=Course+of+Theoretical+Physics&rft.pages=Chapter+12&rft.edition=4th+Revised+English&rft.pub=Pergamon+Press&rft.date=1951&rft.isbn=0-08-025072-6&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1970" class="citation book cs1">Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1970). <i>Gravitation</i>. <a href="/wiki/W.H._Freeman" class="mw-redirect" title="W.H. Freeman">W.H. Freeman</a>. Chapters 31 and 32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0"><bdi>0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=Chapters+31+and+32&rft.pub=W.H.+Freeman&rft.date=1970&rft.isbn=0-7167-0344-0&rft.aulast=Misner&rft.aufirst=C.+W.&rft.au=Thorne%2C+K.+S.&rft.au=Wheeler%2C+J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1972" class="citation book cs1">Weinberg, S. (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: Principles and Applications of the General Theory of Relativity</i></a></span>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. Chapter 8. <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%3A+Principles+and+Applications+of+the+General+Theory+of+Relativity&rft.pages=Chapter+8&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=0-471-92567-5&rft.aulast=Weinberg&rft.aufirst=S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgravitationcosmo00stev_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorWheeler2000" class="citation book cs1">Taylor, E. F.; Wheeler, J. A. (2000). <i>Exploring Black Holes: Introduction to General Relativity</i>. <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-38423-X" title="Special:BookSources/0-201-38423-X"><bdi>0-201-38423-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exploring+Black+Holes%3A+Introduction+to+General+Relativity&rft.pub=Addison-Wesley&rft.date=2000&rft.isbn=0-201-38423-X&rft.aulast=Taylor&rft.aufirst=E.+F.&rft.au=Wheeler%2C+J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeinzleSteinbauer2002" class="citation journal cs1">Heinzle, J. M.; Steinbauer, R. (2002). "Remarks on the distributional Schwarzschild geometry". <i><a href="/wiki/Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">Journal of Mathematical Physics</a></i>. <b>43</b> (3): 1493–1508. <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/0112047">gr-qc/0112047</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/2002JMP....43.1493H">2002JMP....43.1493H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1448684">10.1063/1.1448684</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:119677857">119677857</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Remarks+on+the+distributional+Schwarzschild+geometry&rft.volume=43&rft.issue=3&rft.pages=1493-1508&rft.date=2002&rft_id=info%3Aarxiv%2Fgr-qc%2F0112047&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119677857%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1448684&rft_id=info%3Abibcode%2F2002JMP....43.1493H&rft.aulast=Heinzle&rft.aufirst=J.+M.&rft.au=Steinbauer%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarzschild+metric" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Black_holes" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3" 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:Black_holes" title="Template:Black holes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Black_holes" title="Template talk:Black holes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Black_holes" title="Special:EditPage/Template:Black holes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Black_holes" style="font-size:114%;margin:0 4em"><a href="/wiki/Black_hole" title="Black hole">Black holes</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3" style="text-align:center;"><div> <ul><li><a href="/wiki/Outline_of_black_holes" title="Outline of black holes">Outline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BTZ_black_hole" title="BTZ black hole">BTZ black hole</a></li> <li><a class="mw-selflink selflink">Schwarzschild</a></li> <li><a href="/wiki/Rotating_black_hole" title="Rotating black hole">Rotating</a></li> <li><a href="/wiki/Charged_black_hole" title="Charged black hole">Charged</a></li> <li><a href="/wiki/Virtual_black_hole" title="Virtual black hole">Virtual</a></li> <li><a href="/wiki/Kugelblitz_(astrophysics)" title="Kugelblitz (astrophysics)">Kugelblitz</a></li> <li><a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">Supermassive</a></li> <li><a href="/wiki/Primordial_black_hole" title="Primordial black hole">Primordial</a></li> <li><a href="/wiki/Direct_collapse_black_hole" title="Direct collapse black hole">Direct collapse</a></li> <li><a href="/wiki/Rogue_black_hole" title="Rogue black hole">Rogue</a></li> <li><a href="/wiki/Malament%E2%80%93Hogarth_spacetime" title="Malament–Hogarth spacetime">Malament–Hogarth spacetime</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="11" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Black_hole_-_Messier_87_crop_max_res.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Black_hole_-_Messier_87_crop_max_res.jpg/80px-Black_hole_-_Messier_87_crop_max_res.jpg" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Black_hole_-_Messier_87_crop_max_res.jpg/120px-Black_hole_-_Messier_87_crop_max_res.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Black_hole_-_Messier_87_crop_max_res.jpg/160px-Black_hole_-_Messier_87_crop_max_res.jpg 2x" data-file-width="4320" data-file-height="4320" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Size</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Micro_black_hole" title="Micro black hole">Micro</a> <ul><li><a href="/wiki/Extremal_black_hole" title="Extremal black hole">Extremal</a></li> <li><a href="/wiki/Black_hole_electron" title="Black hole electron">Electron</a></li></ul></li> <li><a href="/wiki/Stellar_black_hole" title="Stellar black hole">Stellar</a> <ul><li><a href="/wiki/Microquasar" title="Microquasar">Microquasar</a></li></ul></li> <li><a href="/wiki/Intermediate-mass_black_hole" title="Intermediate-mass black hole">Intermediate-mass</a></li> <li><a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">Supermassive</a> <ul><li><a href="/wiki/Active_galactic_nucleus" title="Active galactic nucleus">Active galactic nucleus</a></li> <li><a href="/wiki/Quasar" title="Quasar">Quasar</a></li> <li><a href="/wiki/Large_quasar_group" title="Large quasar group">LQG</a></li> <li><a href="/wiki/Blazar" title="Blazar">Blazar</a></li> <li><a href="/wiki/OVV_quasar" title="OVV quasar">OVV</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Formation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Stellar_evolution" title="Stellar evolution">Stellar evolution</a></li> <li><a href="/wiki/Gravitational_collapse" title="Gravitational collapse">Gravitational collapse</a></li> <li><a href="/wiki/Neutron_star" title="Neutron star">Neutron star</a> <ul><li><a href="/wiki/Template:Neutron_star" title="Template:Neutron star">Related links</a></li></ul></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit" title="Tolman–Oppenheimer–Volkoff limit">Tolman–Oppenheimer–Volkoff limit</a></li> <li><a href="/wiki/White_dwarf" title="White dwarf">White dwarf</a> <ul><li><a href="/wiki/Template:White_dwarf" title="Template:White dwarf">Related links</a></li></ul></li> <li><a href="/wiki/Supernova" title="Supernova">Supernova</a> <ul><li><a href="/wiki/Micronova" title="Micronova">Micronova</a></li> <li><a href="/wiki/Superluminous_supernova" title="Superluminous supernova">Hypernova</a></li> <li><a href="/wiki/Template:Supernovae" title="Template:Supernovae">Related links</a></li></ul></li> <li><a href="/wiki/Gamma-ray_burst" title="Gamma-ray burst">Gamma-ray burst</a></li> <li><a href="/wiki/Binary_black_hole" title="Binary black hole">Binary black hole</a></li> <li><a href="/wiki/Quark_star" title="Quark star">Quark star</a></li> <li><a href="/wiki/Supermassive_star" class="mw-redirect" title="Supermassive star">Supermassive star</a></li> <li><a href="/wiki/Quasi-star" title="Quasi-star">Quasi-star</a></li> <li><a href="/wiki/Dark_star_(dark_matter)" title="Dark star (dark matter)">Supermassive dark star</a></li> <li><a href="/wiki/X-ray_binary" title="X-ray binary">X-ray binary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysical_jet" title="Astrophysical jet">Astrophysical jet</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Gravitational singularity</a> <ul><li><a href="/wiki/Ring_singularity" title="Ring singularity">Ring singularity</a></li> <li><a href="/wiki/Penrose%E2%80%93Hawking_singularity_theorems" title="Penrose–Hawking singularity theorems">Theorems</a></li></ul></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Photon_sphere" title="Photon sphere">Photon sphere</a></li> <li><a href="/wiki/Innermost_stable_circular_orbit" title="Innermost stable circular orbit">Innermost stable circular orbit</a></li> <li><a href="/wiki/Ergosphere" title="Ergosphere">Ergosphere</a> <ul><li><a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a></li> <li><a href="/wiki/Blandford%E2%80%93Znajek_process" title="Blandford–Znajek process">Blandford–Znajek process</a></li></ul></li> <li><a href="/wiki/Accretion_disk" title="Accretion disk">Accretion disk</a></li> <li><a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lens</a> <ul><li><a href="/wiki/Gravitational_microlensing" title="Gravitational microlensing">Microlens</a></li></ul></li> <li><a href="/wiki/Bondi_accretion" title="Bondi accretion">Bondi accretion</a></li> <li><a href="/wiki/M%E2%80%93sigma_relation" title="M–sigma relation">M–sigma relation</a></li> <li><a href="/wiki/Quasi-periodic_oscillation" class="mw-redirect" title="Quasi-periodic oscillation">Quasi-periodic oscillation</a></li> <li><a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Bekenstein_bound" title="Bekenstein bound">Bekenstein bound</a></li> <li><a href="/wiki/Bousso%27s_holographic_bound" title="Bousso's holographic bound">Bousso's holographic bound</a> <ul><li><a href="/wiki/Immirzi_parameter" title="Immirzi parameter">Immirzi parameter</a></li></ul></li> <li><a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a></li> <li><a href="/wiki/Spaghettification" title="Spaghettification">Spaghettification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Issues</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole_complementarity" title="Black hole complementarity">Black hole complementarity</a></li> <li><a href="/wiki/Black_hole_information_paradox" title="Black hole information paradox">Information paradox</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship</a></li> <li><a href="/wiki/ER_%3D_EPR" title="ER = EPR">ER = EPR</a></li> <li><a href="/wiki/Binary_black_hole#Final_parsec_problem" title="Binary black hole">Final parsec problem</a></li> <li><a href="/wiki/Firewall_(physics)" title="Firewall (physics)">Firewall (physics)</a></li> <li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/No-hair_theorem" title="No-hair theorem">No-hair theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Metrics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Schwarzschild</a> (<a href="/wiki/Derivation_of_the_Schwarzschild_solution" title="Derivation of the Schwarzschild solution">Derivation</a>)</li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</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/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Hayward_metric" title="Hayward metric">Hayward</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Alternatives</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonsingular_black_hole_models" title="Nonsingular black hole models">Nonsingular black hole models</a></li> <li><a href="/wiki/Black_star_(semiclassical_gravity)" title="Black star (semiclassical gravity)">Black star</a></li> <li><a href="/wiki/Dark_star_(Newtonian_mechanics)" title="Dark star (Newtonian mechanics)">Dark star</a></li> <li><a href="/wiki/Dark-energy_star" title="Dark-energy star">Dark-energy star</a></li> <li><a href="/wiki/Gravastar" title="Gravastar">Gravastar</a></li> <li><a href="/wiki/Magnetospheric_eternally_collapsing_object" title="Magnetospheric eternally collapsing object">Magnetospheric eternally collapsing object</a></li> <li><a href="/wiki/Planck_star" title="Planck star">Planck star</a></li> <li><a href="/wiki/Q_star" title="Q star">Q star</a></li> <li><a href="/wiki/Fuzzball_(string_theory)" title="Fuzzball (string theory)">Fuzzball</a></li> <li><a href="/wiki/Geon_(physics)" title="Geon (physics)">Geon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Analogs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Optical_black_hole" title="Optical black hole">Optical black hole</a></li> <li><a href="/wiki/Sonic_black_hole" title="Sonic black hole">Sonic black hole</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Lists_of_black_holes" title="Lists of black holes">Lists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_black_holes" title="List of black holes">Black holes</a></li> <li><a href="/wiki/List_of_most_massive_black_holes" title="List of most massive black holes">Most massive</a></li> <li><a href="/wiki/List_of_nearest_known_black_holes" title="List of nearest known black holes">Nearest</a></li> <li><a href="/wiki/List_of_quasars" title="List of quasars">Quasars</a></li> <li><a href="/wiki/List_of_microquasars" title="List of microquasars">Microquasars</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Outline_of_black_holes" title="Outline of black holes">Outline of black holes</a></li> <li><a href="/wiki/Black_Hole_Initiative" title="Black Hole Initiative">Black Hole Initiative</a></li> <li><a href="/wiki/Black_hole_starship" title="Black hole starship">Black hole starship</a></li> <li><a href="/wiki/Black_holes_in_fiction" title="Black holes in fiction">Black holes in fiction</a></li> <li><a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a></li> <li><a href="/wiki/Big_Bounce" title="Big Bounce">Big Bounce</a></li> <li><a href="/wiki/Compact_star" class="mw-redirect" title="Compact star">Compact star</a></li> <li><a href="/wiki/Exotic_star" title="Exotic star">Exotic star</a> <ul><li><a href="/wiki/Quark_star" title="Quark star">Quark star</a></li> <li><a href="/wiki/Preon_star" class="mw-redirect" title="Preon star">Preon star</a></li></ul></li> <li><a href="/wiki/Gravitational_waves" class="mw-redirect" title="Gravitational waves">Gravitational waves</a></li> <li><a href="/wiki/Gamma-ray_burst_progenitors" title="Gamma-ray burst progenitors">Gamma-ray burst progenitors</a></li> <li><a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">Gravity well</a></li> <li><a href="/wiki/Hypercompact_stellar_system" title="Hypercompact stellar system">Hypercompact stellar system</a></li> <li><a href="/wiki/Membrane_paradigm" title="Membrane paradigm">Membrane paradigm</a></li> <li><a href="/wiki/Naked_singularity" title="Naked singularity">Naked singularity</a></li> <li><a href="/wiki/Population_III_star" class="mw-redirect" title="Population III star">Population III star</a></li> <li><a href="/wiki/Supermassive_star" class="mw-redirect" title="Supermassive star">Supermassive star</a></li> <li><a href="/wiki/Quasi-star" title="Quasi-star">Quasi-star</a></li> <li><a href="/wiki/Dark_star_(dark_matter)" title="Dark star (dark matter)">Supermassive dark star</a></li> <li><a href="/wiki/Rossi_X-ray_Timing_Explorer" title="Rossi X-ray Timing Explorer">Rossi X-ray Timing Explorer</a></li> <li><a href="/wiki/Superluminal_motion" title="Superluminal motion">Superluminal motion</a></li> <li><a href="/wiki/Timeline_of_black_hole_physics" title="Timeline of black hole physics">Timeline of black hole physics</a></li> <li><a href="/wiki/White_hole" title="White hole">White hole</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/Tidal_disruption_event" title="Tidal disruption event">Tidal disruption event</a></li> <li><a href="/wiki/Planet_Nine" title="Planet Nine">Planet Nine</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Notable</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cygnus_X-1" title="Cygnus X-1">Cygnus X-1</a></li> <li><a href="/wiki/XTE_J1650-500" class="mw-redirect" title="XTE J1650-500">XTE J1650-500</a></li> <li><a href="/wiki/XTE_J1118%2B480" title="XTE J1118+480">XTE J1118+480</a></li> <li><a href="/wiki/A0620-00" title="A0620-00">A0620-00</a></li> <li><a href="/wiki/Sagittarius_A*" title="Sagittarius A*">Sagittarius A*</a></li> <li><a href="/wiki/Centaurus_A" title="Centaurus A">Centaurus A</a></li> <li><a href="/wiki/Phoenix_Cluster" title="Phoenix Cluster">Phoenix Cluster</a></li> <li><a href="/wiki/PKS_1302-102" title="PKS 1302-102">PKS 1302-102</a></li> <li><a href="/wiki/OJ_287" title="OJ 287">OJ 287</a></li> <li><a href="/wiki/SDSS_J0849%2B1114" title="SDSS J0849+1114">SDSS J0849+1114</a></li> <li><a href="/wiki/TON_618" title="TON 618">TON 618</a></li> <li><a href="/wiki/MS_0735.6%2B7421" title="MS 0735.6+7421">MS 0735.6+7421</a></li> <li><a href="/wiki/NeVe_1" title="NeVe 1">NeVe 1</a></li> <li><a href="/wiki/Hercules_A" title="Hercules A">Hercules A</a></li> <li><a href="/wiki/3C_273" title="3C 273">3C 273</a></li> <li><a href="/wiki/Q0906%2B6930" title="Q0906+6930">Q0906+6930</a></li> <li><a href="/wiki/Markarian_501" title="Markarian 501">Markarian 501</a></li> <li><a href="/wiki/ULAS_J1342%2B0928" title="ULAS J1342+0928">ULAS J1342+0928</a></li> <li><a href="/wiki/PSO_J030947.49%2B271757.31" title="PSO J030947.49+271757.31">PSO J030947.49+271757.31</a></li> <li><a href="/wiki/AT2018hyz" title="AT2018hyz">AT2018hyz</a></li> <li><a href="/wiki/Swift_J1644%2B57" title="Swift J1644+57">Swift J1644+57</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3" style="text-align:center;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Black_holes" title="Category:Black holes">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Black_holes" class="extiw" title="commons:Category:Black holes">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Relativity" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a class="mw-selflink selflink">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Schwarzschild_metric&oldid=1258114939">https://en.wikipedia.org/w/index.php?title=Schwarzschild_metric&oldid=1258114939</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Exact_solutions_in_general_relativity" title="Category:Exact solutions in general relativity">Exact solutions in general relativity</a></li><li><a href="/wiki/Category:Black_holes" title="Category:Black holes">Black holes</a></li><li><a href="/wiki/Category:1916_in_science" title="Category:1916 in science">1916 in science</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Harv_and_Sfn_no-target_errors" title="Category:Harv and Sfn no-target errors">Harv and Sfn no-target errors</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_November_2010" title="Category:Articles with unsourced statements from November 2010">Articles with unsourced statements from November 2010</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 November 2024, at 05:54<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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1,\n [\"CITEREFHughstonTod1990\"] = 1,\n [\"CITEREFKox1992\"] = 1,\n [\"CITEREFKruskal1960\"] = 1,\n [\"CITEREFLandauLifshitz1951\"] = 1,\n [\"CITEREFLuminet1979\"] = 1,\n [\"CITEREFMisnerThorneWheeler1970\"] = 1,\n [\"CITEREFMisnerThorneWheeler1973\"] = 1,\n [\"CITEREFNi2017\"] = 1,\n [\"CITEREFSbierski2015\"] = 1,\n [\"CITEREFSchwarzschild1916\"] = 3,\n [\"CITEREFSneppen2021\"] = 1,\n [\"CITEREFSynge1950\"] = 1,\n [\"CITEREFSynge1974\"] = 1,\n [\"CITEREFSzekeres1960\"] = 1,\n [\"CITEREFTaylorWheeler2000\"] = 1,\n [\"CITEREFTennent1971\"] = 1,\n [\"CITEREFWeinberg1972\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Black holes\"] = 1,\n [\"Citation needed\"] = 1,\n [\"Cite arXiv\"] = 5,\n [\"Cite book\"] = 16,\n [\"Cite journal\"] = 16,\n [\"Details\"] = 1,\n [\"Further\"] = 1,\n [\"General relativity sidebar\"] = 1,\n [\"Harv\"] = 1,\n [\"MacTutor Biography\"] = 1,\n [\"Math\"] = 52,\n [\"Mvar\"] = 17,\n [\"Pi\"] = 1,\n [\"Reflist\"] = 1,\n [\"Relativity\"] = 1,\n [\"Sfrac\"] = 2,\n [\"Short description\"] = 1,\n [\"Sub\"] = 1,\n [\"Tmath\"] = 8,\n [\"Val\"] = 4,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-ctx5l","timestamp":"20241125095824","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Schwarzschild metric","url":"https:\/\/en.wikipedia.org\/wiki\/Schwarzschild_metric","sameAs":"http:\/\/www.wikidata.org\/entity\/Q742969","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q742969","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-07-10T15:04:52Z","dateModified":"2024-11-18T05:54:14Z","headline":"spherically symmetric static vacuum solution of Einstein\u2019s field equations with zero 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Schwarzschild metric - Wikipedia