Kerr metric - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Kerr metric - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"ed1969b3-a0ae-4951-9301-5d117f04067e","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Kerr_metric","wgTitle":"Kerr metric","wgCurRevisionId":1277991679,"wgRevisionId":1277991679,"wgArticleId":456715,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description matches Wikidata","Articles needing additional references from February 2011","All articles needing additional references","Exact solutions in general relativity","Black holes","Metric tensors"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Kerr_metric","wgRelevantArticleId":456715,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1068747","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.22"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Kerr metric - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Kerr_metric"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Kerr_metric&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Kerr_metric"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Kerr_metric rootpage-Kerr_metric skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Kerr+metric" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Kerr+metric" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Kerr+metric" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Kerr+metric" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Overview</span> </div> </a> <ul id="toc-Overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Metric"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Metric</span> </div> </a> <button aria-controls="toc-Metric-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Metric subsection</span> </button> <ul id="toc-Metric-sublist" class="vector-toc-list"> <li id="toc-Boyer–Lindquist_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Boyer–Lindquist_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Boyer–Lindquist coordinates</span> </div> </a> <ul id="toc-Boyer–Lindquist_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kerr–Schild_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kerr–Schild_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kerr–Schild coordinates</span> </div> </a> <ul id="toc-Kerr–Schild_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Soliton_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Soliton_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Soliton coordinates</span> </div> </a> <ul id="toc-Soliton_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mass_of_rotational_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mass_of_rotational_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mass of rotational energy</span> </div> </a> <ul id="toc-Mass_of_rotational_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wave_operator" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wave_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Wave operator</span> </div> </a> <ul id="toc-Wave_operator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frame_dragging" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frame_dragging"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Frame dragging</span> </div> </a> <ul id="toc-Frame_dragging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Important_surfaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Important_surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Important surfaces</span> </div> </a> <ul id="toc-Important_surfaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ergosphere_and_the_Penrose_process" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ergosphere_and_the_Penrose_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Ergosphere and the Penrose process</span> </div> </a> <ul id="toc-Ergosphere_and_the_Penrose_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Features" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Features"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Features</span> </div> </a> <button aria-controls="toc-Features-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Features subsection</span> </button> <ul id="toc-Features-sublist" class="vector-toc-list"> <li id="toc-Trajectory_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trajectory_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Trajectory equations</span> </div> </a> <ul id="toc-Trajectory_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symmetries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Symmetries</span> </div> </a> <ul id="toc-Symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Overextreme_Kerr_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Overextreme_Kerr_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Overextreme Kerr solutions</span> </div> </a> <ul id="toc-Overextreme_Kerr_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kerr_black_holes_as_wormholes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kerr_black_holes_as_wormholes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Kerr black holes as wormholes</span> </div> </a> <ul id="toc-Kerr_black_holes_as_wormholes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anti-universe_region" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Anti-universe_region"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Anti-universe region</span> </div> </a> <button aria-controls="toc-Anti-universe_region-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Anti-universe region subsection</span> </button> <ul id="toc-Anti-universe_region-sublist" class="vector-toc-list"> <li id="toc-The_ring_singularity_and_beyond" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_ring_singularity_and_beyond"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>The ring singularity and beyond</span> </div> </a> <ul id="toc-The_ring_singularity_and_beyond-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_timelike_curves_and_the_Cauchy_horizon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_timelike_curves_and_the_Cauchy_horizon"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>Closed timelike curves and the Cauchy horizon</span> </div> </a> <ul id="toc-Closed_timelike_curves_and_the_Cauchy_horizon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_other_exact_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_other_exact_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Relation to other exact solutions</span> </div> </a> <ul id="toc-Relation_to_other_exact_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multipole_moments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multipole_moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Multipole moments</span> </div> </a> <ul id="toc-Multipole_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Open_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Open_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Open problems</span> </div> </a> <ul id="toc-Open_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kerr metric</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 22 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-22" 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">22 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B1%D9%8A%D8%A9_%D9%83%D9%8A%D8%B1" title="مترية كير – Arabic" lang="ar" hreflang="ar" data-title="مترية كير" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A8trica_de_Kerr" title="Mètrica de Kerr – Catalan" lang="ca" hreflang="ca" data-title="Mètrica de Kerr" 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/Kerrova_metrika" title="Kerrova metrika – Czech" lang="cs" hreflang="cs" data-title="Kerrova metrika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kerr-Metrik" title="Kerr-Metrik – German" lang="de" hreflang="de" data-title="Kerr-Metrik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AA%D8%B1%DB%8C%DA%A9_%DA%A9%D8%B1" title="متریک کر – Persian" lang="fa" hreflang="fa" data-title="متریک کر" 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/Trou_noir_de_Kerr#Métrique_de_Kerr" title="Trou noir de Kerr – French" lang="fr" hreflang="fr" data-title="Trou noir de Kerr" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BB%A4_%EA%B3%84%EB%9F%89" title="커 계량 – Korean" lang="ko" hreflang="ko" data-title="커 계량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Metrica_de_Kerr" title="Metrica de Kerr – Interlingua" lang="ia" hreflang="ia" data-title="Metrica de Kerr" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metrica_di_Kerr" title="Metrica di Kerr – Italian" lang="it" hreflang="it" data-title="Metrica di Kerr" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kerr-metrika" title="Kerr-metrika – Hungarian" lang="hu" hreflang="hu" data-title="Kerr-metrika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%86%E0%B5%BC_%E0%B4%AE%E0%B5%86%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8D" title="കെർ മെട്രിക് – Malayalam" lang="ml" hreflang="ml" data-title="കെർ മെട്രിക്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kerrmetriek" title="Kerrmetriek – Dutch" lang="nl" hreflang="nl" data-title="Kerrmetriek" 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%E8%A7%A3" title="カー解 – Japanese" lang="ja" hreflang="ja" data-title="カー解" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metryka_Kerra" title="Metryka Kerra – Polish" lang="pl" hreflang="pl" data-title="Metryka Kerra" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A9trica_de_Kerr" title="Métrica de Kerr – Portuguese" lang="pt" hreflang="pt" data-title="Métrica de Kerr" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0_%D0%9A%D0%B5%D1%80%D1%80%D0%B0" title="Метрика Керра – Russian" lang="ru" hreflang="ru" data-title="Метрика Керра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kerrova_metrika" title="Kerrova metrika – Slovak" lang="sk" hreflang="sk" data-title="Kerrova metrika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kerrmetrik" title="Kerrmetrik – Swedish" lang="sv" hreflang="sv" data-title="Kerrmetrik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0_%D0%9A%D0%B5%D1%80%D1%80%D0%B0" title="Метрика Керра – Ukrainian" lang="uk" hreflang="uk" data-title="Метрика Керра" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%DB%8C%D8%B1_%D9%85%DB%8C%D9%B9%D8%B1%DA%A9" title="کیر میٹرک – Urdu" lang="ur" hreflang="ur" data-title="کیر میٹرک" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%AAtric_Kerr" title="Mêtric Kerr – Vietnamese" lang="vi" hreflang="vi" data-title="Mêtric Kerr" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%8B%E7%88%BE%E5%BA%A6%E8%A6%8F" title="克爾度規 – Chinese" lang="zh" hreflang="zh" data-title="克爾度規" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1068747#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kerr_metric" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Kerr_metric" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kerr_metric"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kerr_metric&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kerr_metric&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Kerr_metric"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kerr_metric&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kerr_metric&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Kerr_metric" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Kerr_metric" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Kerr_metric&oldid=1277991679" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Kerr_metric&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Kerr_metric&id=1277991679&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKerr_metric"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKerr_metric"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Kerr_metric&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Kerr_metric&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1068747" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Exact solution for the Einstein field equations</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><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><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:Spacetime_lattice_analogy.svg" class="mw-file-description" title="Spacetime curvature schematic"><img alt="Spacetime curvature schematic" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/220px-Spacetime_lattice_analogy.svg.png" decoding="async" width="220" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/330px-Spacetime_lattice_analogy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/440px-Spacetime_lattice_analogy.svg.png 2x" data-file-width="1260" data-file-height="469" /></a></span><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}" /></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a class="mw-selflink selflink">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/20px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="13" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:General_relativity" title="Category:General relativity">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Kerr metric</b> or <b>Kerr geometry</b> describes the geometry of empty <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> around a rotating uncharged <a href="/wiki/Axially_symmetric" class="mw-redirect" title="Axially symmetric">axially symmetric</a> <a href="/wiki/Black_hole" title="Black hole">black hole</a> with a quasispherical <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a>. The Kerr <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> is an <a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">exact solution</a> of 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>; these equations are highly <a href="/wiki/Nonlinear_system" title="Nonlinear system">non-linear</a>, which makes exact solutions very difficult to find. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a <i>charged</i>, spherical, non-rotating body, the <a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström metric</a>, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, <i>rotating</i> black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by <a href="/wiki/Roy_Kerr" title="Roy Kerr">Roy Kerr</a>.<sup id="cite_ref-kerr_1963_1-0" class="reference"><a href="#cite_note-kerr_1963-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-melia2009_2-0" class="reference"><a href="#cite_note-melia2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 69–81">: 69–81 </span></sup> The natural extension to a charged, rotating black hole, the <a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman metric</a>, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table, where <i>Q</i> represents the body's <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> and <i>J</i> represents its spin <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>: </p> <dl><dd><table class="wikitable"> <tbody><tr> <th> </th> <th>Non-rotating (<i>J</i> = 0) </th> <th>Rotating (<i>J</i> ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span>) </th></tr> <tr> <th>Uncharged (<i>Q</i> = 0) </th> <td><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> </td> <td>Kerr </td></tr> <tr> <th>Charged (<i>Q</i> ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /></span>) </th> <td><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a> </td> <td><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a> </td></tr> </tbody></table></dd></dl> <p>According to the Kerr metric, a rotating body should exhibit <a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> (also known as <a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>), a distinctive prediction of general relativity. The first measurement of this frame dragging effect was done in 2011 by the <a href="/wiki/Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> experiment. Roughly speaking, this effect predicts that objects coming close to a rotating mass will be entrained to participate in its rotation, not because of any applied force or torque that can be felt, but rather because of the swirling curvature of spacetime itself associated with rotating bodies. In the case of a rotating black hole, at close enough distances, all objects – even light – <i>must</i> rotate with the black hole; the region where this holds is called the <a href="/wiki/Ergosphere" title="Ergosphere">ergosphere</a>. </p><p>The light from distant sources can travel around the event horizon several times (if close enough); <a href="/wiki/Strong_gravitational_lensing" title="Strong gravitational lensing">creating multiple images of the same object</a>. To a distant viewer, the apparent perpendicular distance between images decreases at a factor of <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><span class="texhtml mvar" style="font-style:italic;">e</span></a><sup>2<a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a></sup> (about 500). However, fast spinning black holes have less distance between multiplicity images.<sup id="cite_ref-Sneppen_3-0" class="reference"><a href="#cite_note-Sneppen-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Rotating black holes have surfaces where the metric seems to have apparent <a href="/wiki/Coordinate_singularity" title="Coordinate singularity">singularities</a>; the size and shape of these surfaces depends on the black hole's <a href="/wiki/Mass" title="Mass">mass</a> and angular momentum. The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. A similar situation obtains when considering the Schwarzschild metric which also appears to result in a singularity at <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2104ecc5278f6cce8963d5d5ab12b5f132f0e470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:2.009ex;" alt="{\displaystyle r=r_{\text{s}}}" /></span>⁠</span> dividing the space above and below <i>r</i><sub>s</sub> into two disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see <i><a href="/wiki/Schwarzschild_metric#Singularities_and_black_holes" title="Schwarzschild metric">Schwarzschild metric § Singularities and black holes</a></i>) – such a coordinate transformation eliminates the apparent singularity where the inner and outer surfaces meet. Objects between these two surfaces must co-rotate with the rotating black hole, as noted above; this feature can in principle be used to extract energy from a rotating black hole, up to its <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a> energy, <i>Mc</i><sup>2</sup>. </p><p>The LIGO experiment that first detected gravitational waves, announced in 2016, also provided the <a href="/wiki/First_observation_of_gravitational_waves" title="First observation of gravitational waves">first direct observation</a> of a pair of Kerr black holes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Metric">Metric</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=2" title="Edit section: Metric"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr metric is commonly expressed in one of two forms, the Boyer–Lindquist form and the Kerr–Schild form. It can be readily derived from the Schwarzschild metric, using the <a href="/wiki/Newman%E2%80%93Janis_algorithm" title="Newman–Janis algorithm">Newman–Janis algorithm</a><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> by <a href="/wiki/Newman%E2%80%93Penrose_formalism" title="Newman–Penrose formalism">Newman–Penrose formalism</a> (also known as the spin–coefficient formalism),<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> <a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> or Ellipsoid coordinate transformation.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Boyer–Lindquist_coordinates"><span id="Boyer.E2.80.93Lindquist_coordinates"></span>Boyer–Lindquist coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=3" title="Edit section: Boyer–Lindquist coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Boyer%E2%80%93Lindquist_coordinates" title="Boyer–Lindquist coordinates">Boyer–Lindquist coordinates</a></div> <p>The Kerr metric describes the geometry of spacetime in the vicinity of a mass <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> rotating with angular momentum <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" /></span>⁠</span>.<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 metric (or equivalently its <a href="/wiki/Line_element" title="Line element">line element</a> for <a href="/wiki/Proper_time" title="Proper time">proper time</a>) in <a href="/wiki/Boyer%E2%80%93Lindquist_coordinates" title="Boyer–Lindquist coordinates">Boyer–Lindquist coordinates</a> is<sup id="cite_ref-zanotti_11-0" class="reference"><a href="#cite_note-zanotti-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-tapir26_12-0" class="reference"><a href="#cite_note-tapir26-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}ds^{2}&=-c^{2}d\tau ^{2}\\&=-\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)c^{2}dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}-{\frac {2r_{\text{s}}ra\sin ^{2}\theta }{\Sigma }}c\,dt\,d\phi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Σ</mi> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Σ</mi> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <mi>a</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <mi>c</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}ds^{2}&=-c^{2}d\tau ^{2}\\&=-\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)c^{2}dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}-{\frac {2r_{\text{s}}ra\sin ^{2}\theta }{\Sigma }}c\,dt\,d\phi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2eb6913226973b785a86aa87b790e42ef77440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:101.837ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}ds^{2}&=-c^{2}d\tau ^{2}\\&=-\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)c^{2}dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}-{\frac {2r_{\text{s}}ra\sin ^{2}\theta }{\Sigma }}c\,dt\,d\phi \end{aligned}}}" /></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>where the coordinates <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,\theta ,\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\theta ,\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/487df509cbe0c1a243d2d0817e27ffd65ef09b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.592ex; height:2.509ex;" alt="{\displaystyle r,\theta ,\phi }" /></span>⁠</span> are standard <a href="/wiki/Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">oblate spheroidal coordinates</a>, which are equivalent to the cartesian coordinates<sup id="cite_ref-visser35_13-0" class="reference"><a href="#cite_note-visser35-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25bfd83966617d56806e4d09474cc4628ff18817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.97ex; height:3.509ex;" alt="{\displaystyle x={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }" /></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce54d6ad71ad3cb71fd15f468e3b78ec5f2dc2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.541ex; height:3.509ex;" alt="{\displaystyle y={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }" /></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r\cos \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=r\cos \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e34b6f795f25bdab1f767630e40ba328fb107e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.858ex; height:2.509ex;" alt="{\displaystyle z=r\cos \theta ,}" /></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc67a392f63faf10fdb2eee39b5f382dbb5a936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.929ex; height:2.009ex;" alt="{\displaystyle r_{\text{s}}}" /></span> is the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild radius</a> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42e959817979c654359405e7c7abe5b05f7ffc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.942ex; height:5.676ex;" alt="{\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},}" /></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table> <p>and where for brevity, the length scales <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,\Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,\Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2ca6b499959e765d9f681ce47f7fff7224b011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.942ex; height:2.509ex;" alt="{\displaystyle a,\Sigma }" /></span>⁠</span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /></span>⁠</span> have been introduced as </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {J}{Mc}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>J</mi> <mrow> <mi>M</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {J}{Mc}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59793ad048f2b82b2ee0ac77106cfafc7984e7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.26ex; height:5.176ex;" alt="{\displaystyle a={\frac {J}{Mc}},}" /></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc91e5668dd995ce76c9eaba88ee5888f48c6d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.681ex; height:3.009ex;" alt="{\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta ,}" /></span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef9930bccda5ea8eac78fe110d751e3eda70356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.726ex; height:3.009ex;" alt="{\displaystyle \Delta =r^{2}-r_{\text{s}}r+a^{2}.}" /></span></td> <td></td> <td class="nowrap"><span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span></td></tr></tbody></table> <p>A key feature to note in the above metric is the cross-term <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt\,d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt\,d\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf42afeb8889b642ec15d4c5467dbfd794c4f543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.044ex; height:2.509ex;" alt="{\displaystyle dt\,d\phi }" /></span>⁠</span>. This implies that there is coupling between time and motion in the plane of rotation that disappears when the black hole's angular momentum goes to zero. </p><p>In the non-relativistic limit where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> (or, equivalently, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc67a392f63faf10fdb2eee39b5f382dbb5a936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.929ex; height:2.009ex;" alt="{\displaystyle r_{\text{s}}}" /></span>⁠</span>) goes to zero, the Kerr metric becomes the orthogonal metric for the <a href="/wiki/Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">oblate spheroidal coordinates</a> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\mathop {\longrightarrow } _{M\to 0}-c^{2}dt^{2}+{\frac {\Sigma }{r^{2}+a^{2}}}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}\right)\sin ^{2}\theta d\phi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <munder> <mrow class="MJX-TeXAtom-OP"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Σ</mi> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Σ</mi> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\mathop {\longrightarrow } _{M\to 0}-c^{2}dt^{2}+{\frac {\Sigma }{r^{2}+a^{2}}}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}\right)\sin ^{2}\theta d\phi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a6f556a275728c309e75d0f9c95a02d4c6d176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.79ex; height:5.676ex;" alt="{\displaystyle g\mathop {\longrightarrow } _{M\to 0}-c^{2}dt^{2}+{\frac {\Sigma }{r^{2}+a^{2}}}dr^{2}+\Sigma d\theta ^{2}+\left(r^{2}+a^{2}\right)\sin ^{2}\theta d\phi ^{2}}" /></span></td> <td></td> <td class="nowrap"><span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Kerr–Schild_coordinates"><span id="Kerr.E2.80.93Schild_coordinates"></span>Kerr–Schild coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=4" title="Edit section: Kerr–Schild coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr metric can be expressed in <a href="/wiki/Kerr%E2%80%93Schild_perturbations" title="Kerr–Schild perturbations">"Kerr–Schild" form</a>, using a particular set of <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> as follows.<sup id="cite_ref-Debney_15-0" class="reference"><a href="#cite_note-Debney-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> These solutions were proposed by <a href="/wiki/Roy_Patrick_Kerr" class="mw-redirect" title="Roy Patrick Kerr">Kerr</a> and <a href="/wiki/Alfred_Schild" title="Alfred Schild">Schild</a> in 1965. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2407844d6745accc61012b556f3e55f583ff6941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:18.421ex; height:2.843ex;" alt="{\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}" /></span></td> <td></td> <td class="nowrap"><span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mi>M</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2035966c7d3235a513be93a39ba48dca91bba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.585ex; height:6.176ex;" alt="{\displaystyle f={\frac {2GMr^{3}}{r^{4}+a^{2}z^{2}}}}" /></span></td> <td></td> <td class="nowrap"><span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b9999283f6d0cf0016e586bd092ebb2ec232d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.636ex; height:6.176ex;" alt="{\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}" /></span></td> <td></td> <td class="nowrap"><span id="math_12" class="reference nourlexpansion" style="font-weight:bold;">12</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{0}=1.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1.</mn> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{0}=1.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6a7d88e8069627cc9be2be66857a11d89bf9a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:6.99ex; height:2.509ex;" alt="{\displaystyle k_{0}=1.\!}" /></span></td> <td></td> <td class="nowrap"><span id="math_13" class="reference nourlexpansion" style="font-weight:bold;">13</span></td></tr></tbody></table> <p>Notice that <b>k</b> is a <a href="/wiki/Unit_vector" title="Unit vector">unit 3-vector</a>, making the 4-vector a <a href="/wiki/Null_vector" title="Null vector">null vector</a>, with respect to both <i>g</i> and <i>η</i>.<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> Here <i>M</i> is the constant mass of the spinning object, <i>η</i> is the <a href="/wiki/Minkowski_space#Standard_basis" title="Minkowski space">Minkowski tensor</a>, and <i>a</i> is a constant rotational parameter of the spinning object. It is understood that the vector <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" /></span>⁠</span> is directed along the positive z-axis. The quantity <i>r</i> is not the radius, but rather is implicitly defined by </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5ec0b32292a09e3bef9acae6e84abbe7ddc143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.357ex; height:6.176ex;" alt="{\displaystyle {\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}=1}" /></span></td> <td></td> <td class="nowrap"><span id="math_14" class="reference nourlexpansion" style="font-weight:bold;">14</span></td></tr></tbody></table> <p>Notice that the quantity <i>r</i> becomes the usual radius <i>R</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f20116e4f9f6877703667939e795f11f742dd44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.273ex; height:4.843ex;" alt="{\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}" /></span></dd></dl> <p>when the rotational parameter <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>⁠</span> approaches zero. In this form of solution, units are selected so that the speed of light is unity (<span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}" /></span>⁠</span>). At large distances from the source (<i>R</i> ≫ <i>a</i>), these equations reduce to the <a href="/wiki/Eddington%E2%80%93Finkelstein_coordinates" title="Eddington–Finkelstein coordinates">Eddington–Finkelstein form</a> of the Schwarzschild metric. </p><p>In the Kerr–Schild form of the Kerr metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.<sup id="cite_ref-Exact_19-0" class="reference"><a href="#cite_note-Exact-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Soliton_coordinates">Soliton coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=5" title="Edit section: Soliton coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the Kerr metric (along with the <a href="/w/index.php?title=Kerr%E2%80%93NUT_metric&action=edit&redlink=1" class="new" title="Kerr–NUT metric (page does not exist)">Kerr–NUT metric</a>) is axially symmetric, it can be cast into a form to which the <a href="/wiki/Belinski%E2%80%93Zakharov_transform" title="Belinski–Zakharov transform">Belinski–Zakharov transform</a> can be applied. This implies that the Kerr black hole has the form of a <a href="/wiki/Gravitational_soliton" title="Gravitational soliton">gravitational soliton</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> </p> <div class="mw-heading mw-heading2"><h2 id="Mass_of_rotational_energy">Mass of rotational energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=6" title="Edit section: Mass of rotational energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the complete rotational energy <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60d083c5cf720acb82f5f42db89774a3014a5939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.925ex; height:3.176ex;" alt="{\displaystyle E_{\rm {rot}}=c^{2}\left(M-M_{\rm {irr}}\right)}" /></span>⁠</span> of a black hole is extracted, for example with the <a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a>,<sup id="cite_ref-mtw_21-0" class="reference"><a href="#cite_note-mtw-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bhat_22-0" class="reference"><a href="#cite_note-bhat-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> the remaining mass cannot shrink below the irreducible mass. Therefore, if a black hole rotates with the spin <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0977500058af49541e1573f49afa61c437ac026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.77ex; height:2.176ex;" alt="{\displaystyle a=M}" /></span>⁠</span>, its total mass-equivalent <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> is higher by a factor of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>⁠</span> in comparison with a corresponding Schwarzschild black hole where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> is equal to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{irr}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>irr</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{irr}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a502252980202b4a5565f289d856bc385d09c368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.233ex; height:2.509ex;" alt="{\displaystyle M_{\text{irr}}}" /></span>⁠</span>. The reason for this is that in order to get a static body to spin, energy needs to be applied to the system. Because of the <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a> this energy also has a mass-equivalent, which adds to the total mass–energy of the system, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span>. </p><p>The total mass equivalent <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> (the gravitating mass) of the body (including its <a href="/wiki/Rotational_energy" title="Rotational energy">rotational energy</a>) and its irreducible mass <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{irr}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>irr</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{irr}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a502252980202b4a5565f289d856bc385d09c368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.233ex; height:2.509ex;" alt="{\displaystyle M_{\text{irr}}}" /></span>⁠</span> are related by<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-tongeren_24-0" class="reference"><a href="#cite_note-tongeren-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2M_{\rm {irr}}^{2}=M^{2}+{\sqrt {M^{4}-J^{2}c^{2}/G^{2}}}\Longrightarrow M^{2}=M_{\rm {irr}}^{2}+{\frac {J^{2}c^{2}}{4M_{\rm {irr}}^{2}G^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">⟹</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2M_{\rm {irr}}^{2}=M^{2}+{\sqrt {M^{4}-J^{2}c^{2}/G^{2}}}\Longrightarrow M^{2}=M_{\rm {irr}}^{2}+{\frac {J^{2}c^{2}}{4M_{\rm {irr}}^{2}G^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fd606fa6abeaee3e8cbcb13b9748523131f21a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:60.869ex; height:6.676ex;" alt="{\displaystyle 2M_{\rm {irr}}^{2}=M^{2}+{\sqrt {M^{4}-J^{2}c^{2}/G^{2}}}\Longrightarrow M^{2}=M_{\rm {irr}}^{2}+{\frac {J^{2}c^{2}}{4M_{\rm {irr}}^{2}G^{2}}}.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Wave_operator">Wave operator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=7" title="Edit section: Wave operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since even a direct check on the Kerr metric involves cumbersome calculations, the <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> components <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e3d0245cc7459cd001ce43d00a03b9419239f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.774ex; height:3.009ex;" alt="{\displaystyle g^{ik}}" /></span>⁠</span> of the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> in Boyer–Lindquist coordinates are shown below in the expression for the square of the <a href="/wiki/Four-gradient" title="Four-gradient">four-gradient</a> <a href="/wiki/Differential_operator" title="Differential operator">operator</a>:<sup id="cite_ref-mtw_21-1" class="reference"><a href="#cite_note-mtw-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{\mu \nu }{\frac {\partial }{\partial x^{\mu }}}{\frac {\partial }{\partial x^{\nu }}}=-{\frac {1}{c^{2}\Delta }}\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\left({\frac {\partial }{\partial t}}\right)^{2}-{\frac {2r_{\text{s}}ra}{c\Sigma \Delta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial {t}}}+{\frac {1}{\Delta \sin ^{2}\theta }}\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)\left({\frac {\partial }{\partial \phi }}\right)^{2}+{\frac {\Delta }{\Sigma }}\left({\frac {\partial }{\partial r}}\right)^{2}+{\frac {1}{\Sigma }}\left({\frac {\partial }{\partial \theta }}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <mi>a</mi> </mrow> <mrow> <mi>c</mi> <mi mathvariant="normal">Σ</mi> <mi mathvariant="normal">Δ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{\mu \nu }{\frac {\partial }{\partial x^{\mu }}}{\frac {\partial }{\partial x^{\nu }}}=-{\frac {1}{c^{2}\Delta }}\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\left({\frac {\partial }{\partial t}}\right)^{2}-{\frac {2r_{\text{s}}ra}{c\Sigma \Delta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial {t}}}+{\frac {1}{\Delta \sin ^{2}\theta }}\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)\left({\frac {\partial }{\partial \phi }}\right)^{2}+{\frac {\Delta }{\Sigma }}\left({\frac {\partial }{\partial r}}\right)^{2}+{\frac {1}{\Sigma }}\left({\frac {\partial }{\partial \theta }}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b529130b9cc3d93af26ea038814bbda2aee127c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:128.565ex; height:6.509ex;" alt="{\displaystyle g^{\mu \nu }{\frac {\partial }{\partial x^{\mu }}}{\frac {\partial }{\partial x^{\nu }}}=-{\frac {1}{c^{2}\Delta }}\left(r^{2}+a^{2}+{\frac {r_{\text{s}}ra^{2}}{\Sigma }}\sin ^{2}\theta \right)\left({\frac {\partial }{\partial t}}\right)^{2}-{\frac {2r_{\text{s}}ra}{c\Sigma \Delta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial {t}}}+{\frac {1}{\Delta \sin ^{2}\theta }}\left(1-{\frac {r_{\text{s}}r}{\Sigma }}\right)\left({\frac {\partial }{\partial \phi }}\right)^{2}+{\frac {\Delta }{\Sigma }}\left({\frac {\partial }{\partial r}}\right)^{2}+{\frac {1}{\Sigma }}\left({\frac {\partial }{\partial \theta }}\right)^{2}}" /></span></td> <td></td> <td class="nowrap"><span id="math_15" class="reference nourlexpansion" style="font-weight:bold;">15</span></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Frame_dragging">Frame dragging</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=8" title="Edit section: Frame dragging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We may rewrite the Kerr metric (<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>) in the following form: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{\mathrm {rr} }dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }\left(d\phi +{\frac {g_{t\phi }}{g_{\phi \phi }}}dt\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mi>θ</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>ϕ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{\mathrm {rr} }dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }\left(d\phi +{\frac {g_{t\phi }}{g_{\phi \phi }}}dt\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf96da5771fe96de6fd1828cf575e98fd555a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.143ex; height:7.676ex;" alt="{\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{\mathrm {rr} }dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }\left(d\phi +{\frac {g_{t\phi }}{g_{\phi \phi }}}dt\right)^{2}.}" /></span></td> <td></td> <td class="nowrap"><span id="math_16" class="reference nourlexpansion" style="font-weight:bold;">16</span></td></tr></tbody></table> <p>This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius <i>r</i> and the <a href="/wiki/Colatitude" title="Colatitude">colatitude</a> <i>θ</i>, where Ω is called the <a href="/wiki/Killing_horizon" title="Killing horizon">Killing horizon</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{\text{s}}rac}{\Sigma \left(r^{2}+a^{2}\right)+r_{\text{s}}ra^{2}\sin ^{2}\theta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <mi>a</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">Σ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>r</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{\text{s}}rac}{\Sigma \left(r^{2}+a^{2}\right)+r_{\text{s}}ra^{2}\sin ^{2}\theta }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d0ba72cca942f14d1801ff93246d905d952b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.281ex; height:6.009ex;" alt="{\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{\text{s}}rac}{\Sigma \left(r^{2}+a^{2}\right)+r_{\text{s}}ra^{2}\sin ^{2}\theta }}.}" /></span></td> <td></td> <td class="nowrap"><span id="math_17" class="reference nourlexpansion" style="font-weight:bold;">17</span></td></tr></tbody></table> <p>Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is called frame-dragging, and has been tested experimentally.<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> Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. An "ice skater", in orbit over the equator and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be torqued spinward. The arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. If she is already rotating at a certain speed when she extends her arms, inertial effects and frame-dragging effects will balance and her spin will not change. Due to the <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>, gravitational effects are locally indistinguishable from inertial effects, so this rotation rate, at which when she extends her arms nothing happens, is her local reference for non-rotation. This frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. A useful metaphor is a <a href="/wiki/Planetary_gear" class="mw-redirect" title="Planetary gear">planetary gear</a> system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. This can also be interpreted through <a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Important_surfaces">Important surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=9" title="Edit section: Important surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kerr-surfaces.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Kerr-surfaces.png/220px-Kerr-surfaces.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Kerr-surfaces.png/330px-Kerr-surfaces.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Kerr-surfaces.png/440px-Kerr-surfaces.png 2x" data-file-width="800" data-file-height="588" /></a><figcaption>Location of the horizons, ergospheres and the ring singularity of the Kerr spacetime in Cartesian Kerr–Schild coordinates.<sup id="cite_ref-visser35_13-1" class="reference"><a href="#cite_note-visser35-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Kerr.black.hole.shadow.and.horizons.gif" title="File:Kerr.black.hole.shadow.and.horizons.gif"><img resource="/wiki/File:Kerr.black.hole.shadow.and.horizons.thumb.gif" src="//upload.wikimedia.org/wikipedia/commons/c/c1/Kerr.black.hole.shadow.and.horizons.thumb.gif" decoding="async" width="220" height="166" class="mw-file-element" data-file-width="220" data-file-height="166" /></a><figcaption>Comparison of the shadow (black) and the important surfaces (white) of a black hole. The spin parameter <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>⁠</span> is animated from <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /></span>⁠</span> to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span>, while the left side of the black hole is rotating towards the observer.<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></figcaption></figure> <p>There are several important surfaces in the Kerr metric (<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>). The inner surface corresponds to an event horizon similar to that observed in the Schwarzschild metric; this occurs where the purely radial component <i>g</i><sub>rr</sub> of the metric goes to infinity. Solving the quadratic equation <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>g</i><sub>rr</sub></span></span>⁠</span> = 0</span> yields the solution: </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_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b810d64ecc677ff2da406996b0b5820a4cff8031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.977ex; height:6.176ex;" alt="{\displaystyle r_{\rm {H}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}}}}{2}}}" /></span></dd></dl> <p>which in natural units (that give <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=M=c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>M</mi> <mo>=</mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=M=c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd1d0521ebbc9ebc0d1814a4d640862c2ef084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.734ex; height:2.176ex;" alt="{\displaystyle G=M=c=1}" /></span>⁠</span>) simplifies to: </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_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c7ac2281ebebb2f453c04d0056b7d62edd3bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.271ex; height:3.676ex;" alt="{\displaystyle r_{\rm {H}}^{\pm }=1\pm {\sqrt {1-a^{2}}}}" /></span></dd></dl> <p>While in the Schwarzschild metric the event horizon is also the place where the purely temporal component <i>g</i><sub>tt</sub> of the metric changes sign from positive to negative, in Kerr metric that happens at a different distance. Again solving a quadratic equation <span class="nowrap"><i>g</i><sub>tt</sub> = 0</span> yields the solution: </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_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8844794068e85147145052553a2d79baca1e300a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.007ex; height:6.176ex;" alt="{\displaystyle r_{\rm {E}}^{\pm }={\frac {r_{\text{s}}\pm {\sqrt {r_{\text{s}}^{2}-4a^{2}\cos ^{2}\theta }}}{2}}}" /></span></dd></dl> <p>or in natural units: </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_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4329d0c67dd5abf79ebe1f652c866280c3bef074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.301ex; height:3.676ex;" alt="{\displaystyle r_{\rm {E}}^{\pm }=1\pm {\sqrt {1-a^{2}\cos ^{2}\theta }}}" /></span></dd></dl> <p>Due to the cos<sup>2</sup><i>θ</i> term in the square root, this outer surface resembles a flattened sphere that touches the inner surface at the poles of the rotation axis, where the colatitude <i>θ</i> equals 0 or <i>π</i>; the space between these two surfaces is called the ergosphere. Within this volume, the purely temporal component <i>g</i><sub>tt</sub> is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character. A moving particle experiences a positive <a href="/wiki/Proper_time" title="Proper time">proper time</a> along its <a href="/wiki/Worldline" class="mw-redirect" title="Worldline">worldline</a>, its path through spacetime. However, this is impossible within the ergosphere, where <i>g</i><sub>tt</sub> is negative, unless the particle is co-rotating around the interior mass <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>⁠</span> with an angular speed at least of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span>⁠</span>. Thus, no particle can move in the direction opposite to central mass's rotation within the ergosphere. </p><p>As with the event horizon in the Schwarzschild metric, the apparent singularity at <i>r</i><sub>H</sub> is due to the choice of coordinates (i.e., it is a <a href="/wiki/Coordinate_singularity" title="Coordinate singularity">coordinate singularity</a>). In fact, the spacetime can be smoothly continued through it by an appropriate choice of coordinates. In turn, the outer boundary of the ergosphere at <i>r</i><sub>E</sub> is not singular by itself even in Kerr coordinates due to non-zero <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt\ d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> <mtext> </mtext> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt\ d\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf76a77e944f8fd666e71dcd7fe655e4640226f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.237ex; height:2.509ex;" alt="{\displaystyle dt\ d\phi }" /></span>⁠</span> term. </p> <div class="mw-heading mw-heading2"><h2 id="Ergosphere_and_the_Penrose_process">Ergosphere and the Penrose process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=10" title="Edit section: Ergosphere and the Penrose process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a></div> <p>A black hole in general is surrounded by a surface, called the event horizon and situated at the <a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a> for a nonrotating black hole, where the escape velocity is equal to the velocity of light. Within this surface, no observer/particle can maintain itself at a constant radius. It is forced to fall inwards, and so this is sometimes called the <i>static limit</i>. </p><p>A rotating black hole has the same static limit at its event horizon but there is an additional surface outside the event horizon named the "ergosurface" given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mi>M</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5817f7bf07e85b49a9f638ba599b9bfc4b8182d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.297ex; height:3.176ex;" alt="{\displaystyle (r-M)^{2}=M^{2}-J^{2}\cos ^{2}\theta }" /></span></dd></dl> <p>in <a href="/wiki/Boyer%E2%80%93Lindquist_coordinates" title="Boyer–Lindquist coordinates">Boyer–Lindquist coordinates</a>, which can be intuitively characterized as the sphere where "the rotational velocity of the surrounding space" is dragged along with the velocity of light. Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate. </p><p>The region outside the event horizon but inside the surface where the rotational velocity is the speed of light, is called the <i>ergosphere</i> (from Greek <i>ergon</i> meaning <i>work</i>). Particles falling within the ergosphere are forced to rotate faster and thereby gain energy. Because they are still outside the event horizon, they may escape the black hole. The net process is that the rotating black hole emits energetic particles at the cost of its own total energy. The possibility of extracting spin energy from a rotating black hole was first proposed by the mathematician <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> in 1969 and is thus called the Penrose process. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as <a href="/wiki/Gamma-ray_burst" title="Gamma-ray burst">gamma-ray bursts</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Features">Features</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=11" title="Edit section: Features"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr geometry exhibits many noteworthy features: the maximal analytic extension includes a sequence of <a href="/wiki/Asymptotically_flat" class="mw-redirect" title="Asymptotically flat">asymptotically flat</a> exterior regions, each associated with an <a href="/wiki/Ergosphere" title="Ergosphere">ergosphere</a>, stationary limit surfaces, <a href="/wiki/Event_horizon" title="Event horizon">event horizons</a>, <a href="/wiki/Cauchy_horizon" title="Cauchy horizon">Cauchy horizons</a>, <a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">closed timelike curves</a>, and a ring-shaped <a href="/wiki/Gravitational_singularity" title="Gravitational singularity">curvature singularity</a>. The <a href="/wiki/Geodesic_equation" class="mw-redirect" title="Geodesic equation">geodesic equation</a> can be solved exactly in closed form. In addition to two <a href="/wiki/Killing_vector_fields" class="mw-redirect" title="Killing vector fields">Killing vector fields</a> (corresponding to <i><a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translation</a></i> and <i>axisymmetry</i>), the Kerr geometry admits a remarkable <a href="/wiki/Killing_tensor" title="Killing tensor">Killing tensor</a>. There is a pair of principal null congruences (one <i>ingoing</i> and one <i>outgoing</i>). The <a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a> is <a href="/wiki/Algebraically_special" class="mw-redirect" title="Algebraically special">algebraically special</a>, in fact it has <a href="/wiki/Petrov_classification" title="Petrov classification">Petrov type</a> <b>D</b>. The <a href="/wiki/Global_spacetime_structure" class="mw-redirect" title="Global spacetime structure">global structure</a> is known. Topologically, the <a href="/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy type</a> of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point. </p><p>Note that the inner Kerr geometry is unstable with regard to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so.<sup id="cite_ref-visser35_13-2" class="reference"><a href="#cite_note-visser35-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This instability also implies that many of the features of the Kerr geometry described above may not be present inside such a black hole.<sup id="cite_ref-pauldavies_27-0" class="reference"><a href="#cite_note-pauldavies-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-visser13_28-0" class="reference"><a href="#cite_note-visser13-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many <a href="/wiki/Photon_sphere" title="Photon sphere">photon spheres</a>, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" /></span>⁠</span>, the inner and outer photon spheres degenerate, so that there is only one photon sphere at a single radius. The greater the spin of a black hole, the farther from each other the inner and outer photon spheres move. A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere. A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere. Orbiting geodesics with some angular momentum perpendicular to the axis of rotation of the black hole will orbit on photon spheres between these two extremes. Because the spacetime is rotating, such orbits exhibit a precession, since there is a shift in the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>⁠</span> variable after completing one period in the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span>⁠</span> variable. </p> <div class="mw-heading mw-heading3"><h3 id="Trajectory_equations">Trajectory equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=12" title="Edit section: Trajectory equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Orbit_around_a_rotating_Kerr_black_hole.gif" title="File:Orbit around a rotating Kerr black hole.gif"><img resource="/wiki/File:Wiki-en-kerr-mini.gif" src="//upload.wikimedia.org/wikipedia/commons/8/83/Wiki-en-kerr-mini.gif" decoding="async" width="220" height="108" class="mw-file-element" data-file-width="220" data-file-height="108" /></a><figcaption>Animation of a test-particle's orbit around a spinning black hole. Left: top view, right: side view.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Generic_geodesic_orbit_around_a_Kerr_black_hole.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Generic_geodesic_orbit_around_a_Kerr_black_hole.png/220px-Generic_geodesic_orbit_around_a_Kerr_black_hole.png" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Generic_geodesic_orbit_around_a_Kerr_black_hole.png/330px-Generic_geodesic_orbit_around_a_Kerr_black_hole.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Generic_geodesic_orbit_around_a_Kerr_black_hole.png/440px-Generic_geodesic_orbit_around_a_Kerr_black_hole.png 2x" data-file-width="1000" data-file-height="878" /></a><figcaption>Another trajectory of a test mass around a spinning (Kerr) black hole. Unlike orbits around a Schwarzschild black hole, the orbit is not confined to a single plane, but will <a href="/wiki/Ergodicity" title="Ergodicity">ergodically</a> fill a <a href="/wiki/Torus" title="Torus">toruslike</a> region around the equator.</figcaption></figure> <p>The <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> for <a href="/wiki/Test_particle" title="Test particle">test particles</a> in the Kerr spacetime are governed by four <a href="/wiki/Constant_of_motion" title="Constant of motion">constants of motion</a>.<sup id="cite_ref-carter_1968_29-0" class="reference"><a href="#cite_note-carter_1968-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The first is the invariant mass <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span>⁠</span> of the test particle, defined by the relation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14059e91022abe20f99990bc30e3eb64ab809e03" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.142ex; height:3.343ex;" alt="{\displaystyle -\mu ^{2}=p^{\alpha }g_{\alpha \beta }p^{\beta },}" /></span> where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc4e1d2e0eb9fbc9821482a97ad563c500f9ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.543ex; height:2.676ex;" alt="{\displaystyle p^{\alpha }}" /></span>⁠</span> is the <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> of the particle. Furthermore, there are two constants of motion given by the time translation and rotation symmetries of Kerr spacetime, the energy <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span>⁠</span>, and the component of the orbital angular momentum parallel to the spin of the black hole <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a5b940110c5e1fe03782a31c5e700939ae20e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;" alt="{\displaystyle L_{z}}" /></span>⁠</span>.<sup id="cite_ref-mtw_21-2" class="reference"><a href="#cite_note-mtw-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bardeen1972_30-0" class="reference"><a href="#cite_note-bardeen1972-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=-p_{t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=-p_{t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e42b5cf1139028e542f83e12af713f9774bfa47f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.324ex; height:2.509ex;" alt="{\displaystyle E=-p_{t},}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}=p_{\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}=p_{\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78034392b9d7e44238cffc6f46ee3c65a34d7636" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.064ex; height:2.843ex;" alt="{\displaystyle L_{z}=p_{\phi }}" /></span> </p><p>Using <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi theory</a>, <a href="/wiki/Brandon_Carter" title="Brandon Carter">Brandon Carter</a> showed that there exists a fourth constant of motion, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span>⁠</span>,<sup id="cite_ref-carter_1968_29-1" class="reference"><a href="#cite_note-carter_1968-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> now referred to as the <a href="/wiki/Carter_constant" title="Carter constant">Carter constant</a>. It is related to the total angular momentum of the particle and is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</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> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f272384c1ea3642dbbae225661405052defe8b0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.177ex; height:7.509ex;" alt="{\displaystyle Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}\right).}" /></span> </p><p>Since there are four (independent) constants of motion for degrees of freedom, the equations of motion for a test particle in Kerr spacetime are <a href="/wiki/Integrable" class="mw-redirect" title="Integrable">integrable</a>. </p><p>Using these constants of motion, the trajectory equations for a test particle can be written (using natural units of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=M=c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>M</mi> <mo>=</mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=M=c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd1d0521ebbc9ebc0d1814a4d640862c2ef084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.734ex; height:2.176ex;" alt="{\displaystyle G=M=c=1}" /></span>⁠</span>),<sup id="cite_ref-carter_1968_29-2" class="reference"><a href="#cite_note-carter_1968-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Sigma {\frac {dr}{d\lambda }}&=\pm {\sqrt {R(r)}}\\\Sigma {\frac {d\theta }{d\lambda }}&=\pm {\sqrt {\Theta (\theta )}}\\\Sigma {\frac {d\phi }{d\lambda }}&=-\left(aE-{\frac {L_{z}}{\sin ^{2}\theta }}\right)+{\frac {a}{\Delta }}P(r)\\\Sigma {\frac {dt}{d\lambda }}&=-a\left(aE\sin ^{2}\theta -L_{z}\right)+{\frac {r^{2}+a^{2}}{\Delta }}P(r)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mi>E</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mi>E</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Sigma {\frac {dr}{d\lambda }}&=\pm {\sqrt {R(r)}}\\\Sigma {\frac {d\theta }{d\lambda }}&=\pm {\sqrt {\Theta (\theta )}}\\\Sigma {\frac {d\phi }{d\lambda }}&=-\left(aE-{\frac {L_{z}}{\sin ^{2}\theta }}\right)+{\frac {a}{\Delta }}P(r)\\\Sigma {\frac {dt}{d\lambda }}&=-a\left(aE\sin ^{2}\theta -L_{z}\right)+{\frac {r^{2}+a^{2}}{\Delta }}P(r)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b4bc01bed4ae21d6bc699ac858d2ed7edde0c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:44.232ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}\Sigma {\frac {dr}{d\lambda }}&=\pm {\sqrt {R(r)}}\\\Sigma {\frac {d\theta }{d\lambda }}&=\pm {\sqrt {\Theta (\theta )}}\\\Sigma {\frac {d\phi }{d\lambda }}&=-\left(aE-{\frac {L_{z}}{\sin ^{2}\theta }}\right)+{\frac {a}{\Delta }}P(r)\\\Sigma {\frac {dt}{d\lambda }}&=-a\left(aE\sin ^{2}\theta -L_{z}\right)+{\frac {r^{2}+a^{2}}{\Delta }}P(r)\end{aligned}}}" /></span> with </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</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> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09569a7cc45c9c4be26e00290578cdfcc3afe49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.946ex; height:6.176ex;" alt="{\displaystyle \Theta (\theta )=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>a</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623d61dfb34ca3bcae10165b5964b831ee4f58c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.876ex; height:3.343ex;" alt="{\displaystyle P(r)=E\left(r^{2}+a^{2}\right)-aL_{z}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mi>E</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>Q</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098249e06686239db98f7b4a302adbba49397c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.043ex; height:3.343ex;" alt="{\displaystyle R(r)=P(r)^{2}-\Delta \left(\mu ^{2}r^{2}+(L_{z}-aE)^{2}+Q\right)}" /></span></li></ul> <p>where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span>⁠</span> is an <a href="/wiki/Affine_parameter" class="mw-redirect" title="Affine parameter">affine parameter</a> such that <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913f9481588c30da0add6b6d3633c0db128633ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.218ex; height:5.509ex;" alt="{\displaystyle {\frac {dx^{\alpha }}{d\lambda }}=p^{\alpha }}" /></span>⁠</span>. In particular, when <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67319256f71b2ecddcb2a1f2a58bef0494135e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu >0}" /></span>⁠</span> the affine parameter <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span>⁠</span>, is related to the proper time <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span>⁠</span> through <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\tau /\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\tau /\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7eb4eff2e84083b0603ee4785c2f255cabaaba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.22ex; height:2.843ex;" alt="{\displaystyle \lambda =\tau /\mu }" /></span>⁠</span>. </p><p>Because of the <a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a>-effect, a zero-angular-momentum observer (ZAMO) is corotating with the angular velocity <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span>⁠</span> which is defined with respect to the bookkeeper's coordinate time <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>⁠</span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The local velocity <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span>⁠</span> of the test-particle is measured relative to a probe corotating with <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span>⁠</span>. The gravitational time-dilation between a ZAMO at fixed <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>⁠</span> and a stationary observer far away from the mass is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>τ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mtext> </mtext> <mi mathvariant="normal">Σ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13242de2235b16944b13822763f6ed661bafaeb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.869ex; height:7.676ex;" alt="{\displaystyle {\frac {t}{\tau }}={\sqrt {\frac {\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta \sin ^{2}\theta }{\Delta \ \Sigma }}}.}" /></span> In Cartesian Kerr–Schild coordinates, the equations for a photon are<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}+i{\ddot {y}}=4iMa{\frac {r}{\Sigma ^{2}}}W\left[{\dot {x}}+i{\dot {y}}-{\frac {x+iy}{r}}{\dot {r}}\right]-M(x+iy)\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C-a^{2}W^{2}}{r\Sigma ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>4</mn> <mi>i</mi> <mi>M</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>W</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>r</mi> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}+i{\ddot {y}}=4iMa{\frac {r}{\Sigma ^{2}}}W\left[{\dot {x}}+i{\dot {y}}-{\frac {x+iy}{r}}{\dot {r}}\right]-M(x+iy)\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C-a^{2}W^{2}}{r\Sigma ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe5085203cdf9ef96c35da0c12f7ef468f7e0c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.035ex; height:6.343ex;" alt="{\displaystyle {\ddot {x}}+i{\ddot {y}}=4iMa{\frac {r}{\Sigma ^{2}}}W\left[{\dot {x}}+i{\dot {y}}-{\frac {x+iy}{r}}{\dot {r}}\right]-M(x+iy)\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C-a^{2}W^{2}}{r\Sigma ^{2}}}}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {z}}=-Mz\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C}{r\Sigma ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>M</mi> <mi>z</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mi>r</mi> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {z}}=-Mz\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C}{r\Sigma ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b4f61db95fe5443a48aa93553c87ef3c05606d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.65ex; height:6.343ex;" alt="{\displaystyle {\ddot {z}}=-Mz\left({\frac {4r^{2}}{\Sigma }}-1\right){\frac {C}{r\Sigma ^{2}}}}" /></span> where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>⁠</span> is analogous to Carter's constant and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" /></span>⁠</span> is a useful quantity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=p_{\theta }^{2}+\left(aE\sin {\theta }-{\frac {L_{z}}{\sin {\theta }}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mi>E</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=p_{\theta }^{2}+\left(aE\sin {\theta }-{\frac {L_{z}}{\sin {\theta }}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea68a39e0a8cfbeab5842b676db9b656d8cb07fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.14ex; height:6.509ex;" alt="{\displaystyle C=p_{\theta }^{2}+\left(aE\sin {\theta }-{\frac {L_{z}}{\sin {\theta }}}\right)^{2}}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\dot {t}}-a\sin ^{2}{\theta }{\dot {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>t</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\dot {t}}-a\sin ^{2}{\theta }{\dot {\phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de0d161d413accd22c046f03134e50173eec4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.202ex; height:3.009ex;" alt="{\displaystyle W={\dot {t}}-a\sin ^{2}{\theta }{\dot {\phi }}}" /></span> </p><p>If we set <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" /></span>⁠</span>, the <a href="/wiki/Schwarzschild_geodesics" title="Schwarzschild geodesics">Schwarzschild geodesics</a> are restored. </p> <div class="mw-heading mw-heading2"><h2 id="Symmetries">Symmetries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=13" title="Edit section: Symmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group of isometries of the Kerr metric is the subgroup of the ten-dimensional <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> which takes the two-dimensional locus of the singularity to itself. It retains the <a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translations</a> (one dimension) and rotations around its axis of rotation (one dimension). Thus it has two dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the component which reverses time and longitude; the component which reflects through the equatorial plane; and the component that does both. </p><p>In physics, symmetries are typically associated with conserved constants of motion, in accordance with <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>. As shown above, the geodesic equations have four conserved quantities: one of which comes from the definition of a geodesic, and two of which arise from the time translation and rotation symmetry of the Kerr geometry. The fourth conserved quantity does not arise from a symmetry in the standard sense and is commonly referred to as a hidden symmetry. </p> <div class="mw-heading mw-heading2"><h2 id="Overextreme_Kerr_solutions">Overextreme Kerr solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=14" title="Edit section: Overextreme Kerr solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The location of the event horizon is determined by the larger root of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}" /></span>⁠</span>. When <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{s}}/2<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{s}}/2<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a2857ae803da4bcaab4f85cf977e85b2f46ff89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.582ex; height:2.843ex;" alt="{\displaystyle r_{\text{s}}/2<a}" /></span>⁠</span> (i.e. <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GM^{2}<Jc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi>J</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GM^{2}<Jc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ff76874dccf2722fd7f9f619de5edbc0e199d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.956ex; height:2.676ex;" alt="{\displaystyle GM^{2}<Jc}" /></span>⁠</span>), there are no (real valued) solutions to this equation, and there is no event horizon. With no event horizons to hide it from the rest of the universe, the black hole ceases to be a black hole and will instead be a <a href="/wiki/Naked_singularity" title="Naked singularity">naked singularity</a>.<sup id="cite_ref-Chandrasekhar_1983_33-0" class="reference"><a href="#cite_note-Chandrasekhar_1983-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kerr_black_holes_as_wormholes">Kerr black holes as wormholes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=15" title="Edit section: Kerr black holes as wormholes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Kerr_metric" title="Special:EditPage/Kerr metric">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">February 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Although the Kerr solution appears to be singular at the roots of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}" /></span>⁠</span>, these are actually coordinate singularities, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>⁠</span> corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a <a href="/wiki/Cauchy_horizon" title="Cauchy horizon">Cauchy horizon</a>. A (future-directed, time-like) curve can start in the exterior and pass through the event horizon. Once having passed through the event horizon, the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>⁠</span> coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Anti-universe_region">Anti-universe region</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=16" title="Edit section: Anti-universe region"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr metric, which describes the spacetime geometry around a rotating black hole, can be extended beyond the inner event horizon. In the Boyer-Lindquist coordinate system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,r,\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t,r,\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa61ce2c0a7b5e0dd50e2627d92d4af74500d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.275ex; height:2.843ex;" alt="{\displaystyle (t,r,\theta ,\phi )}" /></span>, this inner horizon is located at </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 r_{-}=M-{\sqrt {M^{2}-a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <mi>M</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{-}=M-{\sqrt {M^{2}-a^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de53b0c0b81758cedf57fe373a9ae8dd65a21429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.589ex; height:3.509ex;" alt="{\displaystyle r_{-}=M-{\sqrt {M^{2}-a^{2}}}.}" /></span> </p><p>As one crosses this inner horizon, the radial coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> continues to decrease, even becoming negative. </p> <div class="mw-heading mw-heading3"><h3 id="The_ring_singularity_and_beyond">The ring singularity and beyond</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=17" title="Edit section: The ring singularity and beyond"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}" /></span>, a peculiar feature arises: a ring singularity. Unlike the point singularity in the Schwarzschild metric (a non-rotating black hole), the Kerr singularity is not a single point but a ring lying in the equatorial plane (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4628b0f731f81bfabcd8edaca00aa186f03bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \theta =\pi /2}" /></span>). This ring singularity acts as a portal to a new region of spacetime. </p><p>If we avoid the equatorial plane (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \neq \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≠</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \neq \pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f474cd2417bf2a22133702b35ccee4e8382d335" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \theta \neq \pi /2}" /></span>), we can smoothly continue the coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span> to negative values. This region 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 r<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6c26a0286c0286de471448db8c9529bb944a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r<0}" /></span> is interpreted as an entirely new, asymptotically flat universe, often called the "anti-universe." This anti-universe has some surprising properties: </p><p><b>Negative ADM Mass:</b> The anti-universe possesses a negative <a href="/wiki/ADM_formalism" title="ADM formalism">Arnowitt-Deser-Misner (ADM) mass</a>, which can be thought of as the total mass-energy of the spacetime as measured at infinity. A negative mass is a highly unusual concept in general relativity, and its physical interpretation is still debated. </p> <div class="mw-heading mw-heading3"><h3 id="Closed_timelike_curves_and_the_Cauchy_horizon">Closed timelike curves and the Cauchy horizon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=18" title="Edit section: Closed timelike curves and the Cauchy horizon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Within the anti-universe, an even stranger phenomenon occurs. The metric component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi \phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39023076c70f7559bbcdeeaeea30b890b71d1a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.301ex; height:2.343ex;" alt="{\displaystyle g_{\phi \phi }}" /></span>, which is related to the azimuthal direction around the ring singularity, can change sign. Specifically, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi \phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39023076c70f7559bbcdeeaeea30b890b71d1a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.301ex; height:2.343ex;" alt="{\displaystyle g_{\phi \phi }}" /></span> is given by: </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 g_{\phi \phi }={\frac {-(r^{2}+a^{2})^{2}+\Delta a^{2}\sin ^{2}\theta }{\Sigma }}\sin ^{2}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi mathvariant="normal">Σ</mi> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }={\frac {-(r^{2}+a^{2})^{2}+\Delta a^{2}\sin ^{2}\theta }{\Sigma }}\sin ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64cc510279a66d83b11aa3523b1c8cbca211d90d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.39ex; height:5.843ex;" alt="{\displaystyle g_{\phi \phi }={\frac {-(r^{2}+a^{2})^{2}+\Delta a^{2}\sin ^{2}\theta }{\Sigma }}\sin ^{2}\theta .}" /></span> </p><p>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi \phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39023076c70f7559bbcdeeaeea30b890b71d1a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.301ex; height:2.343ex;" alt="{\displaystyle g_{\phi \phi }}" /></span> becomes negative, the coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> becomes timelike, and a linear combination of the coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> 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> becomes spacelike. This leads to the existence of closed timelike curves (CTCs). A CTC is a path through spacetime where an object could travel back to its own past, violating causality. </p><p>The boundary where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi \phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39023076c70f7559bbcdeeaeea30b890b71d1a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.301ex; height:2.343ex;" alt="{\displaystyle g_{\phi \phi }}" /></span> changes sign and CTCs first appear is called the <a href="/wiki/Cauchy_horizon" title="Cauchy horizon">Cauchy horizon</a>. It is defined by the condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\phi \phi }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\phi \phi }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff3ee0f984175844643548c87a5cb20bcd011fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.561ex; height:2.843ex;" alt="{\displaystyle g_{\phi \phi }=0}" /></span>, which gives </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 (r^{2}+a^{2})^{2}=a^{2}\Delta \sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r^{2}+a^{2})^{2}=a^{2}\Delta \sin ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de04543d2edef08325dfca92824274e826fb8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.184ex; height:3.176ex;" alt="{\displaystyle (r^{2}+a^{2})^{2}=a^{2}\Delta \sin ^{2}\theta }" /></span> </p><p>The Cauchy horizon acts as a boundary beyond which the familiar notions of cause and effect break down. The presence of CTCs raises fundamental questions about the predictability and consistency of the laws of physics in these extreme regions of spacetime. </p><p>The anti-universe region of the extended Kerr metric is a fascinating and perplexing theoretical construct. It presents a scenario with a negative mass, reversed time orientation, and the possibility of time travel through closed timelike curves.<sup id="cite_ref-pauldavies_27-1" class="reference"><a href="#cite_note-pauldavies-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-visser13_28-1" class="reference"><a href="#cite_note-visser13-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> While the physical reality of the anti-universe remains uncertain, its study provides valuable insights into the nature of spacetime, gravity, and the limits of our current understanding of the universe. </p><p>While it is expected that the exterior region of the Kerr solution is stable, and that all rotating black holes will eventually approach a Kerr metric, the interior region of the solution appears to be unstable, much like a pencil balanced on its point.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-visser35_13-3" class="reference"><a href="#cite_note-visser35-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This is related to the idea of <a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">cosmic censorship</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_exact_solutions">Relation to other exact solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=19" title="Edit section: Relation to other exact solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr geometry is a particular example of a <a href="/wiki/Stationary_spacetime" title="Stationary spacetime">stationary</a> <a href="/wiki/Circular_symmetry#Three_dimensions" title="Circular symmetry">axially symmetric</a> <a href="/wiki/Vacuum_solution" class="mw-redirect" title="Vacuum solution">vacuum solution</a> to the <a href="/wiki/Einstein_field_equation" class="mw-redirect" title="Einstein field equation">Einstein field equation</a>. The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the <a href="/w/index.php?title=Ernst_vacuum&action=edit&redlink=1" class="new" title="Ernst vacuum (page does not exist)">Ernst vacuums</a>. </p><p>The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the <a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman electrovacuum</a> models a (rotating) black hole endowed with an electric charge, while the <a href="/w/index.php?title=Kerr%E2%80%93Vaidya_null_dust&action=edit&redlink=1" class="new" title="Kerr–Vaidya null dust (page does not exist)">Kerr–Vaidya null dust</a> models a (rotating) hole with infalling electromagnetic radiation. </p><p>The special case <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" /></span>⁠</span> of the Kerr metric yields the Schwarzschild metric, which models a <i>nonrotating</i> black hole which is <a href="/wiki/Static_spacetime" title="Static spacetime">static</a> and <a href="/wiki/Spherically_symmetric" class="mw-redirect" title="Spherically symmetric">spherically symmetric</a>, in the <a href="/wiki/Schwarzschild_coordinates" title="Schwarzschild coordinates">Schwarzschild coordinates</a>. (In this case, every Geroch moment but the mass vanishes.) </p><p>The <i>interior</i> of the Kerr geometry, or rather a portion of it, is locally <a href="/wiki/Isometry" title="Isometry">isometric</a> to the <a href="/w/index.php?title=Chandrasekhar%E2%80%93Ferrari_CPW_vacuum&action=edit&redlink=1" class="new" title="Chandrasekhar–Ferrari CPW vacuum (page does not exist)">Chandrasekhar–Ferrari CPW vacuum</a>, an example of a <a href="/w/index.php?title=Colliding_plane_wave&action=edit&redlink=1" class="new" title="Colliding plane wave (page does not exist)">colliding plane wave</a> model. This is particularly interesting, because the <a href="/wiki/Global_spacetime_structure" class="mw-redirect" title="Global spacetime structure">global structure</a> of this CPW solution is quite different from that of the Kerr geometry, and in principle, an experimenter could hope to study the geometry of (the outer portion of) the Kerr interior by arranging the collision of two suitable <a href="/wiki/Gravitational_plane_waves" class="mw-redirect" title="Gravitational plane waves">gravitational plane waves</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Multipole_moments">Multipole moments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=20" title="Edit section: Multipole moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each <a href="/wiki/Asymptotically_flat" class="mw-redirect" title="Asymptotically flat">asymptotically flat</a> Ernst vacuum can be characterized by giving the infinite sequence of relativistic <a href="/wiki/Multipole_moment" class="mw-redirect" title="Multipole moment">multipole moments</a>, the first two of which can be interpreted as the <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of the source of the field. There are alternative formulations of relativistic multipole moments due to Hansen, Thorne, and Geroch, which turn out to agree with each other. The relativistic multipole moments of the Kerr geometry were computed by Hansen; they turn out to be </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 M_{n}=M[ia]^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>M</mi> <mo stretchy="false">[</mo> <mi>i</mi> <mi>a</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{n}=M[ia]^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893371be1a5049a3794f9a6c348145599d2e5db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.558ex; height:2.843ex;" alt="{\displaystyle M_{n}=M[ia]^{n}}" /></span></dd></dl> <p>Thus, the special case of the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild vacuum</a> (<span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" /></span>⁠</span>) gives the "monopole <a href="/wiki/Point_source" title="Point source">point source</a>" of general relativity.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p><i>Weyl multipole moments</i> arise from treating a certain metric function (formally corresponding to Newtonian gravitational potential) which appears the Weyl–Papapetrou chart for the Ernst family of all stationary axisymmetric vacuum solutions using the standard euclidean scalar <a href="/wiki/Multipole_moment" class="mw-redirect" title="Multipole moment">multipole moments</a>. They are distinct from the moments computed by Hansen, above. In a sense, the Weyl moments only (indirectly) characterize the "mass distribution" of an isolated source, and they turn out to depend only on the <i>even order</i> relativistic moments. In the case of solutions symmetric across the equatorial plane the <i>odd order</i> Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=M,\qquad a_{1}=0,\qquad a_{2}=M\left({\frac {M^{2}}{3}}-a^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>M</mi> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=M,\qquad a_{1}=0,\qquad a_{2}=M\left({\frac {M^{2}}{3}}-a^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d79fea21a6f1fb14d4cee88b5a8a1fa547f3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.874ex; height:6.343ex;" alt="{\displaystyle a_{0}=M,\qquad a_{1}=0,\qquad a_{2}=M\left({\frac {M^{2}}{3}}-a^{2}\right)}" /></span></dd></dl> <p>In particular, we see that the Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is the <a href="/w/index.php?title=Chazy%E2%80%93Curzon_vacuum&action=edit&redlink=1" class="new" title="Chazy–Curzon vacuum (page does not exist)">Chazy–Curzon vacuum</a> solution, not the Schwarzschild vacuum solution, which arises from the Newtonian potential of a certain finite length uniform density thin <i>rod</i>. </p><p>In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize the Weyl moments to <i>mass multipole moments</i> and <i>momentum multipole moments</i>, characterizing respectively the distribution of mass and of <a href="/wiki/Momentum" title="Momentum">momentum</a> of the source. These are multi-indexed quantities whose suitably symmetrized and anti-symmetrized parts can be related to the real and imaginary parts of the relativistic moments for the full nonlinear theory in a rather complicated manner. </p><p>Perez and Moreschi have given an alternative notion of "monopole solutions" by expanding the standard NP tetrad of the Ernst vacuums in powers of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>⁠</span> (the radial coordinate in the Weyl–Papapetrou chart). According to this formulation: </p> <ul><li>the isolated mass monopole source with <i>zero</i> angular momentum is the <i>Schwarzschild vacuum</i> family (one parameter),</li> <li>the isolated mass monopole source with <i>radial</i> angular momentum is the <i><a href="/wiki/Taub%E2%80%93NUT_vacuum" class="mw-redirect" title="Taub–NUT vacuum">Taub–NUT vacuum</a></i> family (two parameters; not quite asymptotically flat),</li> <li>the isolated mass monopole source with <i>axial</i> angular momentum is the <i>Kerr vacuum</i> family (two parameters).</li></ul> <p>In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity. </p> <div class="mw-heading mw-heading2"><h2 id="Open_problems">Open problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=21" title="Edit section: Open problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr geometry is often used as a model of a <a href="/wiki/Rotating_black_hole" title="Rotating black hole">rotating black hole</a> but if the solution is held to be valid only outside some compact region (subject to certain restrictions), in principle, it should be able to be used as an <a href="/wiki/Exterior_solution" class="mw-redirect" title="Exterior solution">exterior solution</a> to model the gravitational field around a rotating massive object other than a black hole such as a <a href="/wiki/Neutron_star" title="Neutron star">neutron star</a>, or the Earth. This works out very nicely for the non-rotating case, where the Schwarzschild vacuum exterior can be matched to a <a href="/wiki/Schwarzschild_fluid" class="mw-redirect" title="Schwarzschild fluid">Schwarzschild fluid</a> interior, and indeed to more general <a href="/wiki/Static_spherically_symmetric_perfect_fluid" title="Static spherically symmetric perfect fluid">static spherically symmetric perfect fluid</a> solutions. However, the problem of finding a rotating perfect-fluid interior which can be matched to a Kerr exterior, or indeed to any asymptotically flat vacuum exterior solution, has proven very difficult. In particular, the <a href="/wiki/Wahlquist_fluid" title="Wahlquist fluid">Wahlquist fluid</a>, which was once thought to be a candidate for matching to a Kerr exterior, is now known not to admit any such matching. At present, it seems that only approximate solutions modeling slowly rotating fluid balls are known (These are the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments). However, the exterior of the <a href="/w/index.php?title=Neugebauer%E2%80%93Meinel_disk&action=edit&redlink=1" class="new" title="Neugebauer–Meinel disk (page does not exist)">Neugebauer–Meinel disk</a>, an exact <a href="/wiki/Dust_solution" title="Dust solution">dust solution</a> which models a rotating thin disk, approaches in a limiting case the <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GM^{2}=cJ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GM^{2}=cJ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9017b7f98e8e83d088df1c00b471444896f5fe3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.956ex; height:2.676ex;" alt="{\displaystyle GM^{2}=cJ}" /></span>⁠</span> Kerr geometry. Physical thin-disk solutions obtained by identifying parts of the Kerr spacetime are also known.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=22" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/40px-Crab_Nebula.jpg" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/60px-Crab_Nebula.jpg 1.5x" data-file-width="3864" data-file-height="3864" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Astronomy" title="Portal:Astronomy">Astronomy portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/60px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li></ul> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a></li> <li><a href="/wiki/De_Sitter-Schwarzschild_metric" class="mw-redirect" title="De Sitter-Schwarzschild metric">De Sitter-Schwarzschild metric</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman metric</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de–Sitter metric</a></li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström metric</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne metric</a></li> <li><a href="/wiki/Spin-flip" title="Spin-flip">Spin-flip</a></li> <li><a href="/wiki/Rotating_black_hole" title="Rotating black hole">Rotating black hole</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=23" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-lower-alpha"> <ol class="references"> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><i>Warning:</i> Do not confuse the relativistic multipole moments computed by Hansen with the Weyl multipole moments discussed below.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-kerr_1963-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-kerr_1963_1-0">^</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="CITEREFKerr1963" class="citation journal cs1"><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr, Roy P.</a> (1963). "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics". <i>Physical Review Letters</i>. <b>11</b> (5): <span class="nowrap">237–</span>238. <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/1963PhRvL..11..237K">1963PhRvL..11..237K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.11.237">10.1103/PhysRevLett.11.237</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Gravitational+Field+of+a+Spinning+Mass+as+an+Example+of+Algebraically+Special+Metrics&rft.volume=11&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E237-%3C%2Fspan%3E238&rft.date=1963&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.11.237&rft_id=info%3Abibcode%2F1963PhRvL..11..237K&rft.aulast=Kerr&rft.aufirst=Roy+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-melia2009-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-melia2009_2-0">^</a></b></span> <span class="reference-text">Melia, Fulvio (2009). "Cracking the Einstein code: relativity and the birth of black hole physics, with an Afterword by Roy Kerr", Princeton University Press, Princeton, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0226519517" title="Special:BookSources/978-0226519517">978-0226519517</a></span> </li> <li id="cite_note-Sneppen-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sneppen_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSneppen2021" class="citation journal cs1">Sneppen, Albert (December 2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270963">"Divergent reflections around the photon sphere of a black hole"</a>. <i>Scientific Reports</i>. <b>11</b> (1). <a href="/wiki/Cosmic_Dawn_Center" class="mw-redirect" title="Cosmic Dawn Center">Cosmic Dawn Center</a>: 14247. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021NatSR..1114247S">2021NatSR..1114247S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-021-93595-w">10.1038/s41598-021-93595-w</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8270963">8270963</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34244573">34244573</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Divergent+reflections+around+the+photon+sphere+of+a+black+hole&rft.volume=11&rft.issue=1&rft.pages=14247&rft.date=2021-12&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8270963%23id-name%3DPMC&rft_id=info%3Apmid%2F34244573&rft_id=info%3Adoi%2F10.1038%2Fs41598-021-93595-w&rft_id=info%3Abibcode%2F2021NatSR..1114247S&rft.aulast=Sneppen&rft.aufirst=Albert&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8270963&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSutter2021" class="citation web cs1">Sutter, Paul (22 July 2021). <a rel="nofollow" class="external text" href="https://www.livescience.com/black-hole-mirror-copies.html">"Black holes warp the universe into a grotesque hall of mirrors"</a>. <i>livescience.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=livescience.com&rft.atitle=Black+holes+warp+the+universe+into+a+grotesque+hall+of+mirrors&rft.date=2021-07-22&rft.aulast=Sutter&rft.aufirst=Paul&rft_id=https%3A%2F%2Fwww.livescience.com%2Fblack-hole-mirror-copies.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbbot2016" class="citation journal cs1">Abbot, B.P. (11 February 2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". <i>Physical Review Letters</i>. <b>116</b> (6): 061102. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.03837">1602.03837</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PhRvL.116f1102A">2016PhRvL.116f1102A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.116.061102">10.1103/PhysRevLett.116.061102</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26918975">26918975</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124959784">124959784</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Observation+of+Gravitational+Waves+from+a+Binary+Black+Hole+Merger&rft.volume=116&rft.issue=6&rft.pages=061102&rft.date=2016-02-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124959784%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2016PhRvL.116f1102A&rft_id=info%3Aarxiv%2F1602.03837&rft_id=info%3Apmid%2F26918975&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.116.061102&rft.aulast=Abbot&rft.aufirst=B.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNewmanJanis1965" class="citation journal cs1">Newman, E. T.; Janis, A. I. (1965-06-01). <a rel="nofollow" class="external text" href="https://aip.scitation.org/doi/10.1063/1.1704350">"Note on the Kerr Spinning-Particle Metric"</a>. <i>Journal of Mathematical Physics</i>. <b>6</b> (6): <span class="nowrap">915–</span>917. <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/1965JMP.....6..915N">1965JMP.....6..915N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1704350">10.1063/1.1704350</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/0022-2488">0022-2488</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Note+on+the+Kerr+Spinning-Particle+Metric&rft.volume=6&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E915-%3C%2Fspan%3E917&rft.date=1965-06-01&rft.issn=0022-2488&rft_id=info%3Adoi%2F10.1063%2F1.1704350&rft_id=info%3Abibcode%2F1965JMP.....6..915N&rft.aulast=Newman&rft.aufirst=E.+T.&rft.au=Janis%2C+A.+I.&rft_id=https%3A%2F%2Faip.scitation.org%2Fdoi%2F10.1063%2F1.1704350&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNewmanAdamo2014" class="citation journal cs1">Newman, Ezra; Adamo, Tim (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.31791">"Kerr–Newman metric"</a>. <i>Scholarpedia</i>. <b>9</b> (10): 31791. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.6626">1410.6626</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014SchpJ...931791N">2014SchpJ...931791N</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/10.4249%2Fscholarpedia.31791">10.4249/scholarpedia.31791</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/1941-6016">1941-6016</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Kerr%E2%80%93Newman+metric&rft.volume=9&rft.issue=10&rft.pages=31791&rft.date=2014&rft_id=info%3Aarxiv%2F1410.6626&rft.issn=1941-6016&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.31791&rft_id=info%3Abibcode%2F2014SchpJ...931791N&rft.aulast=Newman&rft.aufirst=Ezra&rft.au=Adamo%2C+Tim&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.31791&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarrison1978" class="citation journal cs1">Harrison, B. Kent (1978-10-30). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.41.1197">"Bäcklund Transformation for the Ernst Equation of General Relativity"</a>. <i>Physical Review Letters</i>. <b>41</b> (18): <span class="nowrap">1197–</span>1200. <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/1978PhRvL..41.1197H">1978PhRvL..41.1197H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.41.1197">10.1103/PhysRevLett.41.1197</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/0031-9007">0031-9007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=B%C3%A4cklund+Transformation+for+the+Ernst+Equation+of+General+Relativity&rft.volume=41&rft.issue=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1197-%3C%2Fspan%3E1200&rft.date=1978-10-30&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.41.1197&rft_id=info%3Abibcode%2F1978PhRvL..41.1197H&rft.aulast=Harrison&rft.aufirst=B.+Kent&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.41.1197&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChou2020" class="citation journal cs1">Chou, Yu-Ching (January 2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7002888">"A radiating Kerr black hole and Hawking radiation"</a>. <i>Heliyon</i>. <b>6</b> (1): e03336. <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/2020Heliy...603336C">2020Heliy...603336C</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/10.1016%2Fj.heliyon.2020.e03336">10.1016/j.heliyon.2020.e03336</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7002888">7002888</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32051884">32051884</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Heliyon&rft.atitle=A+radiating+Kerr+black+hole+and+Hawking+radiation&rft.volume=6&rft.issue=1&rft.pages=e03336&rft.date=2020-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7002888%23id-name%3DPMC&rft_id=info%3Apmid%2F32051884&rft_id=info%3Adoi%2F10.1016%2Fj.heliyon.2020.e03336&rft_id=info%3Abibcode%2F2020Heliy...603336C&rft.aulast=Chou&rft.aufirst=Yu-Ching&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7002888&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLandauLifshitz1975" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, L. D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a> (1975). <i>The Classical Theory of Fields</i>. Course of Theoretical Physics. Vol. 2 (Revised 4th English ed.). New York: Pergamon Press. pp. <span class="nowrap">321–</span>330. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-018176-9" title="Special:BookSources/978-0-08-018176-9"><bdi>978-0-08-018176-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classical+Theory+of+Fields&rft.place=New+York&rft.series=Course+of+Theoretical+Physics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E321-%3C%2Fspan%3E330&rft.edition=Revised+4th+English&rft.pub=Pergamon+Press&rft.date=1975&rft.isbn=978-0-08-018176-9&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-zanotti-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-zanotti_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRezzollaZanotti2013" class="citation book cs1">Rezzolla, Luciano; Zanotti, Olindo (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aS1oAgAAQBAJ&pg=PA57"><i>Relativistic Hydrodynamics</i></a>. Oxford University Press. pp. 55–57 [eqns. 1.249 to 1.265]. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852890-6" title="Special:BookSources/978-0-19-852890-6"><bdi>978-0-19-852890-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Hydrodynamics&rft.pages=55-57+eqns.+1.249+to+1.265&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=978-0-19-852890-6&rft.aulast=Rezzolla&rft.aufirst=Luciano&rft.au=Zanotti%2C+Olindo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaS1oAgAAQBAJ%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-tapir26-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-tapir26_12-0">^</a></b></span> <span class="reference-text">Christopher M. Hirata: <a rel="nofollow" class="external text" href="http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec26.pdf#page=5">Lecture XXVI: Kerr black holes: I. Metric structure and regularity of particle orbits</a>, p. 1, Eq. 1</span> </li> <li id="cite_note-visser35-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-visser35_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-visser35_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-visser35_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-visser35_13-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVisser2007" class="citation arxiv cs1">Visser, Matt (2007). "The Kerr spacetime: A brief introduction". p. 15, Eq. 60–61, p. 24, p. 35. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.0622v3">0706.0622v3</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/gr-qc">gr-qc</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Kerr+spacetime%3A+A+brief+introduction&rft.pages=p.-15%2C+Eq.-60-61%2C+p.-24%2C+p.-35&rft.date=2007&rft_id=info%3Aarxiv%2F0706.0622v3&rft.aulast=Visser&rft.aufirst=Matt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoyerLindquist1967" class="citation journal cs1">Boyer, Robert H.; Lindquist, Richard W. (1967). "Maximal Analytic Extension of the Kerr Metric". <i>J. Math. Phys</i>. <b>8</b> (2): <span class="nowrap">265–</span>281. <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/1967JMP.....8..265B">1967JMP.....8..265B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1705193">10.1063/1.1705193</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Math.+Phys.&rft.atitle=Maximal+Analytic+Extension+of+the+Kerr+Metric&rft.volume=8&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E265-%3C%2Fspan%3E281&rft.date=1967&rft_id=info%3Adoi%2F10.1063%2F1.1705193&rft_id=info%3Abibcode%2F1967JMP.....8..265B&rft.aulast=Boyer&rft.aufirst=Robert+H.&rft.au=Lindquist%2C+Richard+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-Debney-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Debney_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDebneyKerrSchild1969" class="citation journal cs1">Debney, G. C.; Kerr, R. P.; Schild, A. (1969). "Solutions of the Einstein and Einstein-Maxwell Equations". <i>Journal of Mathematical Physics</i>. <b>10</b> (10): <span class="nowrap">1842–</span>1854. <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/1969JMP....10.1842D">1969JMP....10.1842D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1664769">10.1063/1.1664769</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Solutions+of+the+Einstein+and+Einstein-Maxwell+Equations&rft.volume=10&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1842-%3C%2Fspan%3E1854&rft.date=1969&rft_id=info%3Adoi%2F10.1063%2F1.1664769&rft_id=info%3Abibcode%2F1969JMP....10.1842D&rft.aulast=Debney&rft.aufirst=G.+C.&rft.au=Kerr%2C+R.+P.&rft.au=Schild%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> Especially see equations (7.10), (7.11) and (7.14).</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBalasinNachbagauer1994" class="citation journal cs1">Balasin, Herbert; Nachbagauer, Herbert (1994). "Distributional energy–momentum tensor of the Kerr–Newman spacetime family". <i>Classical and Quantum Gravity</i>. <b>11</b> (6): <span class="nowrap">1453–</span>1461. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9312028">gr-qc/9312028</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994CQGra..11.1453B">1994CQGra..11.1453B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F11%2F6%2F010">10.1088/0264-9381/11/6/010</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6041750">6041750</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Distributional+energy%E2%80%93momentum+tensor+of+the+Kerr%E2%80%93Newman+spacetime+family&rft.volume=11&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1453-%3C%2Fspan%3E1461&rft.date=1994&rft_id=info%3Aarxiv%2Fgr-qc%2F9312028&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6041750%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F11%2F6%2F010&rft_id=info%3Abibcode%2F1994CQGra..11.1453B&rft.aulast=Balasin&rft.aufirst=Herbert&rft.au=Nachbagauer%2C+Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Berman, Marcelo. "Energy of Black Holes and Hawking's Universe" in <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=DGwYf8cOCq4C&dq=%22Kerr-Newman%22+and+cartesian&pg=PA148">Trends in Black Hole Research</a></i>, page 148 (Kreitler ed., Nova Publishers 2006).</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="CITEREFVisser2008" class="citation arxiv cs1">Visser, Matt (14 January 2008). "The Kerr spacetime: A brief introduction". p. 12. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.0622">0706.0622</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/gr-qc">gr-qc</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Kerr+spacetime%3A+A+brief+introduction&rft.pages=12&rft.date=2008-01-14&rft_id=info%3Aarxiv%2F0706.0622&rft.aulast=Visser&rft.aufirst=Matt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-Exact-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Exact_19-0">^</a></b></span> <span class="reference-text">Stephani, Hans et al. <i>Exact Solutions of Einstein's Field Equations</i> (Cambridge University Press 2003). See <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SiWXP8FjTFEC&dq=%22Kerr-Schild%22+and+%22determinant+of+the+metric%22&pg=PA485">page 485</a> regarding determinant of metric tensor. See <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SiWXP8FjTFEC&dq=%22Kerr+newman+is+a+special+case%22&pg=PA325">page 325</a> regarding generalizations.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBelinskiVerdaguer2001" class="citation book cs1">Belinski, V.; Verdaguer, E. (2001). <i>Gravitational Solitons</i>. Cambridge Monographs on Mathematical Physics. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521805865" title="Special:BookSources/978-0521805865"><bdi>978-0521805865</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitational+Solitons&rft.series=Cambridge+Monographs+on+Mathematical+Physics&rft.pub=Cambridge+University+Press&rft.date=2001&rft.isbn=978-0521805865&rft.aulast=Belinski&rft.aufirst=V.&rft.au=Verdaguer%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://www.mobt3ath.com/uplode/book/book-35341.pdf">PDF</a></span> </li> <li id="cite_note-mtw-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-mtw_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mtw_21-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mtw_21-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Misner, Thorne & Wheeler: <a rel="nofollow" class="external text" href="https://www.pdf-archive.com/2016/03/21/gravitation-misner-thorne-wheeler">Gravitation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170822180641/https://www.pdf-archive.com/2016/03/21/gravitation-misner-thorne-wheeler/">Archived</a> 2017-08-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, pages 899, 900, 908</span> </li> <li id="cite_note-bhat-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-bhat_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBhatDhurandharDadhich1985" class="citation journal cs1">Bhat, Manjiri; Dhurandhar, Sanjeev; Dadhich, Naresh (1985). "Energetics of the Kerr–Newman black hole by the penrose process". <i>Journal of Astrophysics and Astronomy</i>. <b>6</b> (2): <span class="nowrap">85–</span>100. <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/1985JApA....6...85B">1985JApA....6...85B</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.512.1400">10.1.1.512.1400</a></span>. <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%2FBF02715080">10.1007/BF02715080</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53513572">53513572</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Astrophysics+and+Astronomy&rft.atitle=Energetics+of+the+Kerr%E2%80%93Newman+black+hole+by+the+penrose+process&rft.volume=6&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E85-%3C%2Fspan%3E100&rft.date=1985&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.512.1400%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53513572%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02715080&rft_id=info%3Abibcode%2F1985JApA....6...85B&rft.aulast=Bhat&rft.aufirst=Manjiri&rft.au=Dhurandhar%2C+Sanjeev&rft.au=Dadhich%2C+Naresh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="/wiki/Thibault_Damour" title="Thibault Damour">Thibault Damour</a>: <a rel="nofollow" class="external text" href="http://lapth.cnrs.fr/pg-nomin/chardon/IRAP_PhD/BlackHolesNice2012.pdf#page=11">Black Holes: Energetics and Thermodynamics</a>, page 11</span> </li> <li id="cite_note-tongeren-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-tongeren_24-0">^</a></b></span> <span class="reference-text">Stijn van Tongeren: <a rel="nofollow" class="external text" href="https://www.staff.science.uu.nl/~proko101/StijnJvanTongeren_bh_talk2.pdf#page=42">Rotating Black Holes</a>, page 42</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWill2011" class="citation journal cs1"><a href="/wiki/Clifford_Martin_Will" title="Clifford Martin Will">Will, Clifford M.</a> (May 2011). "Finally, results from Gravity Probe B". <i>Physics</i>. <b>4</b>: 43. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1106.1198">1106.1198</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhyOJ...4...43W">2011PhyOJ...4...43W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysics.4.43">10.1103/Physics.4.43</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119237335">119237335</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics&rft.atitle=Finally%2C+results+from+Gravity+Probe+B&rft.volume=4&rft.pages=43&rft.date=2011-05&rft_id=info%3Aarxiv%2F1106.1198&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119237335%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysics.4.43&rft_id=info%3Abibcode%2F2011PhyOJ...4...43W&rft.aulast=Will&rft.aufirst=Clifford+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFde_Vries" class="citation web cs1">de Vries, Andreas. <a rel="nofollow" class="external text" href="http://haegar.fh-swf.de/publikationen/pascal.pdf#page=8">"Shadows of rotating black holes"</a> <span class="cs1-format">(PDF)</span>. p. 8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Shadows+of+rotating+black+holes&rft.pages=8&rft.aulast=de+Vries&rft.aufirst=Andreas&rft_id=http%3A%2F%2Fhaegar.fh-swf.de%2Fpublikationen%2Fpascal.pdf%23page%3D8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-pauldavies-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-pauldavies_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pauldavies_27-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Paul_Davies" title="Paul Davies">Paul Davies</a>: <a rel="nofollow" class="external text" href="https://archive.today/20170708235257/https://books.google.at/books?id=mOgIGyD1uSIC&pg=PT291&lpg=PT291&dq=kerr+closed+loop+past&source=bl&ots=duxTYw68Fj&sig=2vWOrEFsA59UXxO6xFSQWwjtDGw&hl=de&sa=X&ved=0ahUKEwjWhJzk6_rUAhXJZ1AKHaGxCI0Q6AEIWTAG%23v=onepage&q=kerr%20closed%20loop%20past&f=false">About Time: Einstein's Unfinished Revolution</a></span> </li> <li id="cite_note-visser13-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-visser13_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-visser13_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThe_LIGO_Scientific_Collaborationthe_Virgo_Collaboration2007" class="citation arxiv cs1">The LIGO Scientific Collaboration; the Virgo Collaboration (2007). "The Kerr spacetime: A brief introduction". p. 13, below eq. 52. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.0622v3">0706.0622v3</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/gr-qc">gr-qc</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Kerr+spacetime%3A+A+brief+introduction&rft.pages=p.+13%2C+below+eq.+52&rft.date=2007&rft_id=info%3Aarxiv%2F0706.0622v3&rft.au=The+LIGO+Scientific+Collaboration&rft.au=the+Virgo+Collaboration&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-carter_1968-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-carter_1968_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-carter_1968_29-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-carter_1968_29-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCarter1968" class="citation journal cs1"><a href="/wiki/Brandon_Carter" title="Brandon Carter">Carter, Brandon</a> (1968). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200225234931/https://pdfs.semanticscholar.org/dd72/c4b894cd582604e7c4b58bfa2120bfae8375.pdf">"Global structure of the Kerr family of gravitational fields"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review</i>. <b>174</b> (5): <span class="nowrap">1559–</span>1571. <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/1968PhRv..174.1559C">1968PhRv..174.1559C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.174.1559">10.1103/PhysRev.174.1559</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123261579">123261579</a>. Archived from <a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/dd72/c4b894cd582604e7c4b58bfa2120bfae8375.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-02-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Global+structure+of+the+Kerr+family+of+gravitational+fields&rft.volume=174&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1559-%3C%2Fspan%3E1571&rft.date=1968&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123261579%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRev.174.1559&rft_id=info%3Abibcode%2F1968PhRv..174.1559C&rft.aulast=Carter&rft.aufirst=Brandon&rft_id=https%3A%2F%2Fpdfs.semanticscholar.org%2Fdd72%2Fc4b894cd582604e7c4b58bfa2120bfae8375.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-bardeen1972-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-bardeen1972_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBardeenPressTeukolsky1972" class="citation journal cs1">Bardeen, James M.; Press, William H.; Teukolsky, Saul A. (1972). "Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation". <i>The Astrophysical Journal</i>. <b>178</b>: 347. <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/1972ApJ...178..347B">1972ApJ...178..347B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F151796">10.1086/151796</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=Rotating+Black+Holes%3A+Locally+Nonrotating+Frames%2C+Energy+Extraction%2C+and+Scalar+Synchrotron+Radiation&rft.volume=178&rft.pages=347&rft.date=1972&rft_id=info%3Adoi%2F10.1086%2F151796&rft_id=info%3Abibcode%2F1972ApJ...178..347B&rft.aulast=Bardeen&rft.aufirst=James+M.&rft.au=Press%2C+William+H.&rft.au=Teukolsky%2C+Saul+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFrolovFrolov2014" class="citation journal cs1">Frolov, Andrei V.; Frolov, Valeri P. (2014). "Rigidly rotating zero-angular-momentum observer surfaces in the Kerr spacetime". <i>Physical Review D</i>. <b>90</b> (12): 124010. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1408.6316">1408.6316</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvD..90l4010F">2014PhRvD..90l4010F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.90.124010">10.1103/PhysRevD.90.124010</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118417747">118417747</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Rigidly+rotating+zero-angular-momentum+observer+surfaces+in+the+Kerr+spacetime&rft.volume=90&rft.issue=12&rft.pages=124010&rft.date=2014&rft_id=info%3Aarxiv%2F1408.6316&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118417747%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.90.124010&rft_id=info%3Abibcode%2F2014PhRvD..90l4010F&rft.aulast=Frolov&rft.aufirst=Andrei+V.&rft.au=Frolov%2C+Valeri+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span>)</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRiazuelo2020" class="citation journal cs1">Riazuelo, Alain (December 2020). "Seeing Relativity -- III. Journeying within the Kerr metric toward the negative gravity region". <i>International Journal of Modern Physics D</i>. <b>29</b> (16): <span class="nowrap">2050109–</span>2050202. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.04384">2008.04384</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020IJMPD..2950109R">2020IJMPD..2950109R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218271820501096">10.1142/S0218271820501096</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/0218-2718">0218-2718</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221095833">221095833</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+D&rft.atitle=Seeing+Relativity+--+III.+Journeying+within+the+Kerr+metric+toward+the+negative+gravity+region&rft.volume=29&rft.issue=16&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2050109-%3C%2Fspan%3E2050202&rft.date=2020-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221095833%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2020IJMPD..2950109R&rft_id=info%3Aarxiv%2F2008.04384&rft.issn=0218-2718&rft_id=info%3Adoi%2F10.1142%2FS0218271820501096&rft.aulast=Riazuelo&rft.aufirst=Alain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-Chandrasekhar_1983-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chandrasekhar_1983_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChandrasekhar1983" class="citation book cs1"><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar, S.</a> (1983). <i>The Mathematical Theory of Black Holes</i>. International Series of Monographs on Physics. Vol. 69. p. 375.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Theory+of+Black+Holes&rft.series=International+Series+of+Monographs+on+Physics&rft.pages=375&rft.date=1983&rft.aulast=Chandrasekhar&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Andrew Hamilton: <a rel="nofollow" class="external text" href="http://jila.colorado.edu/~ajsh/insidebh/penrose.html#kerr">Black hole Penrose diagrams</a> (JILA Colorado)</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Penrose 1968</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBičákLedvinka1993" class="citation journal cs1">Bičák, Jří; Ledvinka, Tomáš (1993). "Relativistic disks as sources of the Kerr metric". <i>Physical Review Letters</i>. <b>71</b> (11): <span class="nowrap">1669–</span>1672. <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/1993PhRvL..71.1669B">1993PhRvL..71.1669B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.71.1669">10.1103/physrevlett.71.1669</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10054468">10054468</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Relativistic+disks+as+sources+of+the+Kerr+metric&rft.volume=71&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1669-%3C%2Fspan%3E1672&rft.date=1993&rft_id=info%3Apmid%2F10054468&rft_id=info%3Adoi%2F10.1103%2Fphysrevlett.71.1669&rft_id=info%3Abibcode%2F1993PhRvL..71.1669B&rft.aulast=Bi%C4%8D%C3%A1k&rft.aufirst=J%C5%99%C3%AD&rft.au=Ledvinka%2C+Tom%C3%A1%C5%A1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_metric&action=edit&section=25" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWiltshire,_David_L.Visser,_MattScott,_Susan_M.2009" class="citation book cs1">Wiltshire, David L.; Visser, Matt; <a href="/wiki/Susan_M._Scott" title="Susan M. Scott">Scott, Susan M.</a>, eds. (2009). <i>The Kerr Spacetime: Rotating Black Holes in General Relativity</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88512-6" title="Special:BookSources/978-0-521-88512-6"><bdi>978-0-521-88512-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Kerr+Spacetime%3A+Rotating+Black+Holes+in+General+Relativity&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-88512-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStephani,_HansKramer,_DietrichMacCallum,_MalcolmHoenselaers,_Cornelius2003" class="citation book cs1">Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003). <i>Exact Solutions of Einstein's Field Equations</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46136-8" title="Special:BookSources/978-0-521-46136-8"><bdi>978-0-521-46136-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exact+Solutions+of+Einstein%27s+Field+Equations&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-46136-8&rft.au=Stephani%2C+Hans&rft.au=Kramer%2C+Dietrich&rft.au=MacCallum%2C+Malcolm&rft.au=Hoenselaers%2C+Cornelius&rft.au=Herlt%2C+Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMeinelAnsorgKleinwachterNeugebauer2008" class="citation book cs1"><a href="/wiki/Reinhard_Meinel" title="Reinhard Meinel">Meinel, Reinhard</a>; Ansorg, Marcus; Kleinwachter, Andreas; Neugebauer, Gernot; Petroff, David (2008). <a rel="nofollow" class="external text" href="http://www.cambridge.org/9780521863834"><i>Relativistic Figures of Equilibrium</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86383-4" title="Special:BookSources/978-0-521-86383-4"><bdi>978-0-521-86383-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Figures+of+Equilibrium&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-86383-4&rft.aulast=Meinel&rft.aufirst=Reinhard&rft.au=Ansorg%2C+Marcus&rft.au=Kleinwachter%2C+Andreas&rft.au=Neugebauer%2C+Gernot&rft.au=Petroff%2C+David&rft_id=http%3A%2F%2Fwww.cambridge.org%2F9780521863834&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'Neill1995" class="citation book cs1">O'Neill, Barrett (1995). <i>The Geometry of Kerr Black Holes</i>. Wellesley, Massachusetts: A. K. Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-019-5" title="Special:BookSources/978-1-56881-019-5"><bdi>978-1-56881-019-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Kerr+Black+Holes&rft.place=Wellesley%2C+Massachusetts&rft.pub=A.+K.+Peters&rft.date=1995&rft.isbn=978-1-56881-019-5&rft.aulast=O%27Neill&rft.aufirst=Barrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFD'Inverno1992" class="citation book cs1">D'Inverno, Ray (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introducingeinst0000dinv"><i>Introducing Einstein's Relativity</i></a></span>. Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-859686-8" title="Special:BookSources/978-0-19-859686-8"><bdi>978-0-19-859686-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Einstein%27s+Relativity&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1992&rft.isbn=978-0-19-859686-8&rft.aulast=D%27Inverno&rft.aufirst=Ray&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroducingeinst0000dinv&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> <i>See chapter 19</i> for a readable introduction at the advanced undergraduate level.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChandrasekhar1992" class="citation book cs1"><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar, S.</a> (1992). <i>The Mathematical Theory of Black Holes</i>. Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850370-5" title="Special:BookSources/978-0-19-850370-5"><bdi>978-0-19-850370-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Theory+of+Black+Holes&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1992&rft.isbn=978-0-19-850370-5&rft.aulast=Chandrasekhar&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> <i>See chapters 6--10</i> for a very thorough study at the advanced graduate level.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriffiths1991" class="citation book cs1">Griffiths, J. B. (1991). <i>Colliding Plane Waves in General Relativity</i>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853209-5" title="Special:BookSources/978-0-19-853209-5"><bdi>978-0-19-853209-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Colliding+Plane+Waves+in+General+Relativity&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=1991&rft.isbn=978-0-19-853209-5&rft.aulast=Griffiths&rft.aufirst=J.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> <i>See chapter 13</i> for the Chandrasekhar/Ferrari CPW model.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdlerBazinSchiffer1975" class="citation book cs1">Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoge0000adle"><i>Introduction to General Relativity</i></a></span> (Second ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-000423-8" title="Special:BookSources/978-0-07-000423-8"><bdi>978-0-07-000423-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+General+Relativity&rft.place=New+York&rft.edition=Second&rft.pub=McGraw-Hill&rft.date=1975&rft.isbn=978-0-07-000423-8&rft.aulast=Adler&rft.aufirst=Ronald&rft.au=Bazin%2C+Maurice&rft.au=Schiffer%2C+Menahem&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoge0000adle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> <i>See chapter 7</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPenrose1968" class="citation book cs1"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, R.</a> (1968). ed C. de Witt and J. Wheeler (ed.). <i>Battelle Rencontres</i>. W. A. Benjamin, New York. p. 222.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Battelle+Rencontres&rft.pages=222&rft.pub=W.+A.+Benjamin%2C+New+York&rft.date=1968&rft.aulast=Penrose&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPerezMoreschi2000" class="citation arxiv cs1">Perez, Alejandro; Moreschi, Osvaldo M. (2000). "Characterizing exact solutions from asymptotic physical concepts". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0012100v1">gr-qc/0012100v1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Characterizing+exact+solutions+from+asymptotic+physical+concepts&rft.date=2000&rft_id=info%3Aarxiv%2Fgr-qc%2F0012100v1&rft.aulast=Perez&rft.aufirst=Alejandro&rft.au=Moreschi%2C+Osvaldo+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> Characterization of three standard families of vacuum solutions as noted above.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSotiriouApostolatos2004" class="citation journal cs1">Sotiriou, Thomas P.; Apostolatos, Theocharis A. (2004). "Corrections and Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes". <i>Class. Quantum Grav</i>. <b>21</b> (24): <span class="nowrap">5727–</span>5733. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0407064">gr-qc/0407064</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004CQGra..21.5727S">2004CQGra..21.5727S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F21%2F24%2F003">10.1088/0264-9381/21/24/003</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16858122">16858122</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Class.+Quantum+Grav.&rft.atitle=Corrections+and+Comments+on+the+Multipole+Moments+of+Axisymmetric+Electrovacuum+Spacetimes&rft.volume=21&rft.issue=24&rft.pages=%3Cspan+class%3D%22nowrap%22%3E5727-%3C%2Fspan%3E5733&rft.date=2004&rft_id=info%3Aarxiv%2Fgr-qc%2F0407064&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16858122%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F21%2F24%2F003&rft_id=info%3Abibcode%2F2004CQGra..21.5727S&rft.aulast=Sotiriou&rft.aufirst=Thomas+P.&rft.au=Apostolatos%2C+Theocharis+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> Gives the relativistic multipole moments for the Ernst vacuums (plus the electromagnetic and gravitational relativistic multipole moments for the charged generalization).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCarter1971" class="citation journal cs1"><a href="/wiki/Brandon_Carter" title="Brandon Carter">Carter, B.</a> (1971). "Axisymmetric Black Hole Has Only Two Degrees of Freedom". <i>Physical Review Letters</i>. <b>26</b> (6): <span class="nowrap">331–</span>333. <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/1971PhRvL..26..331C">1971PhRvL..26..331C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.26.331">10.1103/PhysRevLett.26.331</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Axisymmetric+Black+Hole+Has+Only+Two+Degrees+of+Freedom&rft.volume=26&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E331-%3C%2Fspan%3E333&rft.date=1971&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.26.331&rft_id=info%3Abibcode%2F1971PhRvL..26..331C&rft.aulast=Carter&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWald1984" class="citation book cs1"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, R. M.</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/generalrelativit0000wald"><i>General Relativity</i></a></span>. Chicago: The University of Chicago Press. pp. <span class="nowrap">312–</span>324. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-87032-8" title="Special:BookSources/978-0-226-87032-8"><bdi>978-0-226-87032-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity&rft.place=Chicago&rft.pages=%3Cspan+class%3D%22nowrap%22%3E312-%3C%2Fspan%3E324&rft.pub=The+University+of+Chicago+Press&rft.date=1984&rft.isbn=978-0-226-87032-8&rft.aulast=Wald&rft.aufirst=R.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeneralrelativit0000wald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKerr,_R._P.Schild,_A.2009" class="citation journal cs1">Kerr, R. P.; Schild, A. (2009). "Republication of: A new class of vacuum solutions of the Einstein field equations". <i>General Relativity and Gravitation</i>. <b>41</b> (10): <span class="nowrap">2485–</span>2499. <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/2009GReGr..41.2485K">2009GReGr..41.2485K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10714-009-0857-z">10.1007/s10714-009-0857-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:361088">361088</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=Republication+of%3A+A+new+class+of+vacuum+solutions+of+the+Einstein+field+equations&rft.volume=41&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2485-%3C%2Fspan%3E2499&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A361088%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10714-009-0857-z&rft_id=info%3Abibcode%2F2009GReGr..41.2485K&rft.au=Kerr%2C+R.+P.&rft.au=Schild%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKrasińskiVerdaguerKerr2009" class="citation journal cs1">Krasiński, Andrzej; Verdaguer, Enric; Kerr, Roy Patrick (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10714-009-0856-0">"Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations"</a>. <i>General Relativity and Gravitation</i>. <b>41</b> (10): <span class="nowrap">2469–</span>2484. <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/2009GReGr..41.2469K">2009GReGr..41.2469K</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/10.1007%2Fs10714-009-0856-0">10.1007/s10714-009-0856-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=Editorial+note+to%3A+R.+P.+Kerr+and+A.+Schild%2C+A+new+class+of+vacuum+solutions+of+the+Einstein+field+equations&rft.volume=41&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2469-%3C%2Fspan%3E2484&rft.date=2009&rft_id=info%3Adoi%2F10.1007%2Fs10714-009-0856-0&rft_id=info%3Abibcode%2F2009GReGr..41.2469K&rft.aulast=Krasi%C5%84ski&rft.aufirst=Andrzej&rft.au=Verdaguer%2C+Enric&rft.au=Kerr%2C+Roy+Patrick&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs10714-009-0856-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+metric" class="Z3988"></span> "... This note is meant to be a guide for those readers who wish to verify all the details [of the derivation of the Kerr solution] ..."</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Black_holes386" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Black_holes" title="Template:Black holes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Black_holes" title="Template talk:Black holes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Black_holes" title="Special:EditPage/Template:Black holes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Black_holes386" style="font-size:114%;margin:0 4em"><a href="/wiki/Black_hole" title="Black hole">Black holes</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3" style="text-align:center;"><div> <ul><li><a href="/wiki/Outline_of_black_holes" title="Outline of black holes">Outline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BTZ_black_hole" title="BTZ black hole">BTZ black hole</a></li> <li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a></li> <li><a href="/wiki/Rotating_black_hole" title="Rotating black hole">Rotating</a></li> <li><a href="/wiki/Charged_black_hole" title="Charged black hole">Charged</a></li> <li><a href="/wiki/Virtual_black_hole" title="Virtual black hole">Virtual</a></li> <li><a href="/wiki/Kugelblitz_(astrophysics)" title="Kugelblitz (astrophysics)">Kugelblitz</a></li> <li><a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">Supermassive</a></li> <li><a href="/wiki/Primordial_black_hole" title="Primordial black hole">Primordial</a></li> <li><a href="/wiki/Direct_collapse_black_hole" title="Direct collapse black hole">Direct collapse</a></li> <li><a href="/wiki/Rogue_black_hole" title="Rogue black hole">Rogue</a></li> <li><a href="/wiki/Malament%E2%80%93Hogarth_spacetime" title="Malament–Hogarth spacetime">Malament–Hogarth spacetime</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="11" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Black_hole_-_Messier_87_crop_max_res.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Black_hole_-_Messier_87_crop_max_res.jpg/120px-Black_hole_-_Messier_87_crop_max_res.jpg" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Black_hole_-_Messier_87_crop_max_res.jpg/250px-Black_hole_-_Messier_87_crop_max_res.jpg 2x" data-file-width="4320" data-file-height="4320" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Size</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Micro_black_hole" title="Micro black hole">Micro</a> <ul><li><a href="/wiki/Extremal_black_hole" title="Extremal black hole">Extremal</a></li> <li><a href="/wiki/Black_hole_electron" title="Black hole electron">Electron</a></li></ul></li> <li><a href="/wiki/Stellar_black_hole" title="Stellar black hole">Stellar</a> <ul><li><a href="/wiki/Microquasar" title="Microquasar">Microquasar</a></li></ul></li> <li><a href="/wiki/Intermediate-mass_black_hole" title="Intermediate-mass black hole">Intermediate-mass</a></li> <li><a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">Supermassive</a> <ul><li><a href="/wiki/Active_galactic_nucleus" title="Active galactic nucleus">Active galactic nucleus</a></li> <li><a href="/wiki/Quasar" title="Quasar">Quasar</a></li> <li><a href="/wiki/Large_quasar_group" title="Large quasar group">LQG</a></li> <li><a href="/wiki/Blazar" title="Blazar">Blazar</a></li> <li><a href="/wiki/OVV_quasar" title="OVV quasar">OVV</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Formation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Stellar_evolution" title="Stellar evolution">Stellar evolution</a></li> <li><a href="/wiki/Gravitational_collapse" title="Gravitational collapse">Gravitational collapse</a></li> <li><a href="/wiki/Neutron_star" title="Neutron star">Neutron star</a> <ul><li><a href="/wiki/Template:Neutron_star" title="Template:Neutron star">Related links</a></li></ul></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit" title="Tolman–Oppenheimer–Volkoff limit">Tolman–Oppenheimer–Volkoff limit</a></li> <li><a href="/wiki/White_dwarf" title="White dwarf">White dwarf</a> <ul><li><a href="/wiki/Template:White_dwarf" title="Template:White dwarf">Related links</a></li></ul></li> <li><a href="/wiki/Supernova" title="Supernova">Supernova</a> <ul><li><a href="/wiki/Micronova" title="Micronova">Micronova</a></li> <li><a href="/wiki/Superluminous_supernova" title="Superluminous supernova">Hypernova</a></li> <li><a href="/wiki/Template:Supernovae" title="Template:Supernovae">Related links</a></li></ul></li> <li><a href="/wiki/Gamma-ray_burst" title="Gamma-ray burst">Gamma-ray burst</a></li> <li><a href="/wiki/Binary_black_hole" title="Binary black hole">Binary black hole</a></li> <li><a href="/wiki/Quark_star" title="Quark star">Quark star</a></li> <li><a href="/wiki/Supermassive_star" class="mw-redirect" title="Supermassive star">Supermassive star</a></li> <li><a href="/wiki/Quasi-star" title="Quasi-star">Quasi-star</a></li> <li><a href="/wiki/Dark_star_(dark_matter)" title="Dark star (dark matter)">Supermassive dark star</a></li> <li><a href="/wiki/X-ray_binary" title="X-ray binary">X-ray binary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysical_jet" title="Astrophysical jet">Astrophysical jet</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Gravitational singularity</a> <ul><li><a href="/wiki/Ring_singularity" title="Ring singularity">Ring singularity</a></li> <li><a href="/wiki/Penrose%E2%80%93Hawking_singularity_theorems" title="Penrose–Hawking singularity theorems">Theorems</a></li></ul></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Photon_sphere" title="Photon sphere">Photon sphere</a></li> <li><a href="/wiki/Innermost_stable_circular_orbit" title="Innermost stable circular orbit">Innermost stable circular orbit</a></li> <li><a href="/wiki/Ergosphere" title="Ergosphere">Ergosphere</a> <ul><li><a href="/wiki/Penrose_process" title="Penrose process">Penrose process</a></li> <li><a href="/wiki/Blandford%E2%80%93Znajek_process" title="Blandford–Znajek process">Blandford–Znajek process</a></li></ul></li> <li><a href="/wiki/Accretion_disk" title="Accretion disk">Accretion disk</a></li> <li><a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lens</a> <ul><li><a href="/wiki/Gravitational_microlensing" title="Gravitational microlensing">Microlens</a></li></ul></li> <li><a href="/wiki/Bondi_accretion" title="Bondi accretion">Bondi accretion</a></li> <li><a href="/wiki/M%E2%80%93sigma_relation" title="M–sigma relation">M–sigma relation</a></li> <li><a href="/wiki/Quasi-periodic_oscillation" class="mw-redirect" title="Quasi-periodic oscillation">Quasi-periodic oscillation</a></li> <li><a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Bekenstein_bound" title="Bekenstein bound">Bekenstein bound</a></li> <li><a href="/wiki/Bousso%27s_holographic_bound" title="Bousso's holographic bound">Bousso's holographic bound</a> <ul><li><a href="/wiki/Immirzi_parameter" title="Immirzi parameter">Immirzi parameter</a></li></ul></li> <li><a href="/wiki/Schwarzschild_radius" title="Schwarzschild radius">Schwarzschild radius</a></li> <li><a href="/wiki/Spaghettification" title="Spaghettification">Spaghettification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Issues</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole_complementarity" title="Black hole complementarity">Black hole complementarity</a></li> <li><a href="/wiki/Black_hole_information_paradox" title="Black hole information paradox">Information paradox</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship</a></li> <li><a href="/wiki/ER_%3D_EPR" title="ER = EPR">ER = EPR</a></li> <li><a href="/wiki/Binary_black_hole#Final_parsec_problem" title="Binary black hole">Final parsec problem</a></li> <li><a href="/wiki/Firewall_(physics)" title="Firewall (physics)">Firewall (physics)</a></li> <li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/No-hair_theorem" title="No-hair theorem">No-hair theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Metrics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Derivation_of_the_Schwarzschild_solution" title="Derivation of the Schwarzschild solution">Derivation</a>)</li> <li><a class="mw-selflink selflink">Kerr</a></li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Hayward_metric" title="Hayward metric">Hayward</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Alternatives</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonsingular_black_hole_models" title="Nonsingular black hole models">Nonsingular black hole models</a></li> <li><a href="/wiki/Black_star_(semiclassical_gravity)" title="Black star (semiclassical gravity)">Black star</a></li> <li><a href="/wiki/Dark_star_(Newtonian_mechanics)" title="Dark star (Newtonian mechanics)">Dark star</a></li> <li><a href="/wiki/Dark-energy_star" title="Dark-energy star">Dark-energy star</a></li> <li><a href="/wiki/Gravastar" title="Gravastar">Gravastar</a></li> <li><a href="/wiki/Magnetospheric_eternally_collapsing_object" title="Magnetospheric eternally collapsing object">Magnetospheric eternally collapsing object</a></li> <li><a href="/wiki/Planck_star" title="Planck star">Planck star</a></li> <li><a href="/wiki/Q_star" title="Q star">Q star</a></li> <li><a href="/wiki/Fuzzball_(string_theory)" title="Fuzzball (string theory)">Fuzzball</a></li> <li><a href="/wiki/Geon_(physics)" title="Geon (physics)">Geon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Analogs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Optical_black_hole" title="Optical black hole">Optical black hole</a></li> <li><a href="/wiki/Sonic_black_hole" title="Sonic black hole">Sonic black hole</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Lists_of_black_holes" title="Lists of black holes">Lists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_black_holes" title="List of black holes">Black holes</a></li> <li><a href="/wiki/List_of_most_massive_black_holes" title="List of most massive black holes">Most massive</a></li> <li><a href="/wiki/List_of_nearest_known_black_holes" title="List of nearest known black holes">Nearest</a></li> <li><a href="/wiki/List_of_quasars" title="List of quasars">Quasars</a></li> <li><a href="/wiki/List_of_microquasars" title="List of microquasars">Microquasars</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Outline_of_black_holes" title="Outline of black holes">Outline of black holes</a></li> <li><a href="/wiki/Black_Hole_Initiative" title="Black Hole Initiative">Black Hole Initiative</a></li> <li><a href="/wiki/Black_hole_starship" title="Black hole starship">Black hole starship</a></li> <li><a href="/wiki/Black_holes_in_fiction" title="Black holes in fiction">Black holes in fiction</a></li> <li><a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a></li> <li><a href="/wiki/Big_Bounce" title="Big Bounce">Big Bounce</a></li> <li><a href="/wiki/Compact_star" class="mw-redirect" title="Compact star">Compact star</a></li> <li><a href="/wiki/Exotic_star" title="Exotic star">Exotic star</a> <ul><li><a href="/wiki/Quark_star" title="Quark star">Quark star</a></li> <li><a href="/wiki/Preon_star" class="mw-redirect" title="Preon star">Preon star</a></li></ul></li> <li><a href="/wiki/Gravitational_waves" class="mw-redirect" title="Gravitational waves">Gravitational waves</a></li> <li><a href="/wiki/Gamma-ray_burst_progenitors" title="Gamma-ray burst progenitors">Gamma-ray burst progenitors</a></li> <li><a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">Gravity well</a></li> <li><a href="/wiki/Hypercompact_stellar_system" title="Hypercompact stellar system">Hypercompact stellar system</a></li> <li><a href="/wiki/Membrane_paradigm" title="Membrane paradigm">Membrane paradigm</a></li> <li><a href="/wiki/Naked_singularity" title="Naked singularity">Naked singularity</a></li> <li><a href="/wiki/Population_III_star" class="mw-redirect" title="Population III star">Population III star</a></li> <li><a href="/wiki/Supermassive_star" class="mw-redirect" title="Supermassive star">Supermassive star</a></li> <li><a href="/wiki/Quasi-star" title="Quasi-star">Quasi-star</a></li> <li><a href="/wiki/Dark_star_(dark_matter)" title="Dark star (dark matter)">Supermassive dark star</a></li> <li><a href="/wiki/Rossi_X-ray_Timing_Explorer" title="Rossi X-ray Timing Explorer">Rossi X-ray Timing Explorer</a></li> <li><a href="/wiki/Superluminal_motion" title="Superluminal motion">Superluminal motion</a></li> <li><a href="/wiki/Timeline_of_black_hole_physics" title="Timeline of black hole physics">Timeline of black hole physics</a></li> <li><a href="/wiki/White_hole" title="White hole">White hole</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/Tidal_disruption_event" title="Tidal disruption event">Tidal disruption event</a></li> <li><a href="/wiki/Planet_Nine" title="Planet Nine">Planet Nine</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Notable</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cygnus_X-1" title="Cygnus X-1">Cygnus X-1</a></li> <li><a href="/wiki/XTE_J1650-500" class="mw-redirect" title="XTE J1650-500">XTE J1650-500</a></li> <li><a href="/wiki/XTE_J1118%2B480" title="XTE J1118+480">XTE J1118+480</a></li> <li><a href="/wiki/A0620-00" title="A0620-00">A0620-00</a></li> <li><a href="/wiki/Sagittarius_A*" title="Sagittarius A*">Sagittarius A*</a></li> <li><a href="/wiki/Centaurus_A" title="Centaurus A">Centaurus A</a></li> <li><a href="/wiki/Phoenix_Cluster" title="Phoenix Cluster">Phoenix Cluster</a></li> <li><a href="/wiki/PKS_1302-102" class="mw-redirect" title="PKS 1302-102">PKS 1302-102</a></li> <li><a href="/wiki/OJ_287" title="OJ 287">OJ 287</a></li> <li><a href="/wiki/SDSS_J0849%2B1114" title="SDSS J0849+1114">SDSS J0849+1114</a></li> <li><a href="/wiki/TON_618" title="TON 618">TON 618</a></li> <li><a href="/wiki/MS_0735.6%2B7421" title="MS 0735.6+7421">MS 0735.6+7421</a></li> <li><a href="/wiki/NeVe_1" title="NeVe 1">NeVe 1</a></li> <li><a href="/wiki/Hercules_A" title="Hercules A">Hercules A</a></li> <li><a href="/wiki/3C_273" title="3C 273">3C 273</a></li> <li><a href="/wiki/Q0906%2B6930" title="Q0906+6930">Q0906+6930</a></li> <li><a href="/wiki/Markarian_501" title="Markarian 501">Markarian 501</a></li> <li><a href="/wiki/ULAS_J1342%2B0928" title="ULAS J1342+0928">ULAS J1342+0928</a></li> <li><a href="/wiki/PSO_J030947.49%2B271757.31" title="PSO J030947.49+271757.31">PSO J030947.49+271757.31</a></li> <li><a href="/wiki/AT2018hyz" title="AT2018hyz">AT2018hyz</a></li> <li><a href="/wiki/Swift_J1644%2B57" title="Swift J1644+57">Swift J1644+57</a></li> <li><a href="/wiki/1ES_1927%2B654" title="1ES 1927+654">1ES 1927+654</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3" style="text-align:center;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Black_holes" title="Category:Black holes">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Black_holes" class="extiw" title="commons:Category:Black holes">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Time_travel41" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_travel" title="Template:Time travel"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_travel" title="Template talk:Time travel"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_travel" title="Special:EditPage/Template:Time travel"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time_travel41" style="font-size:114%;margin:0 4em"><a href="/wiki/Time_travel" title="Time travel">Time travel</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">Chronology protection conjecture</a></li> <li><a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">Closed timelike curve</a></li> <li><a href="/wiki/Novikov_self-consistency_principle" title="Novikov self-consistency principle">Novikov self-consistency principle</a></li> <li><a href="/wiki/Self-fulfilling_prophecy#Causal_loop" title="Self-fulfilling prophecy">Self-fulfilling prophecy</a></li> <li><a href="/wiki/Quantum_mechanics_of_time_travel" title="Quantum mechanics of time travel">Quantum mechanics of time travel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophy</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/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of space and time</a></li> <li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a></li> <li><a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">Eternalism</a></li> <li><a href="/wiki/Fatalism" title="Fatalism">Fatalism</a></li> <li><a href="/wiki/Free_will" title="Free will">Free will</a></li> <li><a href="/wiki/Predestination" title="Predestination">Predestination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Causality</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Causal_loop" class="mw-redirect" title="Causal loop">Causal loop</a></li> <li><a href="/wiki/Causality_(physics)" title="Causality (physics)">Causality (physics)</a></li> <li><a href="/wiki/Causal_structure" title="Causal structure">Causal structure</a></li> <li><a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">Chronology protection conjecture</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship hypothesis</a></li> <li><a href="/wiki/The_chicken_or_the_egg" class="mw-redirect" title="The chicken or the egg">The chicken or the egg</a></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds interpretation</a></li> <li><a href="/wiki/Grandfather_paradox" class="mw-redirect" title="Grandfather paradox">Grandfather paradox</a></li> <li><a href="/wiki/Quantum_mechanics_of_time_travel" title="Quantum mechanics of time travel">Quantum mechanics of time travel</a></li> <li><a href="/wiki/Time_viewer" title="Time viewer">Time viewer</a></li> <li><a href="/wiki/Temporal_paradox" title="Temporal paradox">Temporal paradox</a></li> <li><a href="/wiki/Temporal_paradox#Grandfather_paradox" title="Temporal paradox">Grandfather paradox</a></li> <li><a href="/wiki/Time_loop" title="Time loop">Time loop</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_travel_in_fiction" title="Time travel in fiction">Time travel in fiction</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_timelines#Fictional" title="List of timelines">Timelines in fiction</a></li> <li><a href="/wiki/List_of_time_travel_works_of_fiction" title="List of time travel works of fiction">Time travel in fiction</a></li> <li><a href="/wiki/List_of_films_featuring_time_loops" title="List of films featuring time loops">Time loops in film</a></li> <li><i>Major works:</i></li> <li><i><a href="/wiki/The_Time_Machine" title="The Time Machine">The Time Machine</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Parallel timelines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_travel_in_fiction#Alternative_future,_history,_timelines,_and_dimensions" title="Time travel in fiction">Alternative future</a></li> <li><a href="/wiki/Alternate_history" title="Alternate history">Alternate history</a></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds interpretation</a></li> <li><a href="/wiki/Multiverse" title="Multiverse">Multiverse</a></li> <li><a href="/wiki/Parallel_universes_in_fiction" title="Parallel universes in fiction">Parallel universes in fiction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Circular <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in <br /> <a href="/wiki/General_relativity" title="General relativity">general relativity</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre metric</a></li> <li><a href="/wiki/BTZ_black_hole" title="BTZ black hole">BTZ black hole</a></li> <li><a href="/wiki/Closed_timelike_curve" title="Closed timelike curve">Closed timelike curves</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel metric</a></li> <li><a class="mw-selflink selflink">Kerr metric</a></li> <li><a href="/wiki/Krasnikov_tube" title="Krasnikov tube">Krasnikov tube</a></li> <li><a href="/wiki/Misner_space" title="Misner space">Misner space</a></li> <li><a href="/wiki/Tipler_cylinder" title="Tipler cylinder">Tipler cylinder</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Wormhole#Traversable_wormholes" title="Wormhole">Traversable wormholes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li> <li><a href="/wiki/Cauchy_horizon" title="Cauchy horizon">Cauchy horizon</a></li> <li><a href="/wiki/Quantum_mechanics_of_time_travel" title="Quantum mechanics of time travel">Quantum mechanics of time travel</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Chronology_protection_conjecture" title="Chronology protection conjecture">Chronology protection conjecture</a></li> <li><a href="/wiki/Retrocausality" title="Retrocausality">Retrocausality</a></li> <li><a href="/wiki/Time_reversal_symmetry" class="mw-redirect" title="Time reversal symmetry">Time reversal symmetry</a></li> <li><a href="/wiki/Wheeler%E2%80%93Feynman_time-symmetric_theory" class="mw-redirect" title="Wheeler–Feynman time-symmetric theory">Wheeler–Feynman time-symmetric theory</a></li> <li><a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a></li> <li><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></li> <li><a href="/wiki/Four-dimensionalism" title="Four-dimensionalism">Four-dimensionalism</a></li> <li><a href="/wiki/Tipler_time_machine" class="mw-redirect" title="Tipler time machine">Tipler time machine</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_fiction" title="Science fiction">Science fiction</a></li> <li><a href="/wiki/Time_machine" title="Time machine">Time machine</a></li> <li><i>Games:</i> <a href="/wiki/List_of_four-dimensional_games" title="List of four-dimensional games">List of four-dimensional games</a></li> <li><a href="/wiki/5D_Chess_with_Multiverse_Time_Travel" title="5D Chess with Multiverse Time Travel">5D Chess with Multiverse Time Travel</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Relativity254" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity254" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a 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 class="mw-selflink selflink">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li> <li><a href="/wiki/Ellis_wormhole" title="Ellis wormhole">Ellis</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Kerr_metric&oldid=1277991679">https://en.wikipedia.org/w/index.php?title=Kerr_metric&oldid=1277991679</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Exact_solutions_in_general_relativity" title="Category:Exact solutions in general relativity">Exact solutions in general relativity</a></li><li><a href="/wiki/Category:Black_holes" title="Category:Black holes">Black holes</a></li><li><a href="/wiki/Category:Metric_tensors" title="Category:Metric tensors">Metric tensors</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_February_2011" title="Category:Articles needing additional references from February 2011">Articles needing additional references from February 2011</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 February 2025, at 22:11<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Kerr_metric&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Kerr metric</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>22 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c6f46dcf-xcxcp","wgBackendResponseTime":246,"wgPageParseReport":{"limitreport":{"cputime":"0.839","walltime":"1.241","ppvisitednodes":{"value":5822,"limit":1000000},"postexpandincludesize":{"value":229904,"limit":2097152},"templateargumentsize":{"value":5918,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":6,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":249632,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 705.958 1 -total"," 33.78% 238.492 2 Template:Reflist"," 22.41% 158.238 21 Template:Cite_journal"," 15.35% 108.356 1 Template:General_relativity_sidebar"," 14.96% 105.598 1 Template:Sidebar_with_collapsible_lists"," 9.36% 66.080 1 Template:Short_description"," 7.78% 54.920 14 Template:Cite_book"," 6.60% 46.611 17 Template:NumBlk"," 6.31% 44.529 1 Template:More_citations_needed_section"," 6.15% 43.434 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"0.393","limit":"10.000"},"limitreport-memusage":{"value":8877777,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-779c5f569f-6dk2n","timestamp":"20250328191141","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Kerr metric","url":"https:\/\/en.wikipedia.org\/wiki\/Kerr_metric","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1068747","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1068747","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-02-06T03:35:12Z","dateModified":"2025-02-27T22:11:30Z","headline":"exact solution for the Einstein field equations"}</script> </body> </html>

CINXE.COM

Kerr metric - Wikipedia