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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Karl Schwarzschild</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 51 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-51" 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">51 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%83%D8%A7%D8%B1%D9%84_%D8%B4%D9%81%D8%A7%D8%B1%D8%AA%D8%B3%D8%B4%D9%8A%D9%84%D8%AF" title="كارل شفارتسشيلد – Arabic" lang="ar" hreflang="ar" data-title="كارل شفارتسشيلد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1%D9%84_%D8%B4%D9%88%D8%A7%D8%B1%D8%AA%D8%B2%D8%B4%DB%8C%D9%84%D8%AF" title="کارل شوارتزشیلد – South Azerbaijani" lang="azb" hreflang="azb" data-title="کارل شوارتزشیلد" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%B2_%E0%A6%B6%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%9C%E0%A6%B6%E0%A6%BF%E0%A6%B2%E0%A7%8D%E0%A6%A1" title="কার্ল শোয়ার্জশিল্ড – Bangla" lang="bn" hreflang="bn" data-title="কার্ল শোয়ার্জশিল্ড" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D0%BB_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D1%8B%D0%BB%D1%8C%D0%B4" title="Карл Шварцшыльд – Belarusian" lang="be" hreflang="be" data-title="Карл Шварцшыльд" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D0%BB_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D0%B4" title="Карл Шварцшилд – Bulgarian" lang="bg" hreflang="bg" data-title="Карл Шварцшилд" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Breton" lang="br" hreflang="br" data-title="Karl Schwarzschild" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Catalan" lang="ca" hreflang="ca" data-title="Karl 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/Karl_Schwarzschild" title="Karl Schwarzschild – Czech" lang="cs" hreflang="cs" data-title="Karl Schwarzschild" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Danish" lang="da" hreflang="da" data-title="Karl Schwarzschild" 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/Karl_Schwarzschild" title="Karl Schwarzschild – German" lang="de" hreflang="de" data-title="Karl Schwarzschild" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CF%81%CE%BB_%CE%A3%CE%B2%CE%AC%CF%81%CF%84%CF%83%CE%B9%CE%BB%CE%BD%CF%84" title="Καρλ Σβάρτσιλντ – Greek" lang="el" hreflang="el" data-title="Καρλ 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/Karl_Schwarzschild" title="Karl Schwarzschild – Spanish" lang="es" hreflang="es" data-title="Karl Schwarzschild" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Esperanto" lang="eo" hreflang="eo" data-title="Karl Schwarzschild" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Basque" lang="eu" hreflang="eu" data-title="Karl Schwarzschild" 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/%DA%A9%D8%A7%D8%B1%D9%84_%D8%B4%D9%88%D8%A7%D8%B1%D8%AA%D8%B2%D8%B4%DB%8C%D9%84%D8%AF" title="کارل شوارتزشیلد – Persian" lang="fa" hreflang="fa" data-title="کارل شوارتزشیلد" 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/Karl_Schwarzschild" title="Karl Schwarzschild – French" lang="fr" hreflang="fr" data-title="Karl Schwarzschild" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Irish" lang="ga" hreflang="ga" data-title="Karl Schwarzschild" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Manx" lang="gv" hreflang="gv" data-title="Karl Schwarzschild" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%B4%EB%A5%BC_%EC%8A%88%EB%B0%94%EB%A5%B4%EC%B8%A0%EC%8B%A4%ED%8A%B8" title="카를 슈바르츠실트 – Korean" lang="ko" hreflang="ko" data-title="카를 슈바르츠실트" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BC%D5%AC_%D5%87%D5%BE%D5%A1%D6%80%D6%81%D5%B7%D5%AB%D5%AC%D5%A4" title="Կառլ Շվարցշիլդ – Armenian" lang="hy" hreflang="hy" data-title="Կառլ Շվարցշիլդ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Indonesian" lang="id" hreflang="id" data-title="Karl 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/Karl_Schwarzschild" title="Karl Schwarzschild – Italian" lang="it" hreflang="it" data-title="Karl 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%A7%D7%A8%D7%9C_%D7%A9%D7%95%D7%95%D7%A8%D7%A6%D7%A9%D7%99%D7%9C%D7%93" title="קרל שוורצשילד – Hebrew" lang="he" hreflang="he" data-title="קרל שוורצשילד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%90%E1%83%A0%E1%83%9A_%E1%83%A8%E1%83%95%E1%83%90%E1%83%A0%E1%83%AA%E1%83%A8%E1%83%98%E1%83%9A%E1%83%93%E1%83%98" title="კარლ შვარცშილდი – Georgian" lang="ka" hreflang="ka" data-title="კარლ შვარცშილდი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Luxembourgish" lang="lb" hreflang="lb" data-title="Karl Schwarzschild" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D0%BB_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D0%B4" title="Карл Шварцшилд – Macedonian" lang="mk" hreflang="mk" data-title="Карл Шварцшилд" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Malagasy" lang="mg" hreflang="mg" data-title="Karl Schwarzschild" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%83%D8%A7%D8%B1%D9%84_%D8%B4%D9%81%D8%A7%D8%B1%D8%AA%D8%B3%D8%B4%D9%8A%D9%84%D8%AF" title="كارل شفارتسشيلد – Egyptian Arabic" lang="arz" hreflang="arz" data-title="كارل شفارتسشيلد" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Dutch" lang="nl" hreflang="nl" data-title="Karl Schwarzschild" 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%AB%E3%83%BC%E3%83%AB%E3%83%BB%E3%82%B7%E3%83%A5%E3%83%B4%E3%82%A1%E3%83%AB%E3%83%84%E3%82%B7%E3%83%AB%E3%83%88" title="カール・シュヴァルツシルト – Japanese" lang="ja" hreflang="ja" data-title="カール・シュヴァルツシルト" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Karl Schwarzschild" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Karl Schwarzschild" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Occitan" lang="oc" hreflang="oc" data-title="Karl Schwarzschild" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Polish" lang="pl" hreflang="pl" data-title="Karl Schwarzschild" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Portuguese" lang="pt" hreflang="pt" data-title="Karl 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/Karl_Schwarzschild" title="Karl Schwarzschild – Romanian" lang="ro" hreflang="ro" data-title="Karl 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 badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D1%8C%D0%B4,_%D0%9A%D0%B0%D1%80%D0%BB" title="Шварцшильд, Карл – Russian" lang="ru" hreflang="ru" data-title="Шварцшильд, Карл" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Simple English" lang="en-simple" hreflang="en-simple" data-title="Karl Schwarzschild" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Slovak" lang="sk" hreflang="sk" data-title="Karl Schwarzschild" 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/Karl_Schwarzschild" title="Karl Schwarzschild – Slovenian" lang="sl" hreflang="sl" data-title="Karl Schwarzschild" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D0%BB_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D0%B4" title="Карл Шварцшилд – Serbian" lang="sr" hreflang="sr" data-title="Карл Шварцшилд" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Karl Schwarzschild" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Finnish" lang="fi" hreflang="fi" data-title="Karl Schwarzschild" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Swedish" lang="sv" hreflang="sv" data-title="Karl Schwarzschild" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AF%81%E0%AE%9A%E0%AF%88%E0%AE%B2%E0%AF%8D%E0%AE%9F%E0%AF%81" title="கார்ல் சுவார்சுசைல்டு – Tamil" lang="ta" hreflang="ta" data-title="கார்ல் சுவார்சுசைல்டு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B8%A5_%E0%B8%8A%E0%B8%A7%E0%B8%B2%E0%B8%97%E0%B8%8B%E0%B9%8C%E0%B8%8A%E0%B8%B4%E0%B8%A5%E0%B8%97%E0%B9%8C" title="คาร์ล ชวาทซ์ชิลท์ – Thai" lang="th" hreflang="th" data-title="คาร์ล ชวาทซ์ชิลท์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Turkish" lang="tr" hreflang="tr" data-title="Karl Schwarzschild" 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%9A%D0%B0%D1%80%D0%BB_%D0%A8%D0%B2%D0%B0%D1%80%D1%86%D1%88%D0%B8%D0%BB%D1%8C%D0%B4" title="Карл Шварцшильд – Ukrainian" lang="uk" hreflang="uk" data-title="Карл Шварцшильд" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Karl_Schwarzschild" title="Karl Schwarzschild – Vietnamese" lang="vi" hreflang="vi" data-title="Karl 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-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%A8%D7%9C_%D7%A9%D7%95%D7%95%D7%90%D7%A8%D7%A6%D7%A9%D7%99%D7%9C%D7%93" title="קארל שווארצשילד – Yiddish" lang="yi" hreflang="yi" data-title="קארל שווארצשילד" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article 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.navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Karl Schwarzschild</div></th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Karl_schwarzschild.portrait.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Karl_schwarzschild.portrait.jpg/190px-Karl_schwarzschild.portrait.jpg" decoding="async" width="190" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Karl_schwarzschild.portrait.jpg/285px-Karl_schwarzschild.portrait.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Karl_schwarzschild.portrait.jpg/380px-Karl_schwarzschild.portrait.jpg 2x" data-file-width="432" data-file-height="600" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1873-10-09</span>)</span>9 October 1873<br /><div style="display:inline" class="birthplace"><a href="/wiki/Frankfurt_am_Main" class="mw-redirect" title="Frankfurt am Main">Frankfurt am Main</a>, <a href="/wiki/German_Empire" title="German Empire">German Empire</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">11 May 1916<span style="display:none">(1916-05-11)</span> (aged 42)<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 class="reference nowrap"><span title="Page / location: xix">: xix </span></sup><br /><div style="display:inline" class="deathplace"><a href="/wiki/Potsdam" title="Potsdam">Potsdam</a>, German Empire</div></td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/Ludwig_Maximilian_University_of_Munich" title="Ludwig Maximilian University of Munich">Ludwig Maximilian University of Munich</a><br /><a href="/wiki/University_of_Strasbourg" title="University of Strasbourg">University of Strasbourg</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Physics" title="Physics">Physics</a><br /><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Hugo_von_Seeliger" title="Hugo von Seeliger">Hugo von Seeliger</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"></td></tr><tr><td colspan="2" class="infobox-subheader" style="font-weight:bold;">Military career</td></tr><tr><th scope="row" class="infobox-label" style="padding-right: 1em">Allegiance</th><td class="infobox-data"><span class="flagicon"><span class="mw-image-border" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Flag_of_Germany_%281867%E2%80%931918%29.svg/23px-Flag_of_Germany_%281867%E2%80%931918%29.svg.png" decoding="async" width="23" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Flag_of_Germany_%281867%E2%80%931918%29.svg/35px-Flag_of_Germany_%281867%E2%80%931918%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Flag_of_Germany_%281867%E2%80%931918%29.svg/45px-Flag_of_Germany_%281867%E2%80%931918%29.svg.png 2x" data-file-width="900" data-file-height="600" /></span></span> </span><a href="/wiki/German_Empire" title="German Empire">German Empire</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-right: 1em">Service <wbr />/ branch</th><td class="infobox-data"><a href="/wiki/German_Army_(German_Empire)" class="mw-redirect" title="German Army (German Empire)">Imperial German Army</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-right: 1em">Years of service</th><td class="infobox-data">1914–1916</td></tr><tr><th scope="row" class="infobox-label" style="padding-right: 1em">Rank</th><td class="infobox-data"><a href="/wiki/Leutnant" title="Leutnant">Lieutenant</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-right: 1em">Battles / wars</th><td class="infobox-data"><a href="/wiki/World_War_I" title="World War I">World War I</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Karl Schwarzschild</b> (<style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">German:</span> <span class="IPA nowrap" lang="de-Latn-fonipa"><a href="/wiki/Help:IPA/Standard_German" title="Help:IPA/Standard German">[kaʁl<span class="wrap"> </span>ˈʃvaʁtsʃɪlt]</a></span> <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/0\/07\/LL-Q188_%28deu%29-Shisma-Schwarzschild.wav\/LL-Q188_%28deu%29-Shisma-Schwarzschild.wav.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"LL-Q188 (deu)-Shisma-Schwarzschild.wav"},"classes":["ext-phonos-PhonosButton","noexcerpt","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/0/07/LL-Q188_%28deu%29-Shisma-Schwarzschild.wav/LL-Q188_%28deu%29-Shisma-Schwarzschild.wav.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label"></span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:LL-Q188_(deu)-Shisma-Schwarzschild.wav" title="File:LL-Q188 (deu)-Shisma-Schwarzschild.wav">ⓘ</a></sup></span></span>; 9 October 1873 – 11 May 1916) was a German <a href="/wiki/Physicist" title="Physicist">physicist</a> and astronomer. </p><p>Schwarzschild provided the first <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solution</a> to the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, for the limited case of a single spherical non-rotating mass, which he accomplished in 1915, the same year that Einstein first introduced general relativity. The <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a>, which makes use of <a href="/wiki/Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</a> and the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a>, leads to a derivation of the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a>, which is the size of the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> of a non-rotating <a href="/wiki/Black_hole" title="Black hole">black hole</a>. </p><p>Schwarzschild accomplished this while serving in the German army during <a href="/wiki/World_War_I" title="World War I">World War I</a>. He died the following year from the <a href="/wiki/Autoimmunity" title="Autoimmunity">autoimmune disease</a> <a href="/wiki/Pemphigus" title="Pemphigus">pemphigus</a>, which he developed while at the <a href="/wiki/Eastern_Front_(World_War_I)" title="Eastern Front (World War I)">Russian front</a>.<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> Various forms of the disease particularly affect people of <a href="/wiki/Ashkenazi_Jews" title="Ashkenazi Jews">Ashkenazi Jewish</a> origin.<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><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Asteroid <a href="/wiki/837_Schwarzschilda" title="837 Schwarzschilda">837 Schwarzschilda</a> is named in his honour, as is the large crater <a href="/wiki/Schwarzschild_(crater)" title="Schwarzschild (crater)"><i>Schwarzschild</i></a>, on the far side of the <a href="/wiki/Moon" title="Moon">Moon</a>.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Life">Life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=1" title="Edit section: Life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Karl Schwarzschild was born on 9 October 1873 in <a href="/wiki/Frankfurt" title="Frankfurt">Frankfurt on Main</a>, the eldest of six boys and one girl,<sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> to <a href="/wiki/History_of_the_Jews_in_Germany" title="History of the Jews in Germany">Jewish</a> parents. His father was active in the <a href="/wiki/Business_sector" title="Business sector">business community</a> of the city, and the family had ancestors in Frankfurt from the sixteenth century onwards.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The family owned two fabric stores in Frankfurt. His brother Alfred became a painter.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The young Schwarzschild attended a Jewish primary school until 11 years of age<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> and then the <a href="/wiki/Lessing-Gymnasium,_Frankfurt" title="Lessing-Gymnasium, Frankfurt">Lessing-Gymnasium</a> (secondary school). He received an all-encompassing education, including subjects like Latin, Ancient Greek, music and art, but developed a special interest in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> early on.<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 fact he was something of a child prodigy, having two papers on binary orbits (<a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>) published before the age of sixteen.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>After graduation in 1890, he attended the <a href="/wiki/University_of_Strasbourg" title="University of Strasbourg">University of Strasbourg</a> to study astronomy. After two years he transferred to the <a href="/wiki/Ludwig_Maximilian_University_of_Munich" title="Ludwig Maximilian University of Munich">Ludwig Maximilian University of Munich</a> where he obtained his doctorate in 1896 for a work on <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>'s theories. </p><p>From 1897, he worked as assistant at the <a href="/wiki/Kuffner_Observatory" title="Kuffner Observatory">Kuffner Observatory</a> in Vienna. His work here concentrated on the <a href="/wiki/Photometry_(astronomy)" title="Photometry (astronomy)">photometry</a> of star clusters and laid the foundations for a formula linking the intensity of the starlight, exposure time, and the resulting contrast on a <a href="/wiki/Photographic_plate" title="Photographic plate">photographic plate</a>. An integral part of that theory is the <a href="/w/index.php?title=Schwarzschild_exponent&action=edit&redlink=1" class="new" title="Schwarzschild exponent (page does not exist)">Schwarzschild exponent</a> (<a href="/wiki/Astrophotography" title="Astrophotography">astrophotography</a>). In 1899, he returned to Munich to complete his <a href="/wiki/Habilitation" title="Habilitation">Habilitation</a>. </p><p>From 1901 until 1909, he was a professor at the prestigious <a href="/wiki/G%C3%B6ttingen_Observatory" title="Göttingen Observatory">Göttingen Observatory</a> within the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>,<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> where he had the opportunity to work with some significant figures, including <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>. Schwarzschild became the director of the observatory. He married Else Rosenbach, a great-granddaughter of <a href="/wiki/Friedrich_W%C3%B6hler" title="Friedrich Wöhler">Friedrich Wöhler</a> and daughter of a professor of surgery at Göttingen, in 1909. Later that year they moved to <a href="/wiki/Potsdam" title="Potsdam">Potsdam</a>, where he took up the post of director of the Astrophysical Observatory. This was then the most prestigious post available for an astronomer in Germany.<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. (July 2021)">citation needed</span></a></i>]</sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Frederick_Stratton,_John_William_Nicholson,_K._Schwarzschild,_Frank_Watson_Dyson_ride_in_automobile.jpg" class="mw-file-description"><img alt="Schwarzschild, third from left in the automobile; possibly during the Fifth Conference of the International Union for Co-operation in Solar Research, held in Bonn, Germany" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg/220px-Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg" decoding="async" width="220" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg/330px-Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg/440px-Frederick_Stratton%2C_John_William_Nicholson%2C_K._Schwarzschild%2C_Frank_Watson_Dyson_ride_in_automobile.jpg 2x" data-file-width="2550" data-file-height="3141" /></a><figcaption>Schwarzschild, third from left in the automobile; possibly during the Fifth Conference of the International Union for Co-operation in Solar Research, held in <a href="/wiki/Bonn,_Germany" class="mw-redirect" title="Bonn, Germany">Bonn, Germany</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg/220px-Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg/330px-Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg/440px-Grave_of_Karl_Schwarzschild_at_Stadtfriedhof_G%C3%B6ttingen_2017_01.jpg 2x" data-file-width="2734" data-file-height="4096" /></a><figcaption>Karl Schwarzschild's grave at <a href="/wiki/Stadtfriedhof_(G%C3%B6ttingen)" title="Stadtfriedhof (Göttingen)">Stadtfriedhof (Göttingen)</a> </figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg/220px-Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg" decoding="async" width="220" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg/330px-Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg/440px-Delegates_to_the_Fourth_Conference_International_Union_for_Cooperation_in_Solar_Research_at_Mount_Wilson_Observatory.jpg 2x" data-file-width="6619" data-file-height="2398" /></a><figcaption>Schwarzschild at the Fourth Conference International Union for Cooperation in Solar Research at <a href="/wiki/Mount_Wilson_Observatory" title="Mount Wilson Observatory">Mount Wilson Observatory</a>, 1910</figcaption></figure> <p>From 1912, Schwarzschild was a member of the <a href="/wiki/Prussian_Academy_of_Sciences" title="Prussian Academy of Sciences">Prussian Academy of Sciences</a>. </p><p>At the outbreak of <a href="/wiki/World_War_I" title="World War I">World War I</a> in 1914, Schwarzschild volunteered for service in the <a href="/wiki/Imperial_German_Army" title="Imperial German Army">German army</a> despite being over 40 years old. He served on both the western and eastern fronts, specifically helping with <a href="/wiki/Ballistics" title="Ballistics">ballistic calculations</a> and rising to the rank of second lieutenant in the artillery.<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>While serving on the front in Russia in 1915, he began to suffer from <a href="/wiki/Pemphigus" title="Pemphigus">pemphigus</a>, a rare and painful autoimmune skin-disease.<sup id="cite_ref-:1_16-0" class="reference"><a href="#cite_note-:1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Nevertheless, he managed to write three outstanding papers, two on the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> and one on <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum theory</a>. His papers on relativity produced the first exact solutions to the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>, and a minor modification of these results gives the well-known solution that now bears his name — the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a>.<sup id="cite_ref-Levy_17-0" class="reference"><a href="#cite_note-Levy-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>In March 1916, Schwarzschild left military service because of his illness and returned to <a href="/wiki/G%C3%B6ttingen" title="Göttingen">Göttingen</a>. Two months later, on May 11, 1916, his struggle with <a href="/wiki/Pemphigus" title="Pemphigus">pemphigus</a> may have led to his death at the age of 42.<sup id="cite_ref-:1_16-1" class="reference"><a href="#cite_note-:1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>He rests in his family grave at the <a href="/wiki/Stadtfriedhof_(G%C3%B6ttingen)" title="Stadtfriedhof (Göttingen)">Stadtfriedhof Göttingen</a>. </p><p>With his wife Else he had three children: </p> <ul><li><a href="/wiki/Agathe_Thornton" title="Agathe Thornton">Agathe Thornton</a> (1910–2006) emigrated to Great Britain in 1933. In 1946, she moved to New Zealand, where she became a classics professor at the <a href="/wiki/University_of_Otago" title="University of Otago">University of Otago</a> in Dunedin.<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></li> <li><a href="/wiki/Martin_Schwarzschild" title="Martin Schwarzschild">Martin Schwarzschild</a> (1912–1997) became a professor of astronomy at <a href="/wiki/Princeton_University" title="Princeton University">Princeton University</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>Alfred Schwarzschild (1914–1944) remained in Nazi Germany and was murdered during the <a href="/wiki/The_Holocaust" title="The Holocaust">Holocaust</a>.<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></li></ul> <div class="mw-heading mw-heading2"><h2 id="Work">Work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=2" title="Edit section: Work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thousands of dissertations, articles, and books have since been devoted to the study of Schwarzschild's solutions to the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. However, although his best known work lies in the area of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, his research interests were extremely broad, including work in <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, observational stellar <a href="/wiki/Photometry_(astronomy)" title="Photometry (astronomy)">photometry</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, instrumental <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, stellar structure, stellar <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Halley%27s_comet" class="mw-redirect" title="Halley's comet">Halley's comet</a>, and <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>.<sup id="cite_ref-Eisenstaedt,_1989_21-0" class="reference"><a href="#cite_note-Eisenstaedt,_1989-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Some of his particular achievements include measurements of <a href="/wiki/Variable_star" title="Variable star">variable stars</a>, using photography, and the improvement of optical systems, through the perturbative investigation of geometrical aberrations. </p> <div class="mw-heading mw-heading3"><h3 id="Physics_of_photography">Physics of photography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=3" title="Edit section: Physics of photography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While at Vienna in 1897, Schwarzschild developed a formula, now known as the <a href="/wiki/Schwarzschild_law" class="mw-redirect" title="Schwarzschild law">Schwarzschild law</a>, to calculate the optical density of photographic material. It involved an exponent now known as the Schwarzschild exponent, which is 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> in the formula: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=f(I\cdot t^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo>⋅</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=f(I\cdot t^{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e97b7577c2d66cd0d65bb1464e2f8850a8b517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.738ex; height:2.843ex;" alt="{\displaystyle i=f(I\cdot t^{p})}"></span> </p><p>(where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is optical density of exposed photographic emulsion, a function 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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>, the intensity of the source being observed, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <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>, the exposure time, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> a constant). This formula was important for enabling more accurate photographic measurements of the intensities of faint astronomical sources. </p> <div class="mw-heading mw-heading3"><h3 id="Electrodynamics">Electrodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=4" title="Edit section: Electrodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</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> Schwarzschild is the first to introduce the correct <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> formalism of the electromagnetic field <sup id="cite_ref-schwarzschil1903_23-0" class="reference"><a href="#cite_note-schwarzschil1903-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=(1/2)\int (H^{2}-E^{2})dV+\int \rho (\phi -{\vec {A}}\cdot {\vec {u}})dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∫</mo> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <mi>V</mi> <mo>+</mo> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=(1/2)\int (H^{2}-E^{2})dV+\int \rho (\phi -{\vec {A}}\cdot {\vec {u}})dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589cfa457a1f5e124638269cfbd4d74d6dbc9523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.547ex; height:5.676ex;" alt="{\displaystyle S=(1/2)\int (H^{2}-E^{2})dV+\int \rho (\phi -{\vec {A}}\cdot {\vec {u}})dV}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}},{\vec {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {E}},{\vec {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a27de7e301460b9d88c4c69904d5d78c5223bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.873ex; height:3.176ex;" alt="{\displaystyle {\vec {E}},{\vec {H}}}"></span> are the electric and applied magnetic fields, <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 {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}"></span> is the vector potential 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 \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 electric potential. </p><p>He also introduced a field free variational formulation of electrodynamics (also known as "action at distance" or "direct interparticle action") based only on the world line of particles as <sup id="cite_ref-schwarzschil1903b_24-0" class="reference"><a href="#cite_note-schwarzschil1903b-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{i}m_{i}\int _{C_{i}}ds_{i}+{\frac {1}{2}}\sum _{i,j}\iint _{C_{i},C_{j}}q_{i}q_{j}\delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)d\mathbf {s} _{i}d\mathbf {s} _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{i}m_{i}\int _{C_{i}}ds_{i}+{\frac {1}{2}}\sum _{i,j}\iint _{C_{i},C_{j}}q_{i}q_{j}\delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)d\mathbf {s} _{i}d\mathbf {s} _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9943a34f11d61616fd369a81e04c6a1d5bdb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.325ex; height:6.676ex;" alt="{\displaystyle S=\sum _{i}m_{i}\int _{C_{i}}ds_{i}+{\frac {1}{2}}\sum _{i,j}\iint _{C_{i},C_{j}}q_{i}q_{j}\delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)d\mathbf {s} _{i}d\mathbf {s} _{j}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd95af722a63223178ee2750f01e310375601c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.946ex; height:2.509ex;" alt="{\displaystyle C_{\alpha }}"></span> are the world lines of the particle, <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\mathbf {s} _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {s} _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d21c4a147df035c229560abc1047ca39aa4f594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.556ex; height:2.509ex;" alt="{\displaystyle d\mathbf {s} _{\alpha }}"></span> the (vectorial) arc element along the world line. Two points on two world lines contribute to the Lagrangian (are coupled) only if they are a zero Minkowskian distance (connected by a light ray), hence the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff33a62564f8cc20113464fe4336fa3790ad50af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.264ex; height:3.009ex;" alt="{\displaystyle \delta \left(\left\Vert P_{i}P_{j}\right\Vert \right)}"></span>. The idea was further developed by <a href="/wiki/Hugo_Tetrode" title="Hugo Tetrode">Hugo Tetrode</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Adriaan_Fokker" title="Adriaan Fokker">Adriaan Fokker</a><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> in the 1920s and <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> and <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> in the 1940s <sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and constitutes an alternative but equivalent formulation of electrodynamics. </p> <div class="mw-heading mw-heading3"><h3 id="Relativity">Relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=5" title="Edit section: Relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Schwarzschild_circular_radii.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Schwarzschild_circular_radii.svg/250px-Schwarzschild_circular_radii.svg.png" decoding="async" width="250" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Schwarzschild_circular_radii.svg/375px-Schwarzschild_circular_radii.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Schwarzschild_circular_radii.svg/500px-Schwarzschild_circular_radii.svg.png 2x" data-file-width="531" data-file-height="560" /></a><figcaption>The <a href="/wiki/Kepler_problem" title="Kepler problem">Kepler problem</a> in general relativity, using the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a></figcaption></figure> <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">Main article: <a href="/wiki/Deriving_the_Schwarzschild_solution" class="mw-redirect" title="Deriving the Schwarzschild solution">Deriving the Schwarzschild solution</a></div> <p>Einstein himself was pleasantly surprised to learn that the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">field equations</a> admitted exact solutions, because of their <i>prima facie</i> complexity, and because he himself had produced only an approximate solution.<sup id="cite_ref-Levy_17-1" class="reference"><a href="#cite_note-Levy-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Einstein's approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury. There, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant "polar-like" coordinate system and was able to produce an exact solution which he first set down in a letter to Einstein of 22 December 1915, written while he was serving in the war stationed on the Russian front. He concluded the letter by writing: "As you see, the war is kindly disposed toward me, allowing me, despite fierce gunfire at a decidedly terrestrial distance, to take this walk into this your land of ideas."<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> In 1916, Einstein wrote to Schwarzschild on this result: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation.</p><div class="templatequotecite">— <cite><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <sup id="cite_ref-Eisenstaedt,_1989_21-1" class="reference"><a href="#cite_note-Eisenstaedt,_1989-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Schwarzschild_interior.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Schwarzschild_interior.jpg/250px-Schwarzschild_interior.jpg" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Schwarzschild_interior.jpg/375px-Schwarzschild_interior.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Schwarzschild_interior.jpg/500px-Schwarzschild_interior.jpg 2x" data-file-width="523" data-file-height="411" /></a><figcaption>Boundary region of Schwarzschild interior and exterior solution</figcaption></figure> <p>Schwarzschild's second paper, which gives what is now known as the "Inner Schwarzschild solution" (in German: "innere Schwarzschild-Lösung"), is valid within a sphere of homogeneous and isotropic distributed molecules within a shell of radius r=R. It is applicable to solids; incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas. </p><p>Schwarzschild's first (spherically symmetric) solution <i>does not</i> contain a coordinate <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> on a surface that is now named after him. In his coordinates, this singularity lies on the sphere of points at a particular radius, called the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mi>M</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3b26bac5d5a787fe4aa7a2b7057ae07dd6eca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.133ex; height:5.676ex;" alt="{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}"></span></dd></dl> <p>where <i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>, <i>M</i> is the mass of the central body, and <i>c</i> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum.<sup id="cite_ref-landau_1975_29-0" class="reference"><a href="#cite_note-landau_1975-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> In cases where the radius of the central body is less than 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_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19598a1629ee9016913aae8f0d3215dc4a030e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.767ex; height:2.509ex;" alt="{\displaystyle R_{s}}"></span> represents the radius within which all massive bodies, and even <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>, must inevitably fall into the central body (ignoring <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">quantum tunnelling</a> effects near the boundary). When the mass density of this central body exceeds a particular limit, it triggers a gravitational collapse which, if it occurs with spherical symmetry, produces what is known as a Schwarzschild <a href="/wiki/Black_hole" title="Black hole">black hole</a>. This occurs, for example, when the mass of a <a href="/wiki/Neutron_star" title="Neutron star">neutron star</a> exceeds the <a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit" title="Tolman–Oppenheimer–Volkoff limit">Tolman–Oppenheimer–Volkoff limit</a> (about three solar masses). </p> <div class="mw-heading mw-heading2"><h2 id="Cultural_references">Cultural references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=6" title="Edit section: Cultural references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Karl Schwarzschild appears as a character in the science fiction short story "Schwarzschild Radius" (1987) by <a href="/wiki/Connie_Willis" title="Connie Willis">Connie Willis</a>. </p><p>Karl Schwarzchild appears as a fictionalized character in the story “Schwarzchild’s Singularity” in the collection "When We Cease to Understand the World" (2020) by <a href="/wiki/Benjam%C3%ADn_Labatut" title="Benjamín Labatut">Benjamín Labatut</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Works">Works</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=7" title="Edit section: Works"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The entire scientific estate of Karl Schwarzschild is stored in a special collection of the <a rel="nofollow" class="external text" href="https://www.sub.uni-goettingen.de/sub-aktuell/">Lower Saxony National- and University Library of Göttingen</a>. </p><p><b>Relativity</b> </p> <ul><li><i><a href="https://en.wikisource.org/wiki/de:%C3%9Cber_das_Gravitationsfeld_eines_Massenpunktes_nach_der_Einsteinschen_Theorie" class="extiw" title="s:de:Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie">Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie.</a></i> Reimer, Berlin 1916, S. 189 ff. (Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften; 1916)</li> <li><i>Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit.</i> Reimer, Berlin 1916, S. 424-434 (Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften; 1916)</li></ul> <p><b>Other papers</b> </p> <ul><li><i>Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs</i>, 1906, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004">Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4</a>, Nummero 1, S. 1-31</li> <li><i>Untersuchungen zur geometrischen Optik II. Theorie der Spiegelteleskope</i>, 1906, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004">Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4</a>, Nummero 2, S. 1-28</li> <li><i>Untersuchungen zur geometrischen Optik III. Über die astrophotographischen Objektive</i>, 1906, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0004">Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 4</a>, Nummero 3, S. 1-54</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58230">Über Differenzformeln zur Durchrechnung optischer Systeme</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1907">1907, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 551-570</li> <li><i>Aktinometrie der Sterne der B. D. bis zur Größe 7.5 in der Zone 0° bis +20° Deklination. Teil A. Unter Mitwirkung von Br. Meyermann, A. Kohlschütter und O. Birck</i>, 1910, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN251726223_0006">Abhandlungen der Gesellschaft der Wissenschaften in Göttingen, Band 6</a>, Numero 6, S. 1-117</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58114">Über das Gleichgewicht der Sonnenatmosphäre</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1906">1906, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 41-53</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D36450">Die Beugung und Polarisation des Lichts durch einen Spalt. I.</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, 1902, Mathematische Annalen, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0055">Band 55</a>, S. 177-247</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57298">Zur Elektrodynamik. I. Zwei Formen des Princips der Action in der Elektronentheorie</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903">1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 126-131</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57299">Zur Elektrodynamik. II. Die elementare elektrodynamische Kraft</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903">1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 132-141</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D57299">Zur Elektrodynamik. III. Ueber die Bewegung des Elektrons</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1903">1903, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 245-278</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D58241">Ueber die Eigenbewegungen der Fixsterne</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1907">1907, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 614-632</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D62941">Ueber die Bestimmung von Vertex und Apex nach der Ellipsoidhypothese aus einer geringeren Anzahl beobachteter Eigenbewegungen</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1908">1908, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 191-200</li> <li>K. Schwarzschild, E. Kron: <i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D63058">Ueber die Helligkeitsverteilung im Schweif des Halley´schen Kometen</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN252457811_1911">1911, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</a>, S. 197-208</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D244900">Die naturwissenschaftlichen Ergebnisse und Ziele der neueren Mechanik.</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, 1904, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X_0013">Jahresbericht der Deutschen Mathematiker-Vereinigung, Band 13</a>, S. 145-156</li> <li><i><a rel="nofollow" class="external text" href="http://dz-srv1.sub.uni-goettingen.de/contentserver/contentserver?command=docconvert&docid=D244507">Über die astronomische Ausbildung der Lehramtskandidaten.</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged December 2017">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></i>, 1907, <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X_0016">Jahresbericht der Deutschen Mathematiker-Vereinigung, Band 16</a>, S. 519-522</li></ul> <p><b>English translations</b> </p> <ul><li><i>On the Gravitational Field of a Point-Mass, According to Einstein's Theory</i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220331132918/http://zelmanov.ptep-online.com/papers/zj-2008-03.pdf">The Abraham Zelmanov Journal, 2008, Volume 1, P. 10-19</a></li> <li><i>On the Gravitational Field of a Sphere of Incompressible Liquid, According to Einstein's Theory</i>, <a rel="nofollow" class="external text" href="http://zelmanov.ptep-online.com/papers/zj-2008-04.pdf">The Abraham Zelmanov Journal, 2008, Volume 1, P. 20-32</a></li> <li><i>On the Permissible Numerical Value of the Curvature of Space</i>, <a rel="nofollow" class="external text" href="http://zelmanov.ptep-online.com/papers/zj-2008-06.pdf">The Abraham Zelmanov Journal, Volume 1, 2008, pp. 64-73</a></li></ul> <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=Karl_Schwarzschild&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_things_named_after_Karl_Schwarzschild" title="List of things named after Karl Schwarzschild">List of things named after Karl Schwarzschild</a></li></ul> <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=Karl_Schwarzschild&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf">Biography of Karl Schwarzschild</a> by Indranu Suhendro, <i>The Abraham Zelmanov Journal</i>, 2008, Volume 1.</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"><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="CITEREFSnygg2012" class="citation book cs1">Snygg, John (2012). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-0-8176-8283-5"><i>A new approach to differential geometry using Clifford's geometric algebra</i></a>. New York: Springer Science. p. 400. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-8283-5">10.1007/978-0-8176-8283-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-8283-5" title="Special:BookSources/978-0-8176-8283-5"><bdi>978-0-8176-8283-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+new+approach+to+differential+geometry+using+Clifford%27s+geometric+algebra&rft.place=New+York&rft.pages=400&rft.pub=Springer+Science&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-8283-5&rft.isbn=978-0-8176-8283-5&rft.aulast=Snygg&rft.aufirst=John&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-0-8176-8283-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" 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="CITEREFAhsan2015" class="citation book cs1">Ahsan, Zafar (2015). <i>Tensors : mathematics of differential geometry and relativity</i>. 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Frankfurt am Main. 2020. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-95542-379-7" title="Special:BookSources/978-3-95542-379-7"><bdi>978-3-95542-379-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1244019080">1244019080</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%22Nachforschung+der+Wahrheit%22+von+der+alten+Lateinschule+zum+Lessing-Gymnasium+in+Frankfurt+am+Main+%3A+Festschrift+zum+500-j%C3%A4hrigen+Jubil%C3%A4um+der+Schule&rft.place=Frankfurt+am+Main&rft.date=2020&rft_id=info%3Aoclcnum%2F1244019080&rft.isbn=978-3-95542-379-7&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F1244019080&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</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="CITEREFSchwarzschild1992" class="citation cs2">Schwarzschild, Karl (1992), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-642-58086-4_2">"Karl Schwarzschild Lectures"</a>, <i>Gesammelte Werke Collected Works</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 29–42, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-58086-4_2">10.1007/978-3-642-58086-4_2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-63467-3" title="Special:BookSources/978-3-642-63467-3"><bdi>978-3-642-63467-3</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2021-05-18</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Gesammelte+Werke+Collected+Works&rft.atitle=Karl+Schwarzschild+Lectures&rft.pages=29-42&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2F978-3-642-58086-4_2&rft.isbn=978-3-642-63467-3&rft.aulast=Schwarzschild&rft.aufirst=Karl&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-642-58086-4_2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1108/rr-08-2015-0205">"MacTutor History of Mathematics Archive"</a>. <i>Reference Reviews</i>. <b>30</b> (1): 27–28. 2016-01-18. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1108%2Frr-08-2015-0205">10.1108/rr-08-2015-0205</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/0950-4125">0950-4125</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reference+Reviews&rft.atitle=MacTutor+History+of+Mathematics+Archive&rft.volume=30&rft.issue=1&rft.pages=27-28&rft.date=2016-01-18&rft_id=info%3Adoi%2F10.1108%2Frr-08-2015-0205&rft.issn=0950-4125&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1108%2Frr-08-2015-0205&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/981916699"><i>Karl Schwarzschild (1873-1916) ein Pionier und Wegbereiter der Astrophysik</i></a>. Klaus Reinsch, Axel Wittmann, Universitätsverlag Göttingen. Göttingen. 2017. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-86395-295-2" title="Special:BookSources/978-3-86395-295-2"><bdi>978-3-86395-295-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/981916699">981916699</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Karl+Schwarzschild+%281873-1916%29+ein+Pionier+und+Wegbereiter+der+Astrophysik&rft.place=G%C3%B6ttingen&rft.date=2017&rft_id=info%3Aoclcnum%2F981916699&rft.isbn=978-3-86395-295-2&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F981916699&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHertzsprung1917" class="citation journal cs1">Hertzsprung, Ejnar (June 1917). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F142329">"Karl Schwarzschild"</a>. <i>The Astrophysical Journal</i>. <b>45</b>: 285. <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/1917ApJ....45..285H">1917ApJ....45..285H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F142329">10.1086/142329</a></span>. <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/0004-637X">0004-637X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Karl+Schwarzschild&rft.volume=45&rft.pages=285&rft.date=1917-06&rft.issn=0004-637X&rft_id=info%3Adoi%2F10.1086%2F142329&rft_id=info%3Abibcode%2F1917ApJ....45..285H&rft.aulast=Hertzsprung&rft.aufirst=Ejnar&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F142329&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwarzschild1992" class="citation cs2">Schwarzschild, Karl (1992), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-642-58086-4_1">"Biography of Karl Schwarzschild (1873-1916)"</a>, <i>Gesammelte Werke Collected Works</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 1–28, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-58086-4_1">10.1007/978-3-642-58086-4_1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-63467-3" title="Special:BookSources/978-3-642-63467-3"><bdi>978-3-642-63467-3</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2021-05-18</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Gesammelte+Werke+Collected+Works&rft.atitle=Biography+of+Karl+Schwarzschild+%281873-1916%29&rft.pages=1-28&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2F978-3-642-58086-4_1&rft.isbn=978-3-642-63467-3&rft.aulast=Schwarzschild&rft.aufirst=Karl&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-642-58086-4_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-:1-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.physicsoftheuniverse.com/scientists_schwarzschild.html">"Karl Schwarzschild - Important Scientists - The Physics of the Universe"</a>. <i>www.physicsoftheuniverse.com</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">25 March</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Knowable+Magazine&rft.atitle=How+black+holes+morphed+from+theory+to+reality&rft.date=2021-01-11&rft_id=info%3Adoi%2F10.1146%2Fknowable-010921-1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250662997%23id-name%3DS2CID&rft.aulast=Levy&rft.aufirst=Adam&rft_id=https%3A%2F%2Fknowablemagazine.org%2Farticle%2Fphysical-world%2F2021%2Fhow-black-holes-morphed-theory-reality&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamTaonga" class="citation web cs1 cs1-prop-foreign-lang-source">Graham, Reg; Taonga, New Zealand Ministry for Culture and Heritage Te Manatu. <a rel="nofollow" class="external text" href="https://teara.govt.nz/mi/photograph/43514/agathe-thornton">"Agathe Thornton"</a>. <i>teara.govt.nz</i> (in Māori)<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=teara.govt.nz&rft.atitle=Agathe+Thornton&rft.aulast=Graham&rft.aufirst=Reg&rft.au=Taonga%2C+New+Zealand+Ministry+for+Culture+and+Heritage+Te+Manatu&rft_id=https%3A%2F%2Fteara.govt.nz%2Fmi%2Fphotograph%2F43514%2Fagathe-thornton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://pr.princeton.edu/news/97/q2/0411schw.html">"Princeton - News - Princeton Astrophysicist Martin Schwarzschild Dies"</a>. <i>pr.princeton.edu</i><span class="reference-accessdate">. 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Springer. p. 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783319200460" title="Special:BookSources/9783319200460"><bdi>9783319200460</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=1st+Karl+Schwarzschild+Meeting+on+Gravitational+Physics&rft.pages=10&rft.pub=Springer&rft.date=2015&rft.isbn=9783319200460&rft.aulast=Nicolini&rft.aufirst=Piero&rft.au=Kaminski%2C+Matthias&rft.au=Mureika%2C+Jonas&rft.au=Bleicher%2C+Marcus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6B_UCgAAQBAJ%26dq%3Dalfred%2Bschwarzschild%2Bholocaust%2Bson%2Bof%2Bkarl%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-Eisenstaedt,_1989-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Eisenstaedt,_1989_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Eisenstaedt,_1989_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Eisenstaedt, “The Early Interpretation of the Schwarzschild Solution,” in D. Howard and J. Stachel (eds), Einstein and the History of General Relativity: Einstein Studies, Vol. 1, pp. 213-234. Boston: Birkhauser, 1989.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Pauli, W.. Theory of Relativity. United States, Dover Publications, 2013.</span> </li> <li id="cite_note-schwarzschil1903-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-schwarzschil1903_23-0">^</a></b></span> <span class="reference-text">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 125</span> </li> <li id="cite_note-schwarzschil1903b-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-schwarzschil1903b_24-0">^</a></b></span> <span class="reference-text">K. Schwarzschild, Nachr. ges. Wiss. Gottingen (1903) 128,132</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">H. Tetrode, Zeitschrift für Physik 10:137, 1922</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">A. D. Fokker, Zeitschrift für Physik 58:386, 1929</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheelerFeynman1949" class="citation journal cs1">Wheeler, John Archibald; Feynman, Richard Phillips (1949-07-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.21.425">"Classical Electrodynamics in Terms of Direct Interparticle Action"</a>. <i>Reviews of Modern Physics</i>. <b>21</b> (3): 425–433. <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/1949RvMP...21..425W">1949RvMP...21..425W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.21.425">10.1103/RevModPhys.21.425</a></span>. <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/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Classical+Electrodynamics+in+Terms+of+Direct+Interparticle+Action&rft.volume=21&rft.issue=3&rft.pages=425-433&rft.date=1949-07-01&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.21.425&rft_id=info%3Abibcode%2F1949RvMP...21..425W&rft.aulast=Wheeler&rft.aufirst=John+Archibald&rft.au=Feynman%2C+Richard+Phillips&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FRevModPhys.21.425&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKarl+Schwarzschild" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://einsteinpapers.press.princeton.edu/vol8-trans/191?highlightText=schwarzschild">Letter from K Schwarzschild to A Einstein dated 22 December 1915</a>, in "The Collected Papers of Albert Einstein, Volume 8: The Berlin Years: Correspondence, 1914-1918 (English translation supplement)", Translated by Ann M. Hentschel, vol.8a, doc.#169.</span> </li> <li id="cite_note-landau_1975-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-landau_1975_29-0">^</a></b></span> <span class="reference-text">Landau 1975.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Karl_Schwarzschild&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Karl_Schwarzschild" class="extiw" title="commons:Category:Karl Schwarzschild">Karl Schwarzschild</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist">German <a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has original text related to this article: <div lang="de" style="margin-left: 10px;"><b><a href="https://en.wikisource.org/wiki/de:Karl_Schwarzschild" class="extiw" title="s:de:Karl Schwarzschild">Works by Karl Schwarzschild</a></b> </div></div></div> </div> <ul><li><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%3AKarl+Schwarzschild" class="Z3988"></span></li> <li>Roberto B. Salgado <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060617222412/http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html">The Light Cone: The Schwarzschild Black Hole</a></li> <li><a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1917ApJ....45..285H">Obituary in the Astrophysical Journal</a>, written by <a href="/wiki/Ejnar_Hertzsprung" title="Ejnar Hertzsprung">Ejnar Hertzsprung</a></li> <li><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=52349">Karl Schwarzschild</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></li> <li><a rel="nofollow" class="external text" href="http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf">Biography of Karl Schwarzschild</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210302153030/http://zelmanov.ptep-online.com/papers/zj-2008-b3.pdf">Archived</a> 2021-03-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by Indranu Suhendro, <i>The Abraham Zelmanov Journal</i>, 2008, Volume 1.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist 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equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a class="mw-selflink selflink">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> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 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.portal-bar+.navbox-styles+.sister-bar{margin-top:-1px}</style><div class="portal-bar noprint metadata noviewer portal-bar-bordered" role="navigation" aria-label="Portals"><span class="portal-bar-header"><a href="/wiki/Wikipedia:Contents/Portals" title="Wikipedia:Contents/Portals">Portals</a>:</span><ul class="portal-bar-content"><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/19px-P_vip.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/28px-P_vip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/69/P_vip.svg/37px-P_vip.svg.png 2x" data-file-width="1911" data-file-height="1944" /></span></span> </span><a href="/wiki/Portal:Biography" title="Portal:Biography">Biography</a></li><li class="portal-bar-item"><span class="nowrap"><span class="mw-image-border" typeof="mw:File"><span><img alt="flag" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Flag_of_Germany.svg/21px-Flag_of_Germany.svg.png" decoding="async" width="21" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Flag_of_Germany.svg/32px-Flag_of_Germany.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Flag_of_Germany.svg/42px-Flag_of_Germany.svg.png 2x" data-file-width="1000" data-file-height="600" /></span></span> </span><a href="/wiki/Portal:Germany" title="Portal:Germany">Germany</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/19px-Crab_Nebula.jpg" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/29px-Crab_Nebula.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/38px-Crab_Nebula.jpg 2x" data-file-width="3864" data-file-height="3864" /></span></span> </span><a href="/wiki/Portal:Astronomy" title="Portal:Astronomy">Astronomy</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><a href="/wiki/File:He1523a.jpg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/16px-He1523a.jpg" decoding="async" width="16" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/25px-He1523a.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/33px-He1523a.jpg 2x" data-file-width="180" data-file-height="207" /></a></span> </span><a href="/wiki/Portal:Stars" title="Portal:Stars">Stars</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/19px-RocketSunIcon.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/29px-RocketSunIcon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/38px-RocketSunIcon.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> </span><a href="/wiki/Portal:Spaceflight" title="Portal:Spaceflight">Spaceflight</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/21px-Earth-moon.jpg" decoding="async" width="21" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/32px-Earth-moon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/42px-Earth-moon.jpg 2x" data-file-width="3000" data-file-height="2400" /></span></span> </span><a href="/wiki/Portal:Outer_space" title="Portal:Outer space">Outer space</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/15px-Solar_system.jpg" decoding="async" width="15" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/23px-Solar_system.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/30px-Solar_system.jpg 2x" data-file-width="4500" data-file-height="5600" /></span></span> </span><a href="/wiki/Portal:Solar_System" title="Portal:Solar System">Solar System</a></li></ul></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 authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q57145#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" 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="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q57145#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q57145#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" 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href="https://viaf.org/viaf/22226062">VIAF</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/194633/">FAST</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.oclc.org/worldcat/entity/E39PBJcwCpYVcBVGQvXvTqmyBP">WorldCat</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/119051524">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/n87810561">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb123988975">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb123988975">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=mub2018994596&CON_LNG=ENG">Czech Republic</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://data.bibliotheken.nl/id/thes/p078246008">Netherlands</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://authority.bibsys.no/authority/rest/authorities/html/97043908">Norway</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000244554&P_CON_LNG=ENG">Latvia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://dbn.bn.org.pl/descriptor-details/9810627564505606">Poland</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007272232005171">Israel</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://cantic.bnc.cat/registre/981058527772106706">Catalonia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://opac.kbr.be/LIBRARY/doc/AUTHORITY/14325169">Belgium</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://ci.nii.ac.jp/author/DA06595421?l=en">CiNii</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.mathgenealogy.org/id.php?id=52349">Mathematics Genealogy Project</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://zbmath.org/authors/?q=ai:schwarzschild.karl">zbMATH</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/MRAuthorID/640869">MathSciNet</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.leopoldina.org/mitgliederverzeichnis/mitglieder/member/Member/show/karl-schwarzschild/">Leopoldina</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.deutsche-biographie.de/pnd119051524.html?language=en">Deutsche Biographie</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.deutsche-digitale-bibliothek.de/person/gnd/119051524">DDB</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/033087067">IdRef</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://snaccooperative.org/ark:/99166/w6ms3vd6">SNAC</a></span></li></ul></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=Karl_Schwarzschild&oldid=1251530029">https://en.wikipedia.org/w/index.php?title=Karl_Schwarzschild&oldid=1251530029</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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