Invariant mass - Wikipedia

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vector-toc-level-2"> <a class="vector-toc-link" href="#Massless_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Massless particles</span> </div> </a> <ul id="toc-Massless_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collider_experiments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Collider_experiments"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Collider experiments</span> </div> </a> <ul id="toc-Collider_experiments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rest_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rest_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Rest energy</span> </div> </a> <ul id="toc-Rest_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Invariant mass</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 41 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-41" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Rusmassa" title="Rusmassa – Afrikaans" lang="af" hreflang="af" data-title="Rusmassa" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%AA%D9%84%D8%A9_%D8%B3%D8%A7%D9%83%D9%86%D8%A9" title="كتلة ساكنة – Arabic" lang="ar" hreflang="ar" data-title="كتلة ساكنة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BF%E0%A6%B0_%E0%A6%AD%E0%A6%B0" title="স্থির ভর – Bangla" lang="bn" hreflang="bn" data-title="স্থির ভর" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Put-pi%C3%A0n_chit-li%C5%8Dng" title="Put-piàn chit-liōng – Minnan" lang="nan" hreflang="nan" data-title="Put-piàn chit-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Massa_en_rep%C3%B2s" title="Massa en repòs – Catalan" lang="ca" hreflang="ca" data-title="Massa en repòs" 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/Klidov%C3%A1_hmotnost" title="Klidová hmotnost – Czech" lang="cs" hreflang="cs" data-title="Klidová hmotnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hvilemasse" title="Hvilemasse – Danish" lang="da" hreflang="da" data-title="Hvilemasse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schwerpunktsenergie" title="Schwerpunktsenergie – German" lang="de" hreflang="de" data-title="Schwerpunktsenergie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Seisumass" title="Seisumass – Estonian" lang="et" hreflang="et" data-title="Seisumass" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Invarianta_maso" title="Invarianta maso – Esperanto" lang="eo" hreflang="eo" data-title="Invarianta maso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%B1%D9%85_%D9%86%D8%A7%D9%85%D8%AA%D8%BA%DB%8C%D8%B1" title="جرم نامتغیر – Persian" lang="fa" hreflang="fa" data-title="جرم نامتغیر" 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/Masse_au_repos" title="Masse au repos – French" lang="fr" hreflang="fr" data-title="Masse au repos" 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/Masa_en_repouso" title="Masa en repouso – Galician" lang="gl" hreflang="gl" data-title="Masa en repouso" 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/%EB%B6%88%EB%B3%80_%EC%A7%88%EB%9F%89" title="불변 질량 – Korean" lang="ko" hreflang="ko" data-title="불변 질량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%9A%E0%A4%B0_%E0%A4%A6%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="निश्चर द्रव्यमान – Hindi" lang="hi" hreflang="hi" data-title="निश्चर द्रव्यमान" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Nevarianta_maso" title="Nevarianta maso – Ido" lang="io" hreflang="io" data-title="Nevarianta maso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Energi_diam" title="Energi diam – Indonesian" lang="id" hreflang="id" data-title="Energi diam" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hv%C3%ADldarmassi" title="Hvíldarmassi – Icelandic" lang="is" hreflang="is" data-title="Hvíldarmassi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Massa_a_riposo" title="Massa a riposo – Italian" lang="it" hreflang="it" data-title="Massa a riposo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%AA_%D7%9E%D7%A0%D7%95%D7%97%D7%94" title="מסת מנוחה – Hebrew" lang="he" hreflang="he" data-title="מסת מנוחה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A3%E1%83%AB%E1%83%A0%E1%83%90%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%90%E1%83%A1%E1%83%90" title="უძრაობის მასა – Georgian" lang="ka" hreflang="ka" data-title="უძრაობის მასა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D1%8B%D0%BD%D1%87_%D0%B0%D0%B1%D0%B0%D0%BB%D0%B4%D0%B0%D0%B3%D1%8B_%D0%BC%D0%B0%D1%81%D1%81%D0%B0" title="Тынч абалдагы масса – Kyrgyz" lang="ky" hreflang="ky" data-title="Тынч абалдагы масса" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rimties_energija" title="Rimties energija – Lithuanian" lang="lt" hreflang="lt" data-title="Rimties energija" 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/Invari%C3%A1ns_t%C3%B6meg" title="Invariáns tömeg – Hungarian" lang="hu" hreflang="hu" data-title="Invariáns tömeg" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rustmassa" title="Rustmassa – Dutch" lang="nl" hreflang="nl" data-title="Rustmassa" 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/%E4%B8%8D%E5%A4%89%E8%B3%AA%E9%87%8F" title="不変質量 – Japanese" lang="ja" hreflang="ja" data-title="不変質量" 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/Hvilemasse" title="Hvilemasse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hvilemasse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvilemasse" title="Kvilemasse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvilemasse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Masa_spoczynkowa" title="Masa spoczynkowa – Polish" lang="pl" hreflang="pl" data-title="Masa spoczynkowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%81%D1%81%D0%B0" title="Инвариантная масса – Russian" lang="ru" hreflang="ru" data-title="Инвариантная масса" 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/Rest_mass" title="Rest mass – Simple English" lang="en-simple" hreflang="en-simple" data-title="Rest mass" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pokojov%C3%A1_hmotnos%C5%A5" title="Pokojová hmotnosť – Slovak" lang="sk" hreflang="sk" data-title="Pokojová hmotnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mirovna_masa" title="Mirovna masa – Slovenian" lang="sl" hreflang="sl" data-title="Mirovna masa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lepomassa" title="Lepomassa – Finnish" lang="fi" hreflang="fi" data-title="Lepomassa" 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/Vilomassa" title="Vilomassa – Swedish" lang="sv" hreflang="sv" data-title="Vilomassa" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Hindi_nagbabagong_masa" title="Hindi nagbabagong masa – Tagalog" lang="tl" hreflang="tl" data-title="Hindi nagbabagong masa" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A1%E0%B8%A7%E0%B8%A5%E0%B8%99%E0%B8%B4%E0%B9%88%E0%B8%87" title="มวลนิ่ง – Thai" lang="th" hreflang="th" data-title="มวลนิ่ง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Durgun_k%C3%BCtle" title="Durgun kütle – Turkish" lang="tr" hreflang="tr" data-title="Durgun kütle" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D0%B2%D0%B0%D1%80%D1%96%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B0%D1%81%D0%B0" title="Інваріантна маса – Ukrainian" lang="uk" hreflang="uk" data-title="Інваріантна маса" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9D%9C%E6%AD%A2%E8%B3%AA%E9%87%8F" title="靜止質量 – Cantonese" lang="yue" hreflang="yue" data-title="靜止質量" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%8D%E5%8F%98%E8%B4%A8%E9%87%8F" title="不变质量 – Chinese" lang="zh" hreflang="zh" data-title="不变质量" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1097654#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Invariant_mass" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Invariant_mass" rel="discussion" title="Discuss improvements to the 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typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Invariant_mass" title="Special:EditPage/Invariant mass">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:Invariant_mass" title="Talk:Invariant mass">talk page</a></b>. <small><i>(<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove these messages</a>)</i></small> <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Invariant_mass" title="Special:EditPage/Invariant mass">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Invariant+mass%22">"Invariant mass"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Invariant+mass%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Invariant+mass%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Invariant+mass%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Invariant+mass%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Invariant+mass%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Cleanup_rewrite plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may need to be rewritten</b> to comply with Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>, as Outdated, incorrect, see talk.<span class="hide-when-compact"> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Invariant_mass&action=edit">You can help</a>. The <a href="/wiki/Talk:Invariant_mass" title="Talk:Invariant mass">talk page</a> may contain suggestions.</span> <span class="date-container"><i>(<span class="date">February 2016</span>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The <b>invariant mass</b>, <b>rest mass</b>, <b>intrinsic mass</b>, <b>proper mass</b>, or in the case of bound systems simply <b>mass</b>, is the portion of the total <a href="/wiki/Mass" title="Mass">mass</a> of an <a href="/wiki/Physical_body" class="mw-redirect" title="Physical body">object</a> or <a href="/wiki/Physical_system" title="Physical system">system</a> of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total <a href="/wiki/Energy" title="Energy">energy</a> and <a href="/wiki/Momentum" title="Momentum">momentum</a> that is the same in all <a href="/wiki/Frames_of_reference" class="mw-redirect" title="Frames of reference">frames of reference</a> related by <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>.<sup id="cite_ref-LawrenceS_1-0" class="reference"><a href="#cite_note-LawrenceS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> If a <a href="/wiki/Center-of-momentum_frame" title="Center-of-momentum frame">center-of-momentum frame</a> exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is non-zero, the total mass (a.k.a. <a href="/wiki/Relativistic_mass" class="mw-redirect" title="Relativistic mass">relativistic mass</a>) of the system is greater than the invariant mass, but the invariant mass remains unchanged. </p><p>Because of <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>, the <b>rest energy</b> of the system is simply the invariant mass times the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared. </p><p>Systems whose <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> is a <a href="/wiki/Minkowski_space#Causal_structure" title="Minkowski space">null vector</a> (for example, a single <a href="/wiki/Photon" title="Photon">photon</a> or many photons moving in exactly the same direction) have <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> invariant mass and are referred to as <i><a href="/wiki/Massless_particle" title="Massless particle">massless</a></i>. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized <a href="/wiki/Tachyon" title="Tachyon">tachyon</a>), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass. </p><p>If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by <i><a href="/wiki/Speed_of_light" title="Speed of light">c</a></i><sup>2</sup>. See <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a> for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy. </p><p>For an isolated <i>massive</i> system, the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the system moves in a straight line with a steady subluminal <a href="/wiki/Velocity" title="Velocity">velocity</a> (with a velocity depending on the <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frame</a> used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by <span class="texhtml"><i>c</i><sup>2</sup></span>. This total energy in the center of momentum frame, is the <i>minimum</i> energy which the system may be observed to have, when seen by various observers from various inertial frames. </p><p>Note that for reasons above, such a rest frame does not exist for single <a href="/wiki/Photon" title="Photon">photons</a>, or rays of <a href="/wiki/Light" title="Light">light</a> moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rest_mass_0_and_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rest_mass_0_and_1.svg/160px-Rest_mass_0_and_1.svg.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rest_mass_0_and_1.svg/240px-Rest_mass_0_and_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rest_mass_0_and_1.svg/320px-Rest_mass_0_and_1.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>Possible 4-momenta of particles. <span style="color:blue;"><b>One</b></span> has zero invariant mass, <span style="color:green;"><b>the other</b></span> is massive</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sum_of_rest_masses">Sum of rest masses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=1" title="Edit section: Sum of rest masses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass). </p><p>Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is <b>not equal</b> to the sum of the rest masses of the particles within it. </p><p>The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound). </p><p>They will often also interact through one or more of the <a href="/wiki/Fundamental_forces" class="mw-redirect" title="Fundamental forces">fundamental forces</a>, giving them a potential energy of interaction, possibly <a href="/wiki/Negative_energy" title="Negative energy">negative</a>. </p> <div class="mw-heading mw-heading2"><h2 id="As_defined_in_particle_physics">As defined in particle physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=2" title="Edit section: As defined in particle physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, the <b>invariant mass</b> <span class="texhtml"><i>m</i><sub>0</sub></span> is equal to the <a href="/wiki/Mass" title="Mass">mass</a> in the rest frame of the particle, and can be calculated by the particle's <a href="/wiki/Energy" title="Energy">energy</a> <span class="texhtml mvar" style="font-style:italic;">E</span> and its <a href="/wiki/Momentum" title="Momentum">momentum</a> <span class="texhtml"><b>p</b></span> as measured in <i>any</i> frame, by the <a href="/wiki/Energy%E2%80%93momentum_relation" title="Energy–momentum relation">energy–momentum relation</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c357c2ed1ab601575255ee1b8b92ab40d156649" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.046ex; height:6.509ex;" alt="{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}"></span> or in <a href="/wiki/Natural_units" title="Natural units">natural units</a> where <span class="texhtml"><i>c</i> = 1</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e09b4fdadca4fddcaa9dad192f268aaaeabe7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.393ex; height:3.509ex;" alt="{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}"></span> </p><p>This invariant mass is the same in all <a href="/wiki/Frame_of_reference" title="Frame of reference">frames of reference</a> (see also <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>). This equation says that the invariant mass is the pseudo-Euclidean length of the <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> <span class="texhtml">(<i>E</i>, <b>p</b>)</span>, calculated using the <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">relativistic version of the Pythagorean theorem</a> which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> for an elementary particle. The Dirac <a href="/wiki/Quantum_operator" class="mw-redirect" title="Quantum operator">quantum operator</a> corresponds to the particle four-momentum vector. </p><p>Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>W</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∑</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mi>c</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e53077c1d80601994b11ce2cfeb4eaa6dc045d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.227ex; height:5.176ex;" alt="{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is the invariant mass of the system of particles, equal to the mass of the decay particle.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c579d4674e2f7108ddf86ba0725eea6525694022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.617ex; height:2.843ex;" alt="{\textstyle \sum E}"></span> is the sum of the energies of the particles</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa4e61965afcaecb19751c37eb339edabc58df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.326ex; height:2.843ex;" alt="{\textstyle \sum \mathbf {p} }"></span> is the vector sum of the <a href="/wiki/Momentum" title="Momentum">momentum</a> of the particles (includes both magnitude and direction of the momenta)</li></ul> <p>The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") <span class="texhtml mvar" style="font-style:italic;">W</span> of the reaction is defined as follows (in natural units): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>∑</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>−</mo> <mo>∑</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mo>∑</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>−</mo> <mo>∑</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1d5ab08a3c5df4531b59741fceb0888523cafc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.83ex; height:5.176ex;" alt="{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}"></span> </p><p>If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle. </p><p>In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the <a href="/wiki/Missing_energy" title="Missing energy">missing energy</a>) the <a href="/wiki/Transverse_mass" title="Transverse mass">transverse mass</a> is used. </p> <div class="mw-heading mw-heading2"><h2 id="Example:_two-particle_collision">Example: two-particle collision</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=3" title="Edit section: Example: two-particle collision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a two-particle collision (or a two-particle decay) the square of the invariant mass (in <a href="/wiki/Natural_units" title="Natural units">natural units</a>) is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</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>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35612052396e82e5d3654cfc1505791b56c3f916" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.803ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Massless_particles">Massless particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=4" title="Edit section: Massless particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The invariant mass of a system made of two massless particles whose momenta form an angle <span class="mwe-math-element"><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> has a convenient expression: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</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>−</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">p</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">p</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</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>−</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c0b0553acbe9c6e2db880847c2a04257a0dbf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.783ex; margin-bottom: -0.222ex; width:46.715ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Collider_experiments">Collider experiments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=5" title="Edit section: Collider experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> and <a href="/wiki/Pseudorapidity" title="Pseudorapidity">pseudorapidity</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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span>. Additionally the transverse momentum, <span class="mwe-math-element"><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_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e552d1d4e5b584da41df89552ec958aa5154170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.648ex; height:2.009ex;" alt="{\displaystyle p_{T}}"></span>, is usually measured. In this case if the particles are massless, or highly relativistic (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\gg m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>≫</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\gg m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7faf74fec88bbbdd3d385775ae055ad114bb8938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.43ex; height:2.176ex;" alt="{\displaystyle E\gg m}"></span>) then the invariant mass becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a20a11d2145e9131b81e90a67d484f5b4335da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.985ex; height:3.176ex;" alt="{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Rest_energy">Rest energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=6" title="Edit section: Rest energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Rest energy</b> (also called <b>rest mass energy</b>) is the energy associated with a particle's invariant mass.<sup id="cite_ref-Hyperphysics_Rest_Energy_2-0" class="reference"><a href="#cite_note-Hyperphysics_Rest_Energy-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Douglas_University_3-0" class="reference"><a href="#cite_note-Douglas_University-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The rest energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> of a <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a> is defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=m_{0}c^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=m_{0}c^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd0e37afe3b2ed97f701cadf87f770814caff73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.67ex; height:3.009ex;" alt="{\displaystyle E_{0}=m_{0}c^{2},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light in vacuum</a>.<sup id="cite_ref-Hyperphysics_Rest_Energy_2-1" class="reference"><a href="#cite_note-Hyperphysics_Rest_Energy-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Douglas_University_3-1" class="reference"><a href="#cite_note-Douglas_University-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In general, only differences in <a href="/wiki/Energy" title="Energy">energy</a> have physical significance.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The concept of rest energy follows from the <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">special theory of relativity</a> that leads to Einstein's famous conclusion about equivalence of energy and mass. See <i><a href="/wiki/Special_relativity#Relativistic_dynamics_and_invariance" title="Special relativity">Special relativity § Relativistic dynamics and invariance</a></i>. </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=Invariant_mass&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Mass in special relativity</a></li> <li><a href="/wiki/Invariant_(physics)" title="Invariant (physics)">Invariant (physics)</a></li> <li><a href="/wiki/Transverse_mass" title="Transverse mass">Transverse mass</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=Invariant_mass&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLandau,_L.D.Lifshitz,_E.M.1975" class="citation book cs1">Landau, L.D.; Lifshitz, E.M. (1975). <i>The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2</i>. Butterworth Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-2768-9" title="Special:BookSources/0-7506-2768-9"><bdi>0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classical+Theory+of+Fields%3A+4-th+revised+English+Edition%3A+Course+of+Theoretical+Physics+Vol.+2&rft.pub=Butterworth+Heinemann&rft.date=1975&rft.isbn=0-7506-2768-9&rft.au=Landau%2C+L.D.&rft.au=Lifshitz%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalzen,_FrancisMartin,_Alan1984" class="citation book cs1"><a href="/wiki/Francis_Halzen" title="Francis Halzen">Halzen, Francis</a>; <a href="/wiki/Alan_Martin_(physicist)" title="Alan Martin (physicist)">Martin, Alan</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quarksleptonsint0000halz"><i>Quarks & Leptons: An Introductory Course in Modern Particle Physics</i></a></span>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-88741-2" title="Special:BookSources/0-471-88741-2"><bdi>0-471-88741-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks+%26+Leptons%3A+An+Introductory+Course+in+Modern+Particle+Physics&rft.pub=John+Wiley+%26+Sons&rft.date=1984&rft.isbn=0-471-88741-2&rft.au=Halzen%2C+Francis&rft.au=Martin%2C+Alan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquarksleptonsint0000halz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invariant_mass&action=edit&section=9" title="Edit section: Citations"><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"><ol class="references"> <li id="cite_note-LawrenceS-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-LawrenceS_1-0">^</a></b></span> <span class="reference-text">Lawrence S. Lerner. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA1073">Physics for Scientists and Engineers, Volume 2, page 1073</a>. 1997.</span> </li> <li id="cite_note-Hyperphysics_Rest_Energy-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hyperphysics_Rest_Energy_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hyperphysics_Rest_Energy_2-1"><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="CITEREFNave" class="citation web cs1">Nave, C.R. <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c2">"Relativistic Energy"</a>. <i>HyperPhysics</i>. Georgia State University<span class="reference-accessdate">. Retrieved <span class="nowrap">28 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=HyperPhysics&rft.atitle=Relativistic+Energy&rft.aulast=Nave&rft.aufirst=C.R.&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2FRelativ%2Freleng.html%23c2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></span> </li> <li id="cite_note-Douglas_University-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Douglas_University_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Douglas_University_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://pressbooks.bccampus.ca/introductorygeneralphysics2phys1207opticsfirst/chapter/28-6-relativistic-energy/access-date=2023-09-07">"13.6 Relativistic Energy or E = m c^2"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=13.6+Relativistic+Energy+or+E+%3D+m+c%5E2&rft_id=https%3A%2F%2Fpressbooks.bccampus.ca%2Fintroductorygeneralphysics2phys1207opticsfirst%2Fchapter%2F28-6-relativistic-energy%2Faccess-date%3D2023-09-07&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillip_L._Reu2007" class="citation report cs1">Phillip L. Reu (March 2007). <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20150623204045/http://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2006/066063.pdf">Development of the Doppler Electron Velocimeter—Theory</a> <span class="cs1-format">(PDF)</span> (Report). Sandia National Laboratories. SAND2006-6063. Archived from <a rel="nofollow" class="external text" href="http://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2006/066063.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2015-06-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Development+of+the+Doppler+Electron+Velocimeter%E2%80%94Theory&rft.pub=Sandia+National+Laboratories&rft.date=2007-03&rft.au=Phillip+L.+Reu&rft_id=http%3A%2F%2Fwww.prod.sandia.gov%2Fcgi-bin%2Ftechlib%2Faccess-control.pl%2F2006%2F066063.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFModellReid1974" class="citation book cs1">Modell, Michael; Reid, Robert C. (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/thermodynamicsit00mode"><i>Thermodynamics and Its Applications</i></a></span>. Englewood Cliffs, NJ: <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-914861-2" title="Special:BookSources/0-13-914861-2"><bdi>0-13-914861-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+and+Its+Applications&rft.place=Englewood+Cliffs%2C+NJ&rft.pub=Prentice-Hall&rft.date=1974&rft.isbn=0-13-914861-2&rft.aulast=Modell&rft.aufirst=Michael&rft.au=Reid%2C+Robert+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthermodynamicsit00mode&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInvariant+mass" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Invariant_mass&oldid=1271741819#Rest_energy">https://en.wikipedia.org/w/index.php?title=Invariant_mass&oldid=1271741819#Rest_energy</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Theory of relativity</a></li><li><a href="/wiki/Category:Mass" title="Category:Mass">Mass</a></li><li><a href="/wiki/Category:Energy_(physics)" title="Category:Energy (physics)">Energy (physics)</a></li><li><a href="/wiki/Category:Physical_quantities" title="Category:Physical quantities">Physical quantities</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_March_2011" title="Category:Articles needing additional references from March 2011">Articles needing additional references from March 2011</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_rewrite_from_February_2016" title="Category:Wikipedia articles needing rewrite from February 2016">Wikipedia articles needing rewrite from February 2016</a></li><li><a href="/wiki/Category:All_articles_needing_rewrite" title="Category:All articles needing rewrite">All articles needing rewrite</a></li><li><a href="/wiki/Category:Articles_with_multiple_maintenance_issues" title="Category:Articles with multiple maintenance issues">Articles with multiple maintenance issues</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 January 2025, at 13:27<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Invariant mass - Wikipedia