User:Christillin/gravitational waves

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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Effects of a passing gravitational wave</span> </div> </a> <ul id="toc-Effects_of_a_passing_gravitational_wave-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources_of_gravitational_waves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources_of_gravitational_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Sources of gravitational waves</span> </div> </a> <button aria-controls="toc-Sources_of_gravitational_waves-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 Sources of gravitational waves subsection</span> </button> <ul id="toc-Sources_of_gravitational_waves-sublist" class="vector-toc-list"> <li id="toc-Power_radiated_by_orbiting_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_radiated_by_orbiting_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Power radiated by orbiting bodies</span> </div> </a> <ul id="toc-Power_radiated_by_orbiting_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_decay_from_gravitational_radiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_decay_from_gravitational_radiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Orbital decay from gravitational radiation</span> </div> </a> <ul id="toc-Orbital_decay_from_gravitational_radiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital_lifetime_limits_from_gravitational_radiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_lifetime_limits_from_gravitational_radiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Orbital lifetime limits from gravitational radiation</span> </div> </a> <ul id="toc-Orbital_lifetime_limits_from_gravitational_radiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wave_amplitudes_from_the_Earth-Sun_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wave_amplitudes_from_the_Earth-Sun_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Wave amplitudes from the Earth-Sun system</span> </div> </a> <ul id="toc-Wave_amplitudes_from_the_Earth-Sun_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radiation_from_other_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radiation_from_other_sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Radiation from other sources</span> </div> </a> <ul id="toc-Radiation_from_other_sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Astrophysics_and_gravitational_waves" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Astrophysics_and_gravitational_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Astrophysics and gravitational waves</span> </div> </a> <button aria-controls="toc-Astrophysics_and_gravitational_waves-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 Astrophysics and gravitational waves subsection</span> </button> <ul id="toc-Astrophysics_and_gravitational_waves-sublist" class="vector-toc-list"> <li id="toc-Energy,_momentum,_and_angular_momentum_carried_by_gravitational_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy,_momentum,_and_angular_momentum_carried_by_gravitational_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Energy, momentum, and angular momentum carried by gravitational waves</span> </div> </a> <ul id="toc-Energy,_momentum,_and_angular_momentum_carried_by_gravitational_waves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Gravitational_wave_detectors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gravitational_wave_detectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Gravitational wave detectors</span> </div> </a> <button aria-controls="toc-Gravitational_wave_detectors-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 Gravitational wave detectors subsection</span> </button> <ul id="toc-Gravitational_wave_detectors-sublist" class="vector-toc-list"> <li id="toc-Einstein@Home" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Einstein@Home"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Einstein@Home</span> </div> </a> <ul id="toc-Einstein@Home-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Mathematics</span> </div> </a> <button aria-controls="toc-Mathematics-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 Mathematics subsection</span> </button> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> <li id="toc-Linear_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Linear approximation</span> </div> </a> <ul id="toc-Linear_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_source" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_source"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Relation to the source</span> </div> </a> <ul id="toc-Relation_to_the_source-sublist" class="vector-toc-list"> </ul> </li> </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">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-namespace">User</span><span class="mw-page-title-separator">:</span><span class="mw-page-title-main">Christillin/gravitational waves</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. 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href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0_%D0%B2%D1%8A%D0%BB%D0%BD%D0%B0" title="Гравитационна вълна – Bulgarian" lang="bg" hreflang="bg" data-title="Гравитационна вълна" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ona_gravitat%C3%B2ria" title="Ona gravitatòria – Catalan" lang="ca" hreflang="ca" data-title="Ona gravitatòria" 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/Gravita%C4%8Dn%C3%AD_vlna" title="Gravitační vlna – Czech" lang="cs" hreflang="cs" data-title="Gravitační vlna" 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/Gravitationswelle" title="Gravitationswelle – German" lang="de" hreflang="de" data-title="Gravitationswelle" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%92%CE%B1%CF%81%CF%85%CF%84%CE%B9%CE%BA%CE%AC_%CE%BA%CF%8D%CE%BC%CE%B1%CF%84%CE%B1" title="Βαρυτικά κύματα – Greek" lang="el" hreflang="el" data-title="Βαρυτικά κύματα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Onda_gravitacional" title="Onda gravitacional – Spanish" lang="es" hreflang="es" data-title="Onda gravitacional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC_%DA%AF%D8%B1%D8%A7%D9%86%D8%B4%DB%8C" title="موج گرانشی – Persian" lang="fa" hreflang="fa" data-title="موج گرانشی" 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/Onde_gravitationnelle" title="Onde gravitationnelle – French" lang="fr" hreflang="fr" data-title="Onde gravitationnelle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Onda_gravitacional" title="Onda gravitacional – Galician" lang="gl" hreflang="gl" data-title="Onda gravitacional" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A4%91%EB%A0%A5%ED%8C%8C" title="중력파 – Korean" lang="ko" hreflang="ko" data-title="중력파" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Onda_gravitazionale" title="Onda gravitazionale – Italian" lang="it" hreflang="it" data-title="Onda gravitazionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%91%D7%99%D7%93%D7%94#גלי_כבידה" title="כבידה – Hebrew" lang="he" hreflang="he" data-title="כבידה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D1%82%D0%BE%D0%BB%D2%9B%D1%8B%D0%BD" title="Гравитациялық толқын – Kazakh" lang="kk" hreflang="kk" data-title="Гравитациялық толқын" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Gravit%C4%81cijas_vi%C4%BC%C5%86i" title="Gravitācijas viļņi – Latvian" lang="lv" hreflang="lv" data-title="Gravitācijas viļņi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Gravitacin%C4%97s_bangos" title="Gravitacinės bangos – Lithuanian" lang="lt" hreflang="lt" data-title="Gravitacinės bangos" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gravit%C3%A1ci%C3%B3s_hull%C3%A1m" title="Gravitációs hullám – Hungarian" lang="hu" hreflang="hu" data-title="Gravitációs hullám" 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%97%E0%B5%81%E0%B4%B0%E0%B5%81%E0%B4%A4%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%BC%E0%B4%B7%E0%B4%A3%E0%B4%A4%E0%B4%B0%E0%B4%82%E0%B4%97%E0%B4%82" title="ഗുരുത്വാകർഷണതരംഗം – Malayalam" lang="ml" hreflang="ml" data-title="ഗുരുത്വാകർഷണതരംഗം" 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/Zwaartekrachtsgolf" title="Zwaartekrachtsgolf – Dutch" lang="nl" hreflang="nl" data-title="Zwaartekrachtsgolf" 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/%E9%87%8D%E5%8A%9B%E6%B3%A2_(%E7%9B%B8%E5%AF%BE%E8%AB%96)" title="重力波 (相対論) – Japanese" lang="ja" hreflang="ja" data-title="重力波 (相対論)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gravitasjonsb%C3%B8lge" title="Gravitasjonsbølge – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gravitasjonsbølge" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Fale_grawitacyjne" title="Fale grawitacyjne – Polish" lang="pl" hreflang="pl" data-title="Fale grawitacyjne" 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/Radia%C3%A7%C3%A3o_gravitacional" title="Radiação gravitacional – Portuguese" lang="pt" hreflang="pt" data-title="Radiação gravitacional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Und%C4%83_gravita%C8%9Bional%C4%83" title="Undă gravitațională – Romanian" lang="ro" hreflang="ro" data-title="Undă gravitațională" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B2%D0%BE%D0%BB%D0%BD%D0%B0" title="Гравитационная волна – Russian" lang="ru" hreflang="ru" data-title="Гравитационная волна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gravitational_waves" title="Gravitational waves – Simple English" lang="en-simple" hreflang="en-simple" data-title="Gravitational waves" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gravitaatios%C3%A4teily" title="Gravitaatiosäteily – Finnish" lang="fi" hreflang="fi" data-title="Gravitaatiosäteily" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gravitationsv%C3%A5g" title="Gravitationsvåg – Swedish" lang="sv" hreflang="sv" data-title="Gravitationsvåg" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A5%E0%B8%B7%E0%B9%88%E0%B8%99%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%82%E0%B8%99%E0%B9%89%E0%B8%A1%E0%B8%96%E0%B9%88%E0%B8%A7%E0%B8%87" title="คลื่นความโน้มถ่วง – Thai" lang="th" hreflang="th" data-title="คลื่นความโน้มถ่วง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B2%D1%96%D1%82%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D1%96_%D1%85%D0%B2%D0%B8%D0%BB%D1%96" title="Гравітаційні хвилі – Ukrainian" lang="uk" hreflang="uk" data-title="Гравітаційні хвилі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%C3%B3ng_h%E1%BA%A5p_d%E1%BA%ABn" title="Sóng hấp dẫn – Vietnamese" 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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 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src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:General_relativity" title="Category:General relativity">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <b>gravitational waves</b> are ripples in the <a href="/wiki/Curvature" title="Curvature">curvature</a> of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> which propagate as a <a href="/wiki/Wave" title="Wave">wave</a>, travelling outward from the source. Predicted to exist by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> in 1916 on the basis of his theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> gravitational waves theoretically transport energy as <b>gravitational radiation</b>. Sources of detectable gravitational waves could possibly include <a href="/wiki/Binary_star" title="Binary star">binary star</a> systems composed of <a href="/wiki/White_dwarfs" class="mw-redirect" title="White dwarfs">white dwarfs</a>, <a href="/wiki/Neutron_stars" class="mw-redirect" title="Neutron stars">neutron stars</a>, or <a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">black holes</a>. The existence of gravitational waves is possibly a consequence of the <a href="/wiki/Lorentz_invariance" class="mw-redirect" title="Lorentz invariance">Lorentz invariance</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> since it brings the concept of a limiting speed of propagation of the physical interactions with it. Gravitational waves cannot exist in the Newtonian theory of gravitation, since in it physical interactions propagate at infinite speed. </p><p>Although gravitational radiation has not been <i>directly</i> detected, there is <i>indirect</i> evidence for its existence. For example, the 1993 <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> was awarded for measurements of the <a href="/wiki/Hulse-Taylor_binary" class="mw-redirect" title="Hulse-Taylor binary">Hulse-Taylor binary</a> system which suggests gravitational waves are more than mathematical anomalies. Various <a href="/wiki/Gravitational-wave_detector" class="mw-redirect" title="Gravitational-wave detector">gravitational wave detectors</a> exist. However, they remain unsuccessful in detecting such phenomena. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Einstein's theory of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, <a href="/wiki/Gravity" title="Gravity">gravity</a> is treated as a phenomenon resulting from the curvature of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. This curvature is caused by the presence of massive objects. Roughly speaking, the more massive the object is, the greater the curvature it produces and hence the more intense the gravity. As massive objects move around in spacetime, the curvature changes to reflect the changed locations of those objects. In <a href="/wiki/Gravitational_wave#Sources_of_gravitational_waves" title="Gravitational wave">certain circumstances</a>, objects that are <a href="/wiki/Acceleration" title="Acceleration">accelerated</a> generate a disturbance in spacetime which spreads, as the metaphor goes, "like ripples on the surface of a pond", although perhaps a better analogy would be <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a>. This disturbance is known as gravitational radiation. Gravitational radiation is thought to travel through the universe at the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, diminishing in strength but never stopping or slowing down.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (November 2011)">dubious</span></a> – <a href="/w/index.php?title=User_talk:Christillin/gravitational_waves&action=edit&redlink=1" class="new" title="User talk:Christillin/gravitational waves (page does not exist)">discuss</a></i>]</sup> </p><p>As waves of gravitational radiation pass a distant observer, that observer will find spacetime distorted by the effects of <a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">strain</a>. Distances between free objects will increase and decrease rhythmically as the wave passes. The magnitude of this effect will decrease the farther the observer is from the source. Binary <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> are predicted to be a strong source of such waves owing to the acceleration of their enormous masses as they <a href="/wiki/Orbit" title="Orbit">orbit</a> each other and yet even those waves are expected to be very weak by the time they reach the Earth, resulting in strains of less than 1 part in 10<sup>20</sup>. Scientists are attempting to demonstrate the existence of these waves with ever more sensitive detectors. The current most sensitive measurement is about one part in 3 x 10<sup>22</sup> (as of 2007) provided by the <a href="/wiki/LIGO" title="LIGO">LIGO</a> detector.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Sources this intense are not expected to last long so it is merely a matter of luck if any are found. Another attempt, still under development, is <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">Laser Interferometer Space Antenna</a>, a joint effort of NASA and ESA. </p><p>Gravitational waves should penetrate regions of space that electromagnetic waves cannot. It is hypothesized that they will be able to provide observers on Earth with information about black holes and other mysterious objects in the distant Universe. Such systems cannot be observed with more traditional means such as <a href="/wiki/Optical_telescope" title="Optical telescope">optical telescopes</a> and <a href="/wiki/Radio_telescope" title="Radio telescope">radio telescopes</a>. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early universe. This is not possible with conventional astronomy, since before <a href="/wiki/Recombination_(cosmology)" title="Recombination (cosmology)">recombination</a> the universe was opaque to electromagnetic radiation.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Precise measurements of gravitational waves will also allow scientists to test the general theory of relativity more thoroughly. </p><p>In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect and there is no credible source for detectable waves of very high frequency. <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen W. Hawking</a> and <a href="/wiki/Werner_Israel" title="Werner Israel">Werner Israel</a> list different frequency bands for gravitational waves that could be plausibly detected, ranging from 10<sup>−7</sup> Hz up to 10<sup>11</sup> Hz.<sup id="cite_ref-HI_4-0" class="reference"><a href="#cite_note-HI-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effects_of_a_passing_gravitational_wave">Effects of a passing gravitational wave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=2" title="Edit section: Effects of a passing gravitational wave"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:GravitationalWave_PlusPolarization.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GravitationalWave_PlusPolarization.gif/150px-GravitationalWave_PlusPolarization.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GravitationalWave_PlusPolarization.gif/225px-GravitationalWave_PlusPolarization.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_PlusPolarization.gif 2x" data-file-width="288" data-file-height="288" /></a><figcaption>The effect of a plus-polarized gravitational wave on a ring of particles.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:GravitationalWave_CrossPolarization.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GravitationalWave_CrossPolarization.gif/150px-GravitationalWave_CrossPolarization.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GravitationalWave_CrossPolarization.gif/225px-GravitationalWave_CrossPolarization.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_CrossPolarization.gif 2x" data-file-width="288" data-file-height="288" /></a><figcaption>The effect of a cross-polarized gravitational wave on a ring of particles.</figcaption></figure> <p>Imagine a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (the surface of your screen). Then a weak gravitational wave arrives, passing through the particles along a line perpendicular to the plane of the particles (i.e. following your line of vision into the screen). The particles will oscillate in a "<a href="/wiki/Cruciform" title="Cruciform">cruciform</a>" manner, as shown in the animations <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The reason for this is unclear. (March 2012)">why?</span></a></i>]</sup>. The area enclosed by the test particles does not change and there is no motion along the direction of propagation. </p><p>The oscillations depicted here in the animation are of course immensely exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in <a href="/wiki/Linearized_gravity" title="Linearized gravity">linearized gravity</a>). However they enable us to visualize the kind of oscillations associated with gravitational waves as produced for example by a pair of masses in a <a href="/wiki/Circular_orbit" title="Circular orbit">circular orbit</a>. In this case the amplitude of the gravitational wave is a constant, but its plane of <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.<sup id="cite_ref-LL75_5-0" class="reference"><a href="#cite_note-LL75-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> If the orbit is elliptical then the gravitational wave's amplitude also varies with time according to an equation called the "<a href="/wiki/Quadrupole" title="Quadrupole">quadrupole</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> </p><p>Like other <a href="/wiki/Wave" title="Wave">waves</a>, there are a few useful characteristics describing a gravitational wave: </p> <ul><li><b>Amplitude</b>: Usually denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b88717e3227b0aeece3422a7523fc93ea38d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.409ex; height:2.176ex;" alt="{\displaystyle h=0.5}"></span> (or 50%). Gravitational waves passing through the Earth are many billions times weaker than this — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\approx 10^{-20}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>≈</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>20</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\approx 10^{-20}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fa8083578595f0c70ac70bcf1bf00604119090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.917ex; height:2.676ex;" alt="{\displaystyle h\approx 10^{-20}}"></span>. Note that this is not the quantity which would be analogous to what is usually called the amplitude of an electromagnetic wave, which would be <span class="mwe-math-element"><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 {dh}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>h</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dh}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f339401eefd979268a02b559cf693c75e343b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.391ex; height:5.509ex;" alt="{\displaystyle {\frac {dh}{dt}}}"></span>.</li> <li><b><a href="/wiki/Frequency" title="Frequency">Frequency</a></b>: Usually denoted <i>f</i>, this is the frequency with which the wave oscillates (1 divided by the amount of time between two successive maximum stretches or squeezes)</li> <li><b><a href="/wiki/Wavelength" title="Wavelength">Wavelength</a></b>: Usually denoted <span class="mwe-math-element"><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>, this is the distance along the wave between points of maximum stretch or squeeze.</li> <li><b><a href="/wiki/Speed" title="Speed">Speed</a></b>: This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>.</li></ul> <p>The speed, wavelength, and frequency of a gravitational wave are related by the equation <i>c = λ f</i>, just like the equation for a <a href="/wiki/Electromagnetic_radiation#Wave_model" title="Electromagnetic radiation">light wave</a>. For example, the animations shown here oscillate roughly once every two seconds. This would correspond to a frequency of 0.5 Hz, and a wavelength of about 600,000 km, or 47 times the diameter of the Earth. </p><p>In the example just discussed, we actually assume something special about the wave. We have assumed that the wave is <a href="/wiki/Linear_polarization" title="Linear polarization">linearly polarized</a>, with a "plus" polarization, written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\,+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\,+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329eaacd25c5bff08b0865722d4a868a2a26e041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.509ex;" alt="{\displaystyle h_{\,+}}"></span>. Polarization of a gravitational wave is just like polarization of a light wave except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\,\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mo>×</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\,\times }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da52980d6a6ad610f7356fdfabf60b88ec5f421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.509ex;" alt="{\displaystyle h_{\,\times }}"></span>, the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of <a href="/wiki/Circular_polarization" title="Circular polarization">circularly polarized</a> waves. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave depends on the angle from the source, as we will see in the next section. </p> <div class="mw-heading mw-heading2"><h2 id="Sources_of_gravitational_waves">Sources of gravitational waves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=3" title="Edit section: Sources of gravitational waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically <a href="/wiki/Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> (like an expanding or contracting sphere) or cylindrically symmetric (like a spinning disk or sphere). A simple example of this principle is provided by the spinning <a href="/wiki/Dumbbell" title="Dumbbell">dumbbell</a>. If the dumbbell spins like wheels on an axle, it will not radiate gravitational waves; if it tumbles end over end like two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. If we imagine an extreme case in which the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off. </p><p>Some more detailed examples: </p> <ul><li>Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) <i><b>will</b></i> radiate.</li> <li>A spinning non-axisymmetric planetoid — say with a large bump or dimple on the equator — <i><b>will</b></i> radiate.</li> <li>A <a href="/wiki/Supernova" title="Supernova">supernova</a> <i><b>will</b></i> radiate except in the unlikely event that the explosion is perfectly symmetric.</li> <li>An isolated non-spinning solid object moving at a constant speed <i><b>will not</b></i> radiate. This can be regarded as a consequence of the principle of <a href="/wiki/Momentum#Conservation_of_linear_momentum" title="Momentum">conservation of linear momentum</a>.</li> <li>A spinning disk <i><b>will not</b></i> radiate. This can be regarded as a consequence of the principle of <a href="/wiki/Angular_momentum#Conservation_of_angular_momentum" title="Angular momentum">conservation of angular momentum</a>. However, it <i>will</i> show <a href="/wiki/Gravitomagnetism" class="mw-redirect" title="Gravitomagnetism">gravitomagnetic</a> effects.</li> <li>A spherically pulsating spherical star (non-zero monopole moment or <a href="/wiki/Mass" title="Mass">mass</a>, but zero quadrupole moment) <i><b>will not</b></i> radiate, in agreement with <a href="/wiki/Birkhoff%27s_theorem_(relativity)" title="Birkhoff's theorem (relativity)">Birkhoff's theorem</a>.</li></ul> <p>More technically, the third time derivative of the <a href="/wiki/Quadrupole_moment" class="mw-redirect" title="Quadrupole moment">quadrupole moment</a> (or the <i>l</i>-th time derivative of the <i>l</i>-th <a href="/wiki/Multipole_expansion" title="Multipole expansion">multipole moment</a>) of an isolated system's <a href="/wiki/Stress-energy_tensor" class="mw-redirect" title="Stress-energy tensor">stress-energy tensor</a> must be nonzero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current necessary for electromagnetic radiation. </p> <div class="mw-heading mw-heading3"><h3 id="Power_radiated_by_orbiting_bodies">Power radiated by orbiting bodies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=4" title="Edit section: Power radiated by orbiting bodies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbit2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f2/Orbit2.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Two stars of dissimilar mass are in <a href="/wiki/Circular_orbits" class="mw-redirect" title="Circular orbits">circular orbits</a>. Each rotates about their common <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> (denoted by the small red cross) in a circle with the larger mass having the smaller orbit.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbit1.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Orbit1.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Two stars of similar mass are in circular orbits about their center of mass</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbit5.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Orbit5.gif/200px-Orbit5.gif" decoding="async" width="200" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Orbit5.gif/300px-Orbit5.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0e/Orbit5.gif 2x" data-file-width="400" data-file-height="200" /></a><figcaption>Two stars of similar mass are in highly <a href="/wiki/Elliptical_orbit" class="mw-redirect" title="Elliptical orbit">elliptical orbits</a> about their center of mass </figcaption></figure> <p>Gravitational waves carry energy away from their sources and, in the case of orbiting bodies, this is associated with an inspiral or decrease in orbit. Imagine for example a simple system of two masses — such as the Earth-Sun system — moving slowly compared to the speed of light in circular orbits. Assume that these two masses orbit each other in a circular orbit in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> plane. To a good approximation, the masses follow simple <a href="/wiki/Planetary_orbit" class="mw-redirect" title="Planetary orbit">Keplerian orbits</a>. However, such an orbit represents a changing quadrupole moment. That is, the system will give off gravitational waves. </p><p>Suppose that the two masses are <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></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 m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span>, and they are separated by a distance <span class="mwe-math-element"><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>. The power given off (radiated) by this system is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {dE}{dt}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {dE}{dt}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eddcbc4a4a854bb764d3d8c61d8d330dd917a590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.958ex; height:6.176ex;" alt="{\displaystyle P={\frac {dE}{dt}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}}"></span> ,<sup id="cite_ref-Gravitational_Radiation_7-0" class="reference"><a href="#cite_note-Gravitational_Radiation-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where <i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>, <i>c</i> is the speed of light in vacuum and where the negative sign means that power is being given off by the system, rather than received. For a system like the Sun and Earth, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is about 1.5<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001110000000000000♠"></span>11</span></sup> m 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 m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></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 m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span> are about 2<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001300000000000000♠"></span>30</span></sup> and 6<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001240000000000000♠"></span>24</span></sup> kg respectively. In this case, the power is about 200 watts. This is truly tiny compared to the <a href="/wiki/Solar_constant#Solar_constant" title="Solar constant">total electromagnetic radiation given off by the Sun</a> (roughly 3.86<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001260000000000000♠"></span>26</span></sup> watts). </p><p>In theory, the loss of energy through gravitational radiation could eventually drop the Earth into the Sun. However, the total energy of the Earth orbiting the Sun (<a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> plus <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a>) is about 1.14<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001360000000000000♠"></span>36</span></sup> <a href="/wiki/Joules" class="mw-redirect" title="Joules">joules</a> of which only 200 joules per second is lost through gravitational radiation, leading to a decay in the orbit by about 1<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2998850000000000000♠"></span>−15</span></sup> meters per day or roughly the diameter of a <a href="/wiki/Proton" title="Proton">proton</a>. At this rate, it would take the Earth approximately 1<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001130000000000000♠"></span>13</span></sup> times more than the current <a href="/wiki/Age_of_the_Universe" class="mw-redirect" title="Age of the Universe">age of the Universe</a> to spiral onto the Sun. This estimate overlooks the decrease in <i>r</i> over time, but the majority of the time the bodies are far apart and only radiating slowly, so the difference is unimportant in this example. In only a few billion years, <a href="/wiki/Earth#Future" title="Earth">the Earth is</a> predicted to be swallowed by the Sun in the red giant stage of its life. </p><p>A more dramatic example of radiated gravitational energy is represented by two solar mass neutron stars orbiting at a distance from each other of 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> m (only 0.63 <a href="/wiki/Light-second" title="Light-second">light-seconds</a> apart). [The Sun is 8 light minutes from the Earth.] Plugging their masses into the above equation shows that the gravitational radiation from them would be 1.38<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001280000000000000♠"></span>28</span></sup> watts, which is about 100 times more than the Sun's electromagnetic radiation. </p> <div class="mw-heading mw-heading3"><h3 id="Orbital_decay_from_gravitational_radiation">Orbital decay from gravitational radiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=5" title="Edit section: Orbital decay from gravitational radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gravitational radiation robs the orbiting bodies of energy. It first circularizes their orbits and then gradually shrinks their radius. As the energy of the orbit is reduced, the distance between the bodies decreases, and they rotate more rapidly. The overall angular momentum is reduced however. This reduction corresponds to the angular momentum carried off by gravitational radiation. The rate of decrease of distance between the bodies versus time is given by:<sup id="cite_ref-Gravitational_Radiation_7-1" class="reference"><a href="#cite_note-Gravitational_Radiation-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dr}{dt}}=-{\frac {64}{5}}\,{\frac {G^{3}}{c^{5}}}\,{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dr}{dt}}=-{\frac {64}{5}}\,{\frac {G^{3}}{c^{5}}}\,{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227a1d186c7096cfe273ff47750707e554b89214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.914ex; height:6.009ex;" alt="{\displaystyle {\frac {dr}{dt}}=-{\frac {64}{5}}\,{\frac {G^{3}}{c^{5}}}\,{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\ }"></span>,</dd></dl> <p>where the variables are the same as in the previous equation. </p><p>The orbit decays at a rate proportional to the inverse third power of the radius. When the radius has shrunk to half its initial value, it is shrinking eight times faster than before. By <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's Third Law</a>, the new rotation rate at this point will be faster by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {8}}=2.828}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> <mo>=</mo> <mn>2.828</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8}}=2.828}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f002848d94d1859b15f659b33c384adb400e41e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:2.843ex;" alt="{\displaystyle {\sqrt {8}}=2.828}"></span>, or nearly three times the previous orbital frequency. As the radius decreases, the power lost to gravitational radiation increases even more. As can be seen from the previous equation, power radiated varies as the inverse fifth power of the radius, or 32 times more in this case. </p><p>If we use the previous values for the Sun and the Earth, we find that the Earth's orbit shrinks by 1.1<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2998800000000000000♠"></span>−20</span></sup> meter per second. This is 3.5<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2998870000000000000♠"></span>−13</span></sup> m per year which is about 1/300 the diameter of a <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>. The effect of gravitational radiation on the size of the Earth's orbit is negligible over the age of the universe. This is not true for closer orbits. </p><p>A more practical example is the orbit of a <a href="/wiki/Solar_analog" title="Solar analog">Sun-like star</a> around a heavy <a href="/wiki/Black_hole" title="Black hole">black hole</a>. Our <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> has a 4 million solar-mass black hole at its center in <a href="/wiki/Sagittarius_A" title="Sagittarius A">Sagittarius A</a>. Such <a href="/wiki/Supermassive_black_holes" class="mw-redirect" title="Supermassive black holes">supermassive black holes</a> are being found in the center of almost all <a href="/wiki/Galaxy" title="Galaxy">galaxies</a>. For this example take a 2 million solar-mass black hole with a solar-mass star orbiting it at a radius of 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001100000000000000♠"></span>10</span></sup> m (63 light-seconds). The mass of the black hole will be 4<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001360000000000000♠"></span>36</span></sup> kg and its <a href="/wiki/Gravitational_radius" class="mw-redirect" title="Gravitational radius">gravitational radius</a> will be 6<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000900000000000000♠"></span>9</span></sup> m. The orbital period will be 1,000 seconds, or a little under 17 minutes. The solar-mass star will draw closer to the black hole by 7.4 meters per second or 7.4 km per orbit. A collision will not be long in coming. </p><p>Assume that a pair of solar-mass <a href="/wiki/Neutron_stars" class="mw-redirect" title="Neutron stars">neutron stars</a> are in circular orbits at a distance of 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> m (189,000 km). This is a little less than 1/7 the diameter of the Sun or 0.63 <a href="/wiki/Light-second" title="Light-second">light-seconds</a>. Their orbital period would be 1,000 seconds. Substituting the new mass and radius in the above formula gives a rate of orbit decrease of 3.7<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="2999400000000000000♠"></span>−6</span></sup> m/s or 3.7 mm per orbit. This is 116 meters per year and is not negligible over cosmic time scales. </p><p>Suppose instead that these two neutron stars were orbiting at a distance of 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000600000000000000♠"></span>6</span></sup> m (1890 km). Their period would be 1 second and their orbital velocity would be about 1/50 of the speed of light. Their orbit would now shrink by 3.7 meters per orbit. A collision is imminent. A runaway loss of energy from the orbit results in an ever more rapid decrease in the distance between the stars. They will eventually merge to form a black hole and cease to radiate gravitational waves. This is referred to as the <a href="/wiki/Inspiral" class="mw-redirect" title="Inspiral">inspiral</a>. </p><p>The above equation can not be applied directly for calculating the lifetime of the orbit, because the rate of change in radius depends on the radius itself, and is thus non-constant with time. The lifetime can be computed by integration of this equation (see next section). </p> <div class="mw-heading mw-heading3"><h3 id="Orbital_lifetime_limits_from_gravitational_radiation">Orbital lifetime limits from gravitational radiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=6" title="Edit section: Orbital lifetime limits from gravitational radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Orbital lifetime is one of the most important properties of gravitational radiation sources. It determines the average number of binary stars in the universe that are close enough to be detected. Short lifetime binaries are strong sources of gravitational radiation but are few in number. Long lifetime binaries are more plentiful but they are weak sources of gravitational waves. LIGO is most sensitive in the frequency band where two neutron stars are about to merge. This time frame is only a few seconds. It takes luck for the detector to see this blink in time out of a million year orbital lifetime. It is predicted that such a merger will only be seen once per decade or so. </p><p>The lifetime of an orbit is given by:<sup id="cite_ref-Gravitational_Radiation_7-2" class="reference"><a href="#cite_note-Gravitational_Radiation-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>256</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92d83a19088a88c225625c19bb4e168c295b3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.007ex; height:6.509ex;" alt="{\displaystyle t={\frac {5}{256}}\,{\frac {c^{5}}{G^{3}}}\,{\frac {r^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}\ }"></span>,</dd></dl> <p>where r is the initial distance between the orbiting bodies. This equation can be derived by integrating the previous equation for the rate of radius decrease. It predicts the time for the radius of the orbit to shrink to zero. As the orbital speed becomes a significant fraction of the speed of light, this equation becomes inaccurate. It is useful for inspirals until the last few milliseconds before the merger of the objects. </p><p>Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001300000000000000♠"></span>30</span></sup> seconds or 1.09<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001230000000000000♠"></span>23</span></sup> years (which is approximately 10<sup>15</sup> times larger than the <a href="/wiki/Age_of_the_universe" title="Age of the universe">age of the universe</a>). The actual figure would be slightly less than that. The Earth will break apart from <a href="/wiki/Roche_limit" title="Roche limit">tidal forces</a> if it orbits closer than a few radii from the sun. This would form a ring around the Sun and instantly stop the emission of gravitational waves. </p><p>If we use a 2 million solar mass black hole with a solar mass star orbiting it at 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001100000000000000♠"></span>10</span></sup> meters, we get a lifetime of 6.50<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> seconds or 20.7 years. </p><p>Assume that a pair of solar mass neutron stars with a diameter of 10 kilometers are in circular orbits at a distance of 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000800000000000000♠"></span>8</span></sup> m (189,000 km). Their lifetime is 1.30<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001130000000000000♠"></span>13</span></sup> seconds or about 414,000 years. Their orbital period will be 1,000 seconds and it could be observed by <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">LISA</a> if they were not too far away. A far greater number of white dwarf binaries exist with orbital periods in this range. White dwarf binaries have masses on the order of our Sun and diameters on the order of our Earth. They cannot get much closer together than 10,000 km before they will merge and cease to radiate gravitational waves. This results in the creation of either a neutron star or a black hole. Until then, their gravitational radiation will be comparable to that of a neutron star binary. LISA is the only gravitational wave experiment which is likely to succeed in detecting such types of binaries. </p><p>If the orbit of a neutron star binary has decayed to 1.89<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7000600000000000000♠"></span>6</span></sup>m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. Its chances of success are quite low despite the large number of such mergers occurring in the universe. No mergers have been seen in the few years that LIGO has been in operation. It is thought that a merger should be seen about once per decade of observing time. </p> <div class="mw-heading mw-heading3"><h3 id="Wave_amplitudes_from_the_Earth-Sun_system">Wave amplitudes from the Earth-Sun system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=7" title="Edit section: Wave amplitudes from the Earth-Sun system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can also think in terms of the amplitude of the wave from a system in circular orbits. Let <span class="mwe-math-element"><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> be the angle between the perpendicular to the plane of the orbit and the line of sight of the observer. Suppose that an observer is outside the system at a distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> from its center of mass. If R is much greater than a wavelength, the two polarizations of the wave will 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 h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {2m_{1}m_{2}}{r}}(1+\cos ^{2}\theta )\cos \left[2\omega (t-R)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {2m_{1}m_{2}}{r}}(1+\cos ^{2}\theta )\cos \left[2\omega (t-R)\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf43dc2a63f724e7924cda2993f971011a069b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:49.791ex; height:6.009ex;" alt="{\displaystyle h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {2m_{1}m_{2}}{r}}(1+\cos ^{2}\theta )\cos \left[2\omega (t-R)\right],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\times }=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}\,(\cos {\theta })\sin \left[2\omega (t-R)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\times }=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}\,(\cos {\theta })\sin \left[2\omega (t-R)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49981871e4c397270bd542a67edb686325ce1d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:44.866ex; height:6.009ex;" alt="{\displaystyle h_{\times }=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}\,(\cos {\theta })\sin \left[2\omega (t-R)\right].}"></span></dd></dl> <p>Here, we use the constant <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of a circular orbit in Newtonian physics: </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 \omega ={\sqrt {G(m_{1}+m_{2})/r^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {G(m_{1}+m_{2})/r^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba400cc7f211d2890c58fa12c3f29e85496c3daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:23.446ex; height:4.843ex;" alt="{\displaystyle \omega ={\sqrt {G(m_{1}+m_{2})/r^{3}}}.}"></span></dd></dl> <p>For example, if the observer is in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> plane then <span class="mwe-math-element"><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>, 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 \cos(\theta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b992d2e432a231875efdab73ed2fafb9c562339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.272ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )=0}"></span>, so the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\times }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597a37a1f408c29aaf9426773a97d5f92c70a049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.335ex; margin-bottom: -0.336ex; width:2.85ex; height:2.509ex;" alt="{\displaystyle h_{\times }}"></span> polarization is always zero. We also see that the frequency of the wave given off is twice the rotation frequency. If we put in numbers for the Earth-Sun system, we find: </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 h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}=-{\frac {1}{R}}\,1.7\times 10^{-10}\,\mathrm {meters} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mn>1.7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">s</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}=-{\frac {1}{R}}\,1.7\times 10^{-10}\,\mathrm {meters} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e0d0c9f35fa90d77c34dbee8e491e9abc7d8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:49.988ex; height:6.009ex;" alt="{\displaystyle h_{+}=-{\frac {1}{R}}\,{\frac {G^{2}}{c^{4}}}\,{\frac {4m_{1}m_{2}}{r}}=-{\frac {1}{R}}\,1.7\times 10^{-10}\,\mathrm {meters} .}"></span></dd></dl> <p>In this case, the minimum distance to find waves is <i>R</i> ≈ 1 <a href="/wiki/Light-year" title="Light-year">light-year</a>, so typical amplitudes will be <i>h</i> ≈ 10<sup>−26</sup>. That is, a ring of particles would stretch or squeeze by just one part in 10<sup>26</sup>. This is well under the detectability limit of all conceivable detectors. </p> <div class="mw-heading mw-heading3"><h3 id="Radiation_from_other_sources">Radiation from other sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=8" title="Edit section: Radiation from other sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the <a href="/wiki/PSR_B1913%2B16" class="mw-redirect" title="PSR B1913+16">Hulse-Taylor binary</a> — a pair of stars, one of which is a <a href="/wiki/Binary_pulsar" title="Binary pulsar">pulsar</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> The characteristics of their orbit can be deduced from the <a href="/wiki/Doppler_shift" class="mw-redirect" title="Doppler shift">Doppler shifting</a> of radio signals given off by the pulsar. Each of the stars has a mass about 1.4 times that of the Sun and the size of their orbit is about 1/75 of the Earth-Sun orbit. This means the distance between the two stars is just a few times larger than the diameter of our own Sun. The combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 10<sup>22</sup> times as much. </p><p>The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. As the energy is carried off, the stars should draw closer to each other. This effect is called an <b><a href="/wiki/Inspiral" class="mw-redirect" title="Inspiral">inspiral</a></b>, and it can be observed in the pulsar's signals. The measurements on the Hulse-Taylor system have been carried out over more than 30 years. It has been shown that the gravitational radiation predicted by general relativity allows these observations to be matched within 0.2 percent. In 1993, <a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Russell Hulse</a> and <a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Joe Taylor</a> were awarded the <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for this work, which was the first indirect evidence for gravitational waves. Unfortunately, the orbital lifetime of this binary system before merger is about 1.84 billion years. This is a substantial fraction of the age of the universe. </p><p>Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or <a href="/wiki/Binary_black_hole" title="Binary black hole">black holes</a>) are in close orbits, they send out intense gravitational waves. As they spiral closer to each other, these waves become more intense. At some point they should become so intense that direct detection by their effect on objects on Earth or in space is possible. This direct detection is the goal of several large scale experiments.<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><p>The only difficulty is that most systems like the Hulse-Taylor binary are so far away. The amplitude of waves given off by the Hulse-Taylor binary as seen on Earth would be roughly <i>h</i> ≈ 10<sup>−26</sup>. There are some sources, however, that astrophysicists expect to find with much larger amplitudes of <i>h</i> ≈ 10<sup>−20</sup>. At least eight other binary pulsars have been discovered.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Astrophysics_and_gravitational_waves">Astrophysics and gravitational waves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=9" title="Edit section: Astrophysics and gravitational waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-physics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-physics">Unsolved problem in physics</span>:</div> <div class="unsolved-body">Can gravitational waves be detected experimentally?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">(more unsolved problems in physics)</a></div> </div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wavy.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b8/Wavy.gif" decoding="async" width="320" height="200" class="mw-file-element" data-file-width="320" data-file-height="200" /></a><figcaption>Two-dimensional representation of gravitational waves generated by two <a href="/wiki/Neutron_star" title="Neutron star">neutron stars</a> orbiting each other.</figcaption></figure> <p>During the past century, <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> has been revolutionized by the use of new methods for observing the universe. Astronomical observations were originally made using <a href="/wiki/Visible_light" class="mw-redirect" title="Visible light">visible light</a>. <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> pioneered the use of telescopes to enhance these observations. However, visible light is only a small portion of the <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a>, and not all objects in the distant universe shine strongly in this particular band. More useful information may be found, for example, in radio wavelengths. Using <a href="/wiki/Radio_telescopes" class="mw-redirect" title="Radio telescopes">radio telescopes</a>, astronomers have found <a href="/wiki/Pulsars" class="mw-redirect" title="Pulsars">pulsars</a>, <a href="/wiki/Quasars" class="mw-redirect" title="Quasars">quasars</a>, and other extreme objects which push the limits of our understanding of physics. Observations in the <a href="/wiki/Microwave" title="Microwave">microwave</a> band have opened our eyes to the <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">faint imprints</a> of the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>, a discovery <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a> called the "greatest discovery of the century, if not all time". Similar advances in observations using <a href="/wiki/Gamma_ray" title="Gamma ray">gamma rays</a>, <a href="/wiki/X-ray" title="X-ray">x-rays</a>, <a href="/wiki/Ultraviolet_light" class="mw-redirect" title="Ultraviolet light">ultraviolet light</a>, and <a href="/wiki/Infrared_light" class="mw-redirect" title="Infrared light">infrared light</a> have also brought new insights to astronomy. As each of these regions of the spectrum has opened, new discoveries have been made that could not have been made otherwise. Astronomers hope that the same holds true of gravitational waves. </p><p>Gravitational waves have two important and unique properties. First, there is no need for any type of matter to be present nearby in order for the waves to be generated by a binary system of uncharged black holes, which would emit no electromagnetic radiation. Second, gravitational waves can pass through any intervening matter without being scattered significantly. Whereas light from distant stars may be blocked out by <a href="/wiki/Interstellar_dust" class="mw-redirect" title="Interstellar dust">interstellar dust</a>, for example, gravitational waves will pass through essentially unimpeded. These two features allow gravitational waves to carry information about astronomical phenomena never before observed by humans. </p><p>The sources of gravitational waves described above are in the low-frequency end of the gravitational-wave spectrum (10<sup>−7</sup> to 10<sup>5</sup> Hz). An astrophysical source at the high-frequency end of the gravitational-wave spectrum (above 10<sup>5</sup> Hz and probably 10<sup>10</sup> Hz) generates<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (May 2009)">clarification needed</span></a></i>]</sup> relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background (see <a href="/wiki/Gravitational_wave_background" title="Gravitational wave background">gravitational wave background</a>).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> At these high frequencies it is potentially possible that the sources may be "man made"<sup id="cite_ref-HI_4-1" class="reference"><a href="#cite_note-HI-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> that is, gravitational waves generated and detected in the laboratory.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BLW06_13-0" class="reference"><a href="#cite_note-BLW06-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Energy,_momentum,_and_angular_momentum_carried_by_gravitational_waves"><span id="Energy.2C_momentum.2C_and_angular_momentum_carried_by_gravitational_waves"></span>Energy, momentum, and angular momentum carried by gravitational waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=10" title="Edit section: Energy, momentum, and angular momentum carried by gravitational waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry <a href="/wiki/Energy" title="Energy">energy</a>, <a href="/wiki/Momentum" title="Momentum">momentum</a>, and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. By carrying these away from a source, waves are able to rob that source of its energy, linear or angular momentum. Gravitational waves perform the same function. Thus, for example, a binary system loses angular momentum as the two orbiting objects spiral towards each other—the angular momentum is radiated away by gravitational waves. </p><p>The waves can also carry off linear momentum, a possibility that has some interesting implications for <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>.<sup id="cite_ref-Merritt2004_14-0" class="reference"><a href="#cite_note-Merritt2004-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> After two supermassive black holes coalesce, emission of linear momentum can produce a "kick" with amplitude as large as 4000 km/s. This is fast enough to eject the coalesced black hole completely from its host galaxy. Even if the kick is too small to eject the black hole completely, it can remove it temporarily from the nucleus of the galaxy, after which it will oscillate about the center, eventually coming to rest.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> A kicked black hole can also carry a star cluster with it, forming a <a href="/wiki/Hyper-compact_stellar_system" class="mw-redirect" title="Hyper-compact stellar system">hyper-compact stellar system</a>.<sup id="cite_ref-MSK09_16-0" class="reference"><a href="#cite_note-MSK09-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "<a href="/wiki/HE0450-2958" title="HE0450-2958">naked quasar</a>". The <a href="/wiki/Quasar" title="Quasar">quasar</a> <a href="/wiki/SDSS_J0927%2B2943" title="SDSS J0927+2943">SDSS J092712.65+294344.0</a> is believed to contain a recoiling supermassive black hole.<sup id="cite_ref-kom_17-0" class="reference"><a href="#cite_note-kom-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Gravitational_wave_detectors">Gravitational wave detectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=11" title="Edit section: Gravitational wave detectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gravitational_wave_detector" class="mw-redirect" title="Gravitational wave detector">Gravitational wave detector</a></div> <p>Though the Hulse-Taylor observations were very important, they give only <i>indirect</i> evidence for gravitational waves. A more conclusive observation would be a <i>direct</i> measurement of the effect of a passing gravitational wave, which could also provide more information about the system which generated it. Any such direct detection is complicated by the <a href="/wiki/Orders_of_magnitude_(length)" title="Orders of magnitude (length)">extraordinarily small</a> effect the waves would produce on a detector. The amplitude of a spherical wave will fall off as the inverse of the distance from the source (the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a362806e220bd4ab2a27e589b42bd01f7a6f28a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle 1/R}"></span> term in the formulas for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> above). Thus, even waves from extreme systems like merging binary black holes die out to very small amplitude by the time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large as <i>h</i> ≈ 10<sup>−20</sup>, but generally no bigger.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2009)">citation needed</span></a></i>]</sup> </p><p>A simple device to detect the expected wave motion is called a <a href="/wiki/Weber_bar" title="Weber bar">Weber bar</a> — a large, solid bar of metal isolated from outside vibrations. This type of instrument was the first type of gravitational wave detector. Strains in space due to an incident gravitational wave excite the bar's <a href="/wiki/Resonant_frequency" class="mw-redirect" title="Resonant frequency">resonant frequency</a> and could thus be amplified to detectable levels. Conceivably, a nearby supernova might be strong enough to be seen without resonant amplification. With this instrument, <a href="/wiki/Joseph_Weber" title="Joseph Weber">Joseph Weber</a> claimed to have detected daily signals of gravitational waves. His results, however, were contested in 1974 by physicists <a href="/wiki/Richard_Garwin" title="Richard Garwin">Richard Garwin</a> and <a href="/wiki/David_Douglass" class="mw-disambig" title="David Douglass">David Douglass</a>. Modern forms of the Weber bar are still operated, <a href="/wiki/Cryogenically" class="mw-redirect" title="Cryogenically">cryogenically</a> cooled, with <a href="/wiki/SQUID" title="SQUID">superconducting quantum interference devices</a> to detect vibration. Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves.<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> </p><p><a href="/wiki/MiniGRAIL" class="mw-redirect" title="MiniGRAIL">MiniGRAIL</a> is a spherical gravitational wave antenna using this principle. It is based at <a href="/wiki/Leiden_University" title="Leiden University">Leiden University</a>, consisting of an exactingly machined 1150 kg sphere cryogenically cooled to 20 mK.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The spherical configuration allows for equal sensitivity in all directions, and is somewhat experimentally simpler than larger linear devices requiring high vacuum. Events are detected by measuring <a href="/wiki/Multipole_moments" class="mw-redirect" title="Multipole moments">deformation of the detector sphere</a>. MiniGRAIL is highly sensitive in the 2–4 kHz range, suitable for detecting gravitational waves from rotating neutron star instabilities or small black hole mergers.<sup id="cite_ref-MiniGRAIL_2000_20-0" class="reference"><a href="#cite_note-MiniGRAIL_2000-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LIGO_schematic_(multilang).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/LIGO_schematic_%28multilang%29.svg/350px-LIGO_schematic_%28multilang%29.svg.png" decoding="async" width="350" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/LIGO_schematic_%28multilang%29.svg/525px-LIGO_schematic_%28multilang%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/LIGO_schematic_%28multilang%29.svg/700px-LIGO_schematic_%28multilang%29.svg.png 2x" data-file-width="1125" data-file-height="675" /></a><figcaption>A schematic diagram of a laser interferometer.</figcaption></figure> <p>A more sensitive class of detector uses laser <a href="/wiki/Interferometry" title="Interferometry">interferometry</a> to measure gravitational-wave induced motion between separated 'free' masses.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is <a href="/wiki/LIGO" title="LIGO">LIGO</a> — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in <a href="/wiki/Livingston,_Louisiana" title="Livingston, Louisiana">Livingston, Louisiana</a>; the other two (in the same vacuum tubes) at the <a href="/wiki/Hanford_site" class="mw-redirect" title="Hanford site">Hanford site</a> in <a href="/wiki/Richland,_Washington" title="Richland, Washington">Richland, Washington</a>. Each consists of two <a href="/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer" title="Fabry–Pérot interferometer">light storage arms</a> which are 2 to 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1m diameter vacuum tubes running the entire 4 kilometers. A passing gravitational wave will slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive. </p><p>Even with such long arms, the strongest gravitational waves will only change the distance between the ends of the arms by at most roughly 10<sup>−18</sup> meters. LIGO should be able to detect gravitational waves as small as <i>h</i> ≈ 5*10<sup>−20</sup>. Upgrades to LIGO and other detectors such as <a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a>, <a href="/wiki/GEO_600" class="mw-redirect" title="GEO 600">GEO 600</a>, and <a href="/wiki/TAMA_300" title="TAMA 300">TAMA 300</a> should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer (<a href="/wiki/LCGT" class="mw-redirect" title="LCGT">LCGT</a>) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year. </p><p>Interferometric detectors are limited at high frequencies by <a href="/wiki/Shot_noise" title="Shot noise">shot noise</a>, which occurs because the lasers produce photons randomly; one analogy is to rainfall—the rate of rainfall, like the laser intensity, is measurable, but the raindrops, like photons, fall at random times, causing fluctuations around the average value. This leads to noise at the output of the detector, much like radio static. In addition, for sufficiently high laser power, the random momentum transferred to the test masses by the laser photons shakes the mirrors, masking signals at low frequencies. Thermal noise (e.g., <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>) is another limit to sensitivity. In addition to these 'stationary' (constant) noise sources, all ground-based detectors are also limited at low frequencies by <a href="/wiki/Seismic" class="mw-redirect" title="Seismic">seismic</a> noise and other forms of environmental vibration, and other 'non-stationary' noise sources; creaks in mechanical structures, lightning or other large electrical disturbances, etc. may also create noise masking an event or may even imitate an event. All these must be taken into account and excluded by analysis before a detection may be considered a true gravitational wave event. </p><p>Space-based interferometers, such as <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">LISA</a> and <a href="/wiki/DECIGO" class="mw-redirect" title="DECIGO">DECIGO</a>, are also being developed. LISA's design calls for three test masses forming an equilateral triangle, with lasers from each spacecraft to each other spacecraft forming two independent interferometers. LISA is planned to occupy a solar orbit trailing the Earth, with each arm of the triangle being five million kilometers. This puts the detector in an <a href="/wiki/Interplanetary_medium" title="Interplanetary medium">excellent vacuum</a> far from Earth-based sources of noise, though it will still be susceptible to shot noise, as well as artifacts caused by <a href="/wiki/Cosmic_ray" title="Cosmic ray">cosmic rays</a> and <a href="/wiki/Solar_wind" title="Solar wind">solar wind</a>. </p><p>There are currently two detectors focusing on detection at the higher end of the gravitational wave spectrum (10<sup>−7</sup> to 10<sup>5</sup> Hz): one at <a href="/wiki/University_of_Birmingham" title="University of Birmingham">University of Birmingham</a>, England, and the other at <a href="/wiki/Istituto_Nazionale_di_Fisica_Nucleare" title="Istituto Nazionale di Fisica Nucleare">INFN</a> Genoa, Italy. A third is under development at <a href="/wiki/Chongqing_University" title="Chongqing University">Chongqing University</a>, China. The Birmingham detector measures changes in the polarization state of a <a href="/wiki/Microwave" title="Microwave">microwave</a> beam circulating in a closed loop about one meter across. Two have been fabricated and they are currently expected to be sensitive to periodic spacetime strains of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\sim {2\times 10^{-13}/{\sqrt {\mathrm {Hz} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>13</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </msqrt> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\sim {2\times 10^{-13}/{\sqrt {\mathrm {Hz} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0face245584bdb04abc7caa1f2dc15cabf6a99fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.794ex; height:3.176ex;" alt="{\displaystyle h\sim {2\times 10^{-13}/{\sqrt {\mathrm {Hz} }}}}"></span>, given as an <a href="/wiki/Spectral_density" title="Spectral density">amplitude spectral density</a>. The INFN Genoa detector is a resonant antenna consisting of two coupled spherical <a href="/wiki/Superconducting" class="mw-redirect" title="Superconducting">superconducting</a> harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. The system is currently expected to have a sensitivity to periodic spacetime strains of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\sim {2\times 10^{-17}/{\sqrt {\mathrm {Hz} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>17</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </msqrt> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\sim {2\times 10^{-17}/{\sqrt {\mathrm {Hz} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11dd7fcda654d54e170af8fa984154299c9ca3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.794ex; height:3.176ex;" alt="{\displaystyle h\sim {2\times 10^{-17}/{\sqrt {\mathrm {Hz} }}}}"></span>, with an expectation to reach a sensitivity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\sim {2\times 10^{-20}/{\sqrt {\mathrm {Hz} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>20</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </msqrt> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\sim {2\times 10^{-20}/{\sqrt {\mathrm {Hz} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88efec631ef35ba906789f6b937868fa2d0605d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.794ex; height:3.176ex;" alt="{\displaystyle h\sim {2\times 10^{-20}/{\sqrt {\mathrm {Hz} }}}}"></span>. The Chongqing University detector is planned to detect relic high-frequency gravitational waves with the predicted typical parameters ?<sub>g</sub> ~ 10<sup>10</sup> Hz (10 GHz) and <i>h</i> ~ 10<sup>−30</sup>-10<sup>−31</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Einstein@Home"><span id="Einstein.40Home"></span>Einstein@Home</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=12" title="Edit section: Einstein@Home"><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/Einstein@Home" title="Einstein@Home">Einstein@Home</a></div> <p>In some sense, the easiest signals to detect should be constant sources. Supernovae and neutron star or black hole mergers should have larger amplitudes and be more interesting, but the waves generated will be more complicated. The waves given off by a spinning, aspherical neutron star would be "<a href="/wiki/Monochrome" title="Monochrome">monochromatic</a>"—like a <a href="/wiki/Pure_tone" title="Pure tone">pure tone</a> in <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>. It would not change very much in amplitude or frequency. </p><p>The <a href="/wiki/Einstein@Home" title="Einstein@Home">Einstein@Home</a> project is a <a href="/wiki/Distributed_computing" title="Distributed computing">distributed computing</a> project similar to <a href="/wiki/SETI@home" title="SETI@home">SETI@home</a> intended to detect this type of simple gravitational wave. By taking data from LIGO and GEO, and sending it out in little pieces to thousands of volunteers for parallel analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=13" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Einstein%27s_equations" class="mw-redirect" title="Einstein's equations">Einstein's equations</a> form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> — denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70b3ee2dea90750949b21cc66fbd3171f5c6b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.591ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }\,}"></span>. The metric holds information regarding how distances are measured in the space under consideration. Because the propagation of gravitational waves through space and time change distances, we will need to use this to find the solution to the wave equation. </p><p>Spacetime curvature is also expressed with respect to a <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c85e631c2ca47eb9c3a6dfaa11a4cdf156e04b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.323ex; height:2.176ex;" alt="{\displaystyle \nabla \,}"></span>, in the form of the <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce4a5b59de7eda449c1f08ed7a84ae5de88884a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.921ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }}"></span>. This curvature is related to the <a href="/wiki/Stress-energy_tensor" class="mw-redirect" title="Stress-energy tensor">stress-energy tensor</a> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463ab8cef859ece28e33b8460ebd4a6699834dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.452ex; height:2.843ex;" alt="{\displaystyle T_{\mu \nu }}"></span> — by the key equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }={\frac {8\pi G_{N}}{c^{4}}}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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }={\frac {8\pi G_{N}}{c^{4}}}T_{\mu \nu }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ef18e216e20b7f2ea24d2a929500f729729842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.708ex; height:5.676ex;" alt="{\displaystyle G_{\mu \nu }={\frac {8\pi G_{N}}{c^{4}}}T_{\mu \nu }\,}"></span> ,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{N}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96cc4c2b25329408cf001ee42a08f71a927070f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.905ex; height:2.509ex;" alt="{\displaystyle G_{N}\,}"></span> is Newton's <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the speed of light. We assume <a href="/wiki/Geometrized_units" class="mw-redirect" title="Geometrized units">geometrized units</a>, so <span class="mwe-math-element"><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_{N}=1=c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{N}=1=c\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b62a2f044f3c552e042c8e263a3f2ae416ba98fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.271ex; height:2.509ex;" alt="{\displaystyle G_{N}=1=c\,}"></span>. </p><p>With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are <a href="/wiki/Wave_equation" title="Wave equation">wave equations</a>. To begin with, we adopt some coordinate system, like <span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/4531ecf2a7beb0f4fbd64497a388da0adfe6037d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.662ex; height:2.843ex;" alt="{\displaystyle (t,r,\theta ,\phi )\,}"></span>. We define the "flat-space metric" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a83fae7dc57e7ccc869c3b15f3c2c957c698e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.637ex; height:2.343ex;" alt="{\displaystyle \eta _{\mu \nu }\,}"></span> to be the quantity which — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have </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 \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \end{bmatrix}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>r</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> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \end{bmatrix}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c272ea91697f1faa475af2e8e402a55293511f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:31.669ex; height:12.843ex;" alt="{\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \end{bmatrix}}\,}"></span> .</dd></dl> <p>This flat-space metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this "flat-space metric". </p><p>Now, we can also think of the physical metric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mu \nu }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70b3ee2dea90750949b21cc66fbd3171f5c6b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.591ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }\,}"></span> as a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, and find its <a href="/wiki/Determinant" title="Determinant">determinant</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \ g\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mtext> </mtext> <mi>g</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \ g\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3276cf998a613a5592dd742d18f4818fb328c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.7ex; height:2.509ex;" alt="{\displaystyle \det \ g\,}"></span>. Finally, we define a quantity </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 {\bar {h}}^{\alpha \beta }\equiv \eta ^{\alpha \beta }-{\sqrt {|\det g|}}g^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>≡</mo> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }\equiv \eta ^{\alpha \beta }-{\sqrt {|\det g|}}g^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175783e4d4be5b90643e92e124ea09dec219676f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.372ex; height:4.843ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }\equiv \eta ^{\alpha \beta }-{\sqrt {|\det g|}}g^{\alpha \beta }\,}"></span> .</dd></dl> <p>This is the crucial field, which will represent the radiation. It is possible (at least in an <a href="/wiki/Asymptotically_flat_spacetime" title="Asymptotically flat spacetime">asymptotically flat spacetime</a>) to choose the coordinates in such a way that this quantity satisfies the "de Donder" gauge conditions (conditions on the coordinates): </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 \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928f757ead50e259b6ef0a259a9987524bcdc0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.71ex; height:3.843ex;" alt="{\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0\,}"></span> ,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" 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 \nabla }"></span> represents the flat-space derivative operator. These equations say that the <a href="/wiki/Divergence" title="Divergence">divergence</a> of the field is zero. The linear Einstein equations can now be written<sup id="cite_ref-RMP80_23-0" class="reference"><a href="#cite_note-RMP80-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mi>π</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd5233255a492cf5eb25462d2c498acf50a0b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.808ex; height:3.343ex;" alt="{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}"></span> ,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box =-\partial _{t}^{2}+\Delta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box =-\partial _{t}^{2}+\Delta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c02d50a5258af745ef47210b3140839b003ab11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.276ex; height:3.176ex;" alt="{\displaystyle \Box =-\partial _{t}^{2}+\Delta \,}"></span> represents the flat-space <a href="/wiki/D%27Alembertian" class="mw-redirect" title="D'Alembertian">d'Alembertian</a> operator, 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 \tau ^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbce7a199291230fce5b970cde0a6c26daf43289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.871ex; height:2.676ex;" alt="{\displaystyle \tau ^{\alpha \beta }\,}"></span> represents the stress-energy tensor plus quadratic terms involving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa36923129ed8bc70907e29c234ae46f69187e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.952ex; height:3.176ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"></span>. This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_approximation">Linear approximation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=14" title="Edit section: Linear approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equations above are valid everywhere — near a black hole, for instance. However, because of the complicated source term, the solution is generally too difficult to find analytically. We can often assume that space is nearly flat, so the metric is nearly equal to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a29c0d2dec611bbb0e99f202e978153b6222ade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.787ex; height:3.176ex;" alt="{\displaystyle \eta ^{\alpha \beta }\,}"></span> tensor. In this case, we can neglect terms quadratic in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa36923129ed8bc70907e29c234ae46f69187e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.952ex; height:3.176ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"></span>, which means that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbce7a199291230fce5b970cde0a6c26daf43289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.871ex; height:2.676ex;" alt="{\displaystyle \tau ^{\alpha \beta }\,}"></span> field reduces to the usual stress-energy tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9189d94d5aa060b85774a632982326ec18cb68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.676ex;" alt="{\displaystyle T^{\alpha \beta }\,}"></span>. That is, Einstein's equations become </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 \Box {\bar {h}}^{\alpha \beta }=-16\pi T^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mi>π</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi T^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d16cb3a8974df41e87a5a9de5d3288d12a378ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.27ex; height:3.343ex;" alt="{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi T^{\alpha \beta }\,}"></span> .</dd></dl> <p>If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stress-energy tensor would be zero, so </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 \Box {\bar {h}}^{\alpha \beta }=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box {\bar {h}}^{\alpha \beta }=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ea3970894cb0a94abe7616e613fc6125cbc4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.021ex; height:3.176ex;" alt="{\displaystyle \Box {\bar {h}}^{\alpha \beta }=0\,}"></span> .</dd></dl> <p>Now, this is the usual homogeneous wave equation — one for each component of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa36923129ed8bc70907e29c234ae46f69187e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.952ex; height:3.176ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }\,}"></span>. Solutions to this equation are well known. For a wave moving away from a point source, the radiated part (meaning the part that dies off as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71ac82f92fcc644fc9ca11d6a5bce653924ee583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.761ex; height:2.843ex;" alt="{\displaystyle 1/r\,}"></span> far from the source) can always be written in the form <span class="mwe-math-element"><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(t-r,\theta ,\phi )/r\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(t-r,\theta ,\phi )/r\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d954d70ab0d5a02083d1d8aea03b3faeaacfcd00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.423ex; height:2.843ex;" alt="{\displaystyle A(t-r,\theta ,\phi )/r\,}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> </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/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}"></span> is just some function. It can be shown<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> that — to a linear approximation — it is always possible to make the field traceless. Now, if we further assume that the source is positioned 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>, the general solution to the wave equation in spherical coordinates is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcl}{\bar {h}}^{\alpha \beta }&=&{\frac {1}{r}}\,{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&A_{+}(t-r,\theta ,\phi )&A_{\times }(t-r,\theta ,\phi )\\0&0&A_{\times }(t-r,\theta ,\phi )&-A_{+}(t-r,\theta ,\phi )\end{bmatrix}}\\\\&\equiv &{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&h_{+}(t-r,r,\theta ,\phi )&h_{\times }(t-r,r,\theta ,\phi )\\0&0&h_{\times }(t-r,r,\theta ,\phi )&-h_{+}(t-r,r,\theta ,\phi )\end{bmatrix}}\end{array}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <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> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <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> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <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> </mtd> <mtd> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <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> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>≡</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}{\bar {h}}^{\alpha \beta }&=&{\frac {1}{r}}\,{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&A_{+}(t-r,\theta ,\phi )&A_{\times }(t-r,\theta ,\phi )\\0&0&A_{\times }(t-r,\theta ,\phi )&-A_{+}(t-r,\theta ,\phi )\end{bmatrix}}\\\\&\equiv &{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&h_{+}(t-r,r,\theta ,\phi )&h_{\times }(t-r,r,\theta ,\phi )\\0&0&h_{\times }(t-r,r,\theta ,\phi )&-h_{+}(t-r,r,\theta ,\phi )\end{bmatrix}}\end{array}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45a8e744422d7168308568b33edbd813ac31b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.005ex; width:58.138ex; height:29.176ex;" alt="{\displaystyle {\begin{array}{lcl}{\bar {h}}^{\alpha \beta }&=&{\frac {1}{r}}\,{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&A_{+}(t-r,\theta ,\phi )&A_{\times }(t-r,\theta ,\phi )\\0&0&A_{\times }(t-r,\theta ,\phi )&-A_{+}(t-r,\theta ,\phi )\end{bmatrix}}\\\\&\equiv &{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&h_{+}(t-r,r,\theta ,\phi )&h_{\times }(t-r,r,\theta ,\phi )\\0&0&h_{\times }(t-r,r,\theta ,\phi )&-h_{+}(t-r,r,\theta ,\phi )\end{bmatrix}}\end{array}}\,}"></span></dd></dl> <p>where we now see the origin of the two polarizations. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_source">Relation to the source</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=15" title="Edit section: Relation to the source"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we know the details of a source — for instance, the parameters of the orbit of a binary — we can relate the source's motion to the gravitational radiation observed far away. With the relation </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 \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mi>π</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd5233255a492cf5eb25462d2c498acf50a0b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.808ex; height:3.343ex;" alt="{\displaystyle \Box {\bar {h}}^{\alpha \beta }=-16\pi \tau ^{\alpha \beta }\,}"></span> ,</dd></dl> <p>we can write the solution in terms of the tensorial <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> for the d'Alembertian operator:<sup id="cite_ref-RMP80_23-1" class="reference"><a href="#cite_note-RMP80-23"><span class="cite-bracket">[</span>23<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 {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-16\pi \int \,G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')\,\tau ^{\gamma \delta }(t',{\vec {x}}')\,dt'\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mi>π</mi> <mo>∫</mo> <mspace width="thinmathspace" /> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>;</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-16\pi \int \,G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')\,\tau ^{\gamma \delta }(t',{\vec {x}}')\,dt'\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35cfb222f7edbb2fa8d953792903d50ee460b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.862ex; height:5.676ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-16\pi \int \,G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')\,\tau ^{\gamma \delta }(t',{\vec {x}}')\,dt'\,d^{3}x'}"></span> .</dd></dl> <p>Though it is possible to expand the Green's function in tensor <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>, it is easier to simply use the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')={\frac {1}{4\pi }}\delta _{\gamma }^{\alpha }\,\delta _{\delta }^{\beta }\,{\frac {\delta (t\pm |{\vec {x}}-{\vec {x}}'|-t')}{|{\vec {x}}-{\vec {x}}'|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>;</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')={\frac {1}{4\pi }}\delta _{\gamma }^{\alpha }\,\delta _{\delta }^{\beta }\,{\frac {\delta (t\pm |{\vec {x}}-{\vec {x}}'|-t')}{|{\vec {x}}-{\vec {x}}'|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55054d27ceadcd914c9952713ef15fbe46ecd7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.658ex; height:6.843ex;" alt="{\displaystyle G_{\gamma \delta }^{\alpha \beta }(t,{\vec {x}};t',{\vec {x}}')={\frac {1}{4\pi }}\delta _{\gamma }^{\alpha }\,\delta _{\delta }^{\beta }\,{\frac {\delta (t\pm |{\vec {x}}-{\vec {x}}'|-t')}{|{\vec {x}}-{\vec {x}}'|}}}"></span> ,</dd></dl> <p>where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so </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 {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-4\int \,{\frac {\tau ^{\alpha \beta }(t-|{\vec {x}}-{\vec {x}}'|,{\vec {x}}')}{|{\vec {x}}-{\vec {x}}'|}}\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mo>∫</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-4\int \,{\frac {\tau ^{\alpha \beta }(t-|{\vec {x}}-{\vec {x}}'|,{\vec {x}}')}{|{\vec {x}}-{\vec {x}}'|}}\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a0c6567b249ed1fd744b4fb70b559e8ba136af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.011ex; height:6.843ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})=-4\int \,{\frac {\tau ^{\alpha \beta }(t-|{\vec {x}}-{\vec {x}}'|,{\vec {x}}')}{|{\vec {x}}-{\vec {x}}'|}}\,d^{3}x'}"></span> .</dd></dl> <p>If the source is confined to a small region very far away, to an excellent approximation we have: </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 {\bar {h}}^{\alpha \beta }(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,\tau ^{\alpha \beta }(t-r,{\vec {x}}')\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,\tau ^{\alpha \beta }(t-r,{\vec {x}}')\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e2e18710a2c4cd0f93f06f8015eb86d83d9d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.969ex; height:5.676ex;" alt="{\displaystyle {\bar {h}}^{\alpha \beta }(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,\tau ^{\alpha \beta }(t-r,{\vec {x}}')\,d^{3}x'}"></span> ,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=|{\vec {x}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=|{\vec {x}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842e3590e731ec69a7fc1d44027bab87f748dbeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.77ex; height:2.843ex;" alt="{\displaystyle r=|{\vec {x}}|}"></span> . </p><p>Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a> and an empty boundary, we can see that </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 \int \,\tau ^{ij}(t-r,{\vec {x}}')\,d^{3}x'=\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </mrow> </msup> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \,\tau ^{ij}(t-r,{\vec {x}}')\,d^{3}x'=\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/964b85f0224a703c9e28569f2a5942d109b36dd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.083ex; height:5.676ex;" alt="{\displaystyle \int \,\tau ^{ij}(t-r,{\vec {x}}')\,d^{3}x'=\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}"></span> ,</dd></dl> <p>Inserting this into the above equation, we arrive at </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 {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </mrow> </msup> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd8591d1c58f683e4df99a7c95663fa1cc78407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.529ex; height:5.676ex;" alt="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,\int \,x'^{i}x'^{j}\nabla _{k}\nabla _{l}\tau ^{kl}(t-r,{\vec {x}}')\,d^{3}x'}"></span> ,</dd></dl> <p>Finally, because we have chosen to work in coordinates for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec286af3687300f6fc3d19f45f52f81de5de61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.323ex; height:3.843ex;" alt="{\displaystyle \nabla _{\beta }\,{\bar {h}}^{\alpha \beta }=0}"></span>, we know that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\beta }\,\tau ^{\alpha \beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\beta }\,\tau ^{\alpha \beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a735f116d6c26f45737c51c8b6bb2f9757ea4e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.242ex; height:3.343ex;" alt="{\displaystyle \nabla _{\beta }\,\tau ^{\alpha \beta }=0}"></span>. With a few simple manipulations, we can use this to prove that </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 \nabla _{0}\nabla _{0}\tau ^{00}=\nabla _{j}\nabla _{k}\tau ^{jk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msup> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{0}\nabla _{0}\tau ^{00}=\nabla _{j}\nabla _{k}\tau ^{jk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1751b37012614396c9b876b0d9c66fb1d91819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.107ex; height:3.343ex;" alt="{\displaystyle \nabla _{0}\nabla _{0}\tau ^{00}=\nabla _{j}\nabla _{k}\tau ^{jk}}"></span> .</dd></dl> <p>With this relation, the expression for the radiated field is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\int \,x'^{i}x'^{j}\tau ^{00}(t-r,{\vec {x}}')\,d^{3}x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </mrow> </msup> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\int \,x'^{i}x'^{j}\tau ^{00}(t-r,{\vec {x}}')\,d^{3}x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196b40e7ca5277b428967ed8cfdc6f00f9d23602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.477ex; height:6.176ex;" alt="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\int \,x'^{i}x'^{j}\tau ^{00}(t-r,{\vec {x}}')\,d^{3}x'}"></span> .</dd></dl> <p>In the linear case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ^{00}=\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msup> <mo>=</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ^{00}=\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae0200eead169e278b7ad807171342748875dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.434ex; height:3.176ex;" alt="{\displaystyle \tau ^{00}=\rho }"></span>, the density of mass-energy. </p><p>To a very good approximation, the density of a simple binary can be described by a pair of delta-functions, which eliminates the integral. Explicitly, if the masses of the two objects are <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></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 M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"></span>, and the positions are <span class="mwe-math-element"><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 {x}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78435b0d85b92bd9ee899b3318b3fa57395252d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}_{1}}"></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 {\vec {x}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024a11e6ca89428b41e10d119a73bc1f62b45ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}_{2}}"></span>, then </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 \rho (t-r,{\vec {x}}')=M_{1}\delta ^{3}({\vec {x}}'-{\vec {x}}_{1}(t-r))+M_{2}\delta ^{3}({\vec {x}}'-{\vec {x}}_{2}(t-r))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (t-r,{\vec {x}}')=M_{1}\delta ^{3}({\vec {x}}'-{\vec {x}}_{1}(t-r))+M_{2}\delta ^{3}({\vec {x}}'-{\vec {x}}_{2}(t-r))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731620f6f91cd1999389fa5b4dd1df25f853e734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.731ex; height:3.176ex;" alt="{\displaystyle \rho (t-r,{\vec {x}}')=M_{1}\delta ^{3}({\vec {x}}'-{\vec {x}}_{1}(t-r))+M_{2}\delta ^{3}({\vec {x}}'-{\vec {x}}_{2}(t-r))}"></span> .</dd></dl> <p>We can use this expression to do the integral above: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\left\{M_{1}x_{1}^{i}(t-r)x_{1}^{j}(t-r)+M_{2}x_{2}^{i}(t-r)x_{2}^{j}(t-r)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\left\{M_{1}x_{1}^{i}(t-r)x_{1}^{j}(t-r)+M_{2}x_{2}^{i}(t-r)x_{2}^{j}(t-r)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48cc754ad7150b1c0892925c797ac8c5ca658fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:68.086ex; height:6.009ex;" alt="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {d^{2}}{dt^{2}}}\,\left\{M_{1}x_{1}^{i}(t-r)x_{1}^{j}(t-r)+M_{2}x_{2}^{i}(t-r)x_{2}^{j}(t-r)\right\}}"></span> .</dd></dl> <p>Using mass-centered coordinates, and assuming a circular binary, this is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {M_{1}M_{2}}{R}}\,n^{i}(t-r)n^{j}(t-r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {M_{1}M_{2}}{R}}\,n^{i}(t-r)n^{j}(t-r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a95c610f8a2bc93be31cbc78bf7e0649096f4a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.536ex; height:5.343ex;" alt="{\displaystyle {\bar {h}}^{ij}(t,{\vec {x}})\approx -{\frac {4}{r}}\,{\frac {M_{1}M_{2}}{R}}\,n^{i}(t-r)n^{j}(t-r)}"></span> ,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\vec {x}}_{1}/|{\vec {x}}_{1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\vec {x}}_{1}/|{\vec {x}}_{1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6291962021b784e3ccb77569ae9bcf97260e3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.717ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}={\vec {x}}_{1}/|{\vec {x}}_{1}|}"></span>. Plugging in the known values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}_{1}(t-r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}_{1}(t-r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6435b264bb5ce154e15518c91ea468ecefc0422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.922ex; height:2.843ex;" alt="{\displaystyle {\vec {x}}_{1}(t-r)}"></span>, we obtain the expressions given above for the radiation from a simple binary. </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=User:Christillin/gravitational_waves&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Linearised_Einstein_field_equations" class="mw-redirect" title="Linearised Einstein field equations">Linearised Einstein field equations</a></li> <li><a href="/wiki/Gravitomagnetism" class="mw-redirect" title="Gravitomagnetism">Gravitomagnetism</a></li> <li><a href="/wiki/Gravitational_wave_astronomy" class="mw-redirect" title="Gravitational wave astronomy">Graviton astronomy</a></li> <li><a href="/wiki/LIGO" title="LIGO">LIGO</a>, <a href="/wiki/VIRGO" class="mw-redirect mw-disambig" title="VIRGO">VIRGO</a>, <a href="/wiki/GEO_600" class="mw-redirect" title="GEO 600">GEO 600</a>, and <a href="/wiki/TAMA_300" title="TAMA 300">TAMA 300</a> — Gravitational wave detectors</li> <li><a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">LISA</a> the proposed Laser Interferometer Space Antenna</li> <li><a href="/wiki/Deci-hertz_Interferometer_Gravitational_wave_Observatory" title="Deci-hertz Interferometer Gravitational wave Observatory">DECIGO</a> "Deci-hertz Interferometer Gravitational wave Observatory", the planned laser interferometric detector in space</li> <li><a href="/wiki/Big_Bang_Observer" title="Big Bang Observer">Big Bang Observer</a> (BBO), proposed successor to LISA</li> <li><a href="/wiki/Sticky_bead_argument" title="Sticky bead argument">Sticky bead argument</a>, for a physical way to see that gravitational radiation should carry energy</li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave spacetime</a>, for an important class of exact solutions modelling gravitational radiation</li> <li><a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a>, for gravitationally induced electromagnetic radiation from black holes</li> <li><a href="/wiki/Spin-flip" title="Spin-flip">Spin-flip</a>, a consequence of gravitational wave emission from binary <a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">supermassive black holes</a></li> <li><a href="/wiki/Gravitational_field" title="Gravitational field">Gravitational field</a></li> <li><a href="/wiki/Orbital_resonance" title="Orbital resonance">Orbital resonance</a></li> <li><a href="/wiki/Tidal_force" title="Tidal force">Tidal force</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.dpf99.library.ucla.edu/session14/barish1412.pdf">http://www.dpf99.library.ucla.edu/session14/barish1412.pdf</a> The Detection of Gravitational Waves using LIGO, B. 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"Early Gravity-Wave Detection Experiments, 1960–1975". <i>Physics in Perspective (Birkhäuser Basel)</i>. <b>6</b> (1): 42–75. <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/2004PhP.....6...42L">2004PhP.....6...42L</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%2Fs00016-003-0179-6">10.1007/s00016-003-0179-6</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:76657516">76657516</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+in+Perspective+%28Birkh%C3%A4user+Basel%29&rft.atitle=Early+Gravity-Wave+Detection+Experiments%2C+1960%E2%80%931975&rft.volume=6&rft.issue=1&rft.pages=42-75&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A76657516%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00016-003-0179-6&rft_id=info%3Abibcode%2F2004PhP.....6...42L&rft.aulast=Levine&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUser%3AChristillin%2Fgravitational+waves" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Unknown parameter <code class="cs1-code">|month=</code> ignored (<a href="/wiki/Help:CS1_errors#parameter_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.minigrail.nl/AboutMiniGRAIL/AboutMiniGRAIL-index.html">Gravitational Radiation Antenna In Leiden</a></span> </li> <li id="cite_note-MiniGRAIL_2000-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-MiniGRAIL_2000_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Waard2000" class="citation journal cs1">de Waard, Arlette (Italy). 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Rome.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Marcel+Grossman+meeting+on+General+Relativity&rft.aulast=de+Waard&rft.aufirst=Arlette&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUser%3AChristillin%2Fgravitational+waves" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|contribution=</code> ignored (<a href="/wiki/Help:CS1_errors#chapter_ignored" title="Help:CS1 errors">help</a>)</span>; <span class="cs1-visible-error citation-comment">Check date values in: <code class="cs1-code">|date=</code> and <code class="cs1-code">|year=</code> / <code class="cs1-code">|date=</code> mismatch (<a href="/wiki/Help:CS1_errors#bad_date" title="Help:CS1 errors">help</a>)</span>; <span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span>; <span class="cs1-visible-error citation-comment">Unknown parameter <code class="cs1-code">|coauthors=</code> ignored (<code class="cs1-code">|author=</code> suggested) (<a href="/wiki/Help:CS1_errors#parameter_ignored_suggest" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">The idea of using laser interferometry for gravitational wave detection was first mentioned by Gerstenstein and Pustovoit 1963 Sov. Phys.–JETP 16 433. Weber mentioned it in an unpublished laboratory notebook. <a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Rainer Weiss</a> first described in detail a practical solution with an analysis of realistic limitations to the technique in R. Weiss (1972). "Electromagetically Coupled Broadband Gravitational Antenna". Quarterly Progress Report, Research Laboratory of Electronics, MIT 105: 54.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://einstein.phys.uwm.edu/">Einstein@Home</a></span> </li> <li id="cite_note-RMP80-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-RMP80_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-RMP80_23-1"><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="CITEREFThorne1980" class="citation journal cs1">Thorne, Kip (1980). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/11159/1/THOrmp80a.pdf">"Multipole expansions of gravitational radiation"</a> <span class="cs1-format">(PDF)</span>. <i>Reviews of Modern Physics</i>. <b>52</b> (2): 299–339. <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/1980RvMP...52..299T">1980RvMP...52..299T</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%2FRevModPhys.52.299">10.1103/RevModPhys.52.299</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Multipole+expansions+of+gravitational+radiation&rft.volume=52&rft.issue=2&rft.pages=299-339&rft.date=1980&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.52.299&rft_id=info%3Abibcode%2F1980RvMP...52..299T&rft.aulast=Thorne&rft.aufirst=Kip&rft_id=https%3A%2F%2Fauthors.library.caltech.edu%2F11159%2F1%2FTHOrmp80a.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUser%3AChristillin%2Fgravitational+waves" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Unknown parameter <code class="cs1-code">|month=</code> ignored (<a href="/wiki/Help:CS1_errors#parameter_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._W._Misner,_K._S._Thorne,_and_J._A._Wheeler1973" class="citation book cs1">C. W. Misner, K. S. Thorne, and J. A. Wheeler (1973). <i>Gravitation</i>. W. H. Freeman and Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman+and+Co.&rft.date=1973&rft.au=C.+W.+Misner%2C+K.+S.+Thorne%2C+and+J.+A.+Wheeler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUser%3AChristillin%2Fgravitational+waves" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> </ol></div></div> <ul><li>Chakrabarty, Indrajit, "<a rel="nofollow" class="external text" href="http://arxiv.org/pdf/physics/9908041">Gravitational Waves: An Introduction</a>". arXiv:physics/9908041 v1, Aug 21, 1999.</li> <li>Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields (Pergamon Press),(1987).</li> <li>Will, Clifford M., <i><a rel="nofollow" class="external text" href="http://www.livingreviews.org/lrr-2006-3/">The Confrontation between General Relativity and Experiment</a></i>. Living Rev. Relativity 9 (2006) 3.</li> <li>Peter Saulson, "Fundamentals of Interferometric Gravitational Wave Detectors", World Scientific, 1994.</li> <li>J. Bicak, W.N. Rudienko, "Gravitacionnyje wolny w OTO i probliema ich obnarużenija", Izdatielstwo Moskovskovo Universitieta, 1987.</li> <li>A. Kułak, "Electromagnetic Detectors of Gravitational Radiation", PhD Thesis, Cracow 1980 (In Polish).</li> <li>P. Tatrocki, "On intuitive description of graviton detector", www.philica.com .</li> <li>P. Tatrocki, "Can the LIGO, VIRGO, GEO600, AIGO, TAMA, LISA detectors really detect?", www.philica.com .</li></ul> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=18" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Michael_Berry_(physicist)" title="Michael Berry (physicist)">Berry, Michael</a>, <i>Principles of cosmology and gravitation</i> (Adam Hilger, Philadelphia, 1989). <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/0-85274-037-9" title="Special:BookSources/0-85274-037-9">0-85274-037-9</a></li> <li><a href="/wiki/Harry_Collins" title="Harry Collins">Collins, Harry</a>, <i>Gravity's Shadow: the search for gravitational waves</i>, University of Chicago Press, 2004.</li> <li><a href="/wiki/Jim_Peebles" title="Jim Peebles">P. J. E. Peebles</a>, <i>Principles of Physical Cosmology</i> (Princeton University Press, Princeton, 1993). <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/0-691-01933-9" title="Special:BookSources/0-691-01933-9">0-691-01933-9</a>.</li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> and Ciufolini, Ignazio, <i>Gravitation and Inertia</i> (Princeton University Press, Princeton, 1995). <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/0-691-03323-4" title="Special:BookSources/0-691-03323-4">0-691-03323-4</a>.</li> <li>Woolf, Harry, ed., <i>Some Strangeness in the Proportion</i> (Addison–Wesley, Reading, Massachusetts, 1980). <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/0-201-09924-1" title="Special:BookSources/0-201-09924-1">0-201-09924-1</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User:Christillin/gravitational_waves&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Gravitational_waves" class="extiw" title="commons:Category:Gravitational waves">Gravitational waves</a> at Wikimedia Commons </p> <ul><li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b007h8gv">Gravitational Waves</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><a rel="nofollow" class="external text" href="http://brownbag.lisascience.org">The LISA Brownbag</a> – Selection of the most significant e-prints related to LISA science</li> <li><a rel="nofollow" class="external text" href="http://www.astroparticle.org/">Astroparticle.org</a>. To know everything about astroparticle physics, including gravitational waves</li> <li><a rel="nofollow" class="external text" href="http://elmer.tapir.caltech.edu/ph237/CourseMaterials.html">Caltech's Physics 237-2002 Gravitational Waves by Kip Thorne</a> <b>Video plus notes:</b> Graduate level but does not assume knowledge of General Relativity, Tensor Analysis, or Differential Geometry; Part 1: Theory (10 lectures), Part 2: Detection (9 lectures)</li> <li><a rel="nofollow" class="external text" href="http://www.astronomycast.com/astronomy/ep-71-gravitational-waves">www.astronomycast.com January 14, 2008 Episode 71: Gravitational Waves</a></li> <li><a rel="nofollow" class="external text" href="http://www.ligo.caltech.edu/">Laser Interferometer Gravitational Wave Observatory</a>. LIGO Laboratory, operated by the <a href="/wiki/California_Institute_of_Technology" title="California Institute of Technology">California Institute of Technology</a> and the <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a></li> <li><a rel="nofollow" class="external text" href="http://www.ligo.org/">The LIGO Scientific Collaboration</a></li> <li><a rel="nofollow" class="external text" href="http://www.ligo.caltech.edu/einstein.ram">Einstein's Messengers</a> – The LIGO Movie by <a href="/wiki/National_Science_Foundation" title="National Science Foundation">NSF</a></li> <li><a rel="nofollow" class="external text" href="http://einstein.phys.uwm.edu/">Home page for Einstein@Home project</a>, a distributed computing project processing raw data from LIGO Laboratory, searching for gravity waves</li> <li><a rel="nofollow" class="external text" href="http://archive.ncsa.uiuc.edu/Cyberia/NumRel/GravWaves.html">The National Center for Supercomputing Applications</a> – a numerical relativity group</li> <li><a rel="nofollow" class="external text" href="https://www.black-holes.org/gwa1.html">Caltech Relativity Tutorial</a> – A basic introduction to gravitational waves, and astrophysical systems giving off gravitational waves</li> <li><a rel="nofollow" class="external text" href="http://arxiv.org/abs/gr-qc/0211084">Resource Letter GrW-1: Gravitational waves</a> – a list of books, journals and web resources compiled by Joan Centrella for research into gravitational waves</li> <li><a rel="nofollow" class="external text" href="http://www.rri.res.in/htmls/tp/GravRad-grim.pdf">Mathematical and Physical Perspectives on Gravitational Radiation</a> – written by B F Schutz of the <a href="/wiki/Max_Planck_Institute" class="mw-redirect" title="Max Planck Institute">Max Planck Institute</a> explaining the significance and background of some key concepts in gravitational radiation</li> <li><a rel="nofollow" class="external text" href="http://www.sciencebits.com/BlackHoleSimulation">Binary BH Merger</a> – estimating the radiated power and merger time of a BH binary using dimensional analysis</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 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observatory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">Detectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Gravitational-wave_observatory#Weber_bars" title="Gravitational-wave observatory">Resonant mass<br /> antennas</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Active</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="/w/index.php?title=NAUTILUS&action=edit&redlink=1" class="new" title="NAUTILUS (page does not exist)">NAUTILUS</a> (<a href="/w/index.php?title=International_Gravitational_Event_Collaboration&action=edit&redlink=1" class="new" title="International Gravitational Event Collaboration (page does not exist)">IGEC</a>)</li> <li><a href="/wiki/AURIGA" title="AURIGA">AURIGA</a> (<a href="/w/index.php?title=International_Gravitational_Event_Collaboration&action=edit&redlink=1" class="new" title="International Gravitational Event Collaboration (page does not exist)">IGEC</a>)</li> <li><a href="/wiki/MiniGrail" title="MiniGrail">MiniGRAIL</a></li> <li><a href="/wiki/Mario_Schenberg_(Gravitational_Wave_Detector)" title="Mario Schenberg (Gravitational Wave Detector)">Mario Schenberg</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Past</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="/w/index.php?title=EXPLORER&action=edit&redlink=1" class="new" title="EXPLORER (page does not exist)">EXPLORER</a> (<a href="/w/index.php?title=International_Gravitational_Event_Collaboration&action=edit&redlink=1" class="new" title="International Gravitational Event Collaboration (page does not exist)">IGEC</a>)</li> <li><a href="/wiki/Allegro_gravitational-wave_detector" title="Allegro gravitational-wave detector">ALLEGRO</a> (<a href="/w/index.php?title=International_Gravitational_Event_Collaboration&action=edit&redlink=1" class="new" title="International Gravitational Event Collaboration (page does not exist)">IGEC</a>)</li> <li><a href="/wiki/NIOBE" title="NIOBE">NIOBE</a> (<a href="/w/index.php?title=International_Gravitational_Event_Collaboration&action=edit&redlink=1" class="new" title="International Gravitational Event Collaboration (page does not exist)">IGEC</a>)</li> <li><a href="/w/index.php?title=Stanford_gravitational_wave_detector&action=edit&redlink=1" class="new" title="Stanford gravitational wave detector (page does not exist)">Stanford gravitational wave detector</a></li> <li><a href="/w/index.php?title=ALTAIR_(gravitational_wave_detector)&action=edit&redlink=1" class="new" title="ALTAIR (gravitational wave detector) (page does not exist)">ALTAIR</a></li> <li><a href="/w/index.php?title=GEOGRAV&action=edit&redlink=1" class="new" title="GEOGRAV (page does not exist)">GEOGRAV</a></li> <li><a href="/w/index.php?title=AGATA_(gravitational_wave_detector)&action=edit&redlink=1" class="new" title="AGATA (gravitational wave detector) (page does not exist)">AGATA</a></li> <li><a href="/wiki/Weber_bar" title="Weber bar">Weber bar</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Proposed</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/Torsion-bar_antenna" title="Torsion-bar antenna">TOBA</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Past proposals</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/MiniGrail" title="MiniGrail">GRAIL</a> (downsized to <a href="/wiki/MiniGrail" title="MiniGrail">MiniGRAIL</a>)</li> <li><a href="/w/index.php?title=Truncated_Icosahedral_Gravitational_Wave_Antenna&action=edit&redlink=1" class="new" title="Truncated Icosahedral Gravitational Wave Antenna (page does not exist)">TIGA</a></li> <li><a href="/w/index.php?title=SFERA&action=edit&redlink=1" class="new" title="SFERA (page does not exist)">SFERA</a></li> <li><a href="/wiki/Mario_Schenberg_(Gravitational_Wave_Detector)" title="Mario Schenberg (Gravitational Wave Detector)">Graviton</a> (downsized to <a href="/wiki/Mario_Schenberg_(Gravitational_Wave_Detector)" title="Mario Schenberg (Gravitational Wave Detector)">Mario Schenberg</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ground-based_interferometric_gravitational-wave_search" title="Ground-based interferometric gravitational-wave search"> Ground-based<br />interferometers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Active</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/AIGO" title="AIGO">AIGO</a> (<a href="/wiki/ACIGA" title="ACIGA">ACIGA</a>)</li> <li><a href="/wiki/CLIO" title="CLIO">CLIO</a></li> <li><a href="/wiki/Holometer" title="Holometer">Fermilab holometer</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO600</a></li> <li><a href="/wiki/LIGO" title="LIGO">Advanced LIGO</a> (<a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">LIGO Scientific Collaboration</a>)</li> <li><a href="/wiki/KAGRA" title="KAGRA">KAGRA</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Advanced Virgo</a> (<a href="/wiki/European_Gravitational_Observatory" title="European Gravitational Observatory">European Gravitational Observatory</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Past</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/TAMA_300" title="TAMA 300">TAMA 300</a></li> <li><a href="/wiki/TAMA_300" title="TAMA 300">TAMA 20, later known as LISM</a></li> <li><a href="/w/index.php?title=TENKO-100&action=edit&redlink=1" class="new" title="TENKO-100 (page does not exist)">TENKO-100</a></li> <li><a href="/wiki/LIGO#Background" title="LIGO">Caltech 40m interferometer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Planned</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/Indian_Initiative_in_Gravitational-wave_Observations" title="Indian Initiative in Gravitational-wave Observations">INDIGO (LIGO-India)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Proposed</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/Cosmic_Explorer_(gravitational_wave_observatory)" title="Cosmic Explorer (gravitational wave observatory)">Cosmic Explorer</a></li> <li><a href="/wiki/Einstein_Telescope" title="Einstein Telescope">Einstein Telescope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Past proposals</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/LIGO-Australia" class="mw-redirect" title="LIGO-Australia">LIGO-Australia</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Space-based<br />interferometers</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Planned</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/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">LISA</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Proposed</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/Big_Bang_Observer" title="Big Bang Observer">Big Bang Observer</a></li> <li><a href="/wiki/Deci-hertz_Interferometer_Gravitational_wave_Observatory" title="Deci-hertz Interferometer Gravitational wave Observatory">DECIGO</a></li> <li><a href="/wiki/TianQin" title="TianQin">TianQin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">Pulsar timing arrays</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/European_Pulsar_Timing_Array" title="European Pulsar Timing Array">EPTA</a></li> <li><a href="/wiki/International_Pulsar_Timing_Array" title="International Pulsar Timing Array">IPTA</a></li> <li><a href="/wiki/North_American_Nanohertz_Observatory_for_Gravitational_Waves" title="North American Nanohertz Observatory for Gravitational Waves">NANOGrav</a></li> <li><a href="/wiki/Parkes_Observatory" title="Parkes Observatory">PPTA</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data analysis</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/Einstein@Home" title="Einstein@Home">Einstein@Home</a></li> <li><a href="/wiki/PyCBC" title="PyCBC">PyCBC</a></li> <li><a href="/wiki/Zooniverse" title="Zooniverse">Zooniverse</a>: Gravity Spy</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Observations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Events</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_gravitational_wave_observations" title="List of gravitational wave observations">List of observations</a></li> <li><a href="/wiki/First_observation_of_gravitational_waves" title="First observation of gravitational waves">First observation (GW150914)</a></li> <li><a href="/wiki/GW151226" title="GW151226">GW151226</a></li> <li><a href="/wiki/GW170104" title="GW170104">GW170104</a></li> <li><a href="/wiki/GW170608" title="GW170608">GW170608</a></li> <li><a href="/wiki/GW170814" title="GW170814">GW170814</a></li> <li><a href="/wiki/GW170817" title="GW170817">GW170817</a> (first <a href="/wiki/Neutron_star_merger" title="Neutron star merger">neutron star merger</a>)</li> <li><a href="/wiki/GW190412" title="GW190412">GW190412</a></li> <li><a href="/wiki/GW190521" title="GW190521">GW190521</a> (first-ever possible light from bh-bh merger)</li> <li><a href="/wiki/GW190814" title="GW190814">GW190814</a> (first-ever "mass gap" collision)</li> <li><a href="/w/index.php?title=GW200105&action=edit&redlink=1" class="new" title="GW200105 (page does not exist)">GW200105</a> (first black hole - neutron star merger)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Methods</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Direct detection <ul><li><a href="/wiki/Interferometric_gravitational-wave_detector" class="mw-redirect" title="Interferometric gravitational-wave detector">Laser interferometers</a></li> <li>Resonant mass detectors</li> <li><i>Proposed: <a href="/wiki/Atom_interferometer" title="Atom interferometer">Atom interferometers</a></i></li></ul></li> <li>Indirect detection <ul><li><a href="/wiki/Cosmic_microwave_background#B-modes" title="Cosmic microwave background">B-modes</a> of <a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">CMB</a></li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">Pulsar timing array</a></li> <li><a href="/wiki/Binary_pulsar" title="Binary pulsar">Binary pulsar</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</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/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests of general relativity</a></li> <li><a href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Metric theories</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Effects / properties</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/Gravitational_wave#Effects_of_passing" title="Gravitational wave">Polarization</a></li> <li><a href="/wiki/Spin-flip" title="Spin-flip">Spin-flip</a></li> <li><a href="/wiki/Redshift" title="Redshift">Redshift</a></li> <li>Travel with <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a></li> <li>h strain</li> <li>Chirp signal (<a href="/wiki/Chirp_mass" title="Chirp mass">chirp mass</a>)</li> <li>Carried <a href="/wiki/Energy" title="Energy">energy</a></li> <li><a href="/wiki/Gravitational_wave_background" title="Gravitational wave background">Gravitational wave background</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types / sources</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/Stochastic" title="Stochastic">Stochastic</a> <ul><li><a href="/wiki/Cosmic_inflation" title="Cosmic inflation">Cosmic inflation</a>-<a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">quantum fluctuation</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li></ul></li> <li>Binary inspiral <ul><li><a href="/wiki/Supermassive_black_hole" title="Supermassive black hole">Supermassive black holes</a></li> <li><a href="/wiki/Stellar_black_hole" title="Stellar black hole">Stellar black holes</a></li> <li><a href="/wiki/Neutron_star" title="Neutron star">Neutron stars</a></li> <li><a href="/wiki/Extreme_mass_ratio_inspiral" title="Extreme mass ratio inspiral">EMRI</a></li></ul></li> <li>Continuous <ul><li>Rotating neutron star</li></ul></li> <li>Burst <ul><li><a href="/wiki/Supernova" title="Supernova">Supernova</a> or from <i>unknown</i> sources</li></ul></li> <li>Hypothesis <ul><li>Colliding <a href="/wiki/Cosmic_string" title="Cosmic string">cosmic string</a> and <i>other unknown</i> sources</li></ul></li></ul> </div></td></tr></tbody></table></div> <p><br /> <a href="/wiki/Category:Binary_stars" title="Category:Binary stars">Category:Binary stars</a> <a href="/wiki/Category:Black_holes" title="Category:Black holes">Category:Black holes</a> <a href="/w/index.php?title=Category:Effects_of_gravitation&action=edit&redlink=1" class="new" title="Category:Effects of gravitation (page does not exist)">Category:Effects of gravitation</a> <a href="/w/index.php?title=Category:Gravitation&action=edit&redlink=1" class="new" title="Category:Gravitation (page does not exist)">Category:Gravitation</a> <a href="/wiki/Category:General_relativity" title="Category:General relativity">Category:General relativity</a> </p>    </div><noscript><img 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