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class="vector-toc-numb">2</span> <span>Relativistic mass</span> </div> </a> <ul id="toc-Relativistic_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_vs._rest_mass" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_vs._rest_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relativistic vs. rest mass</span> </div> </a> <ul id="toc-Relativistic_vs._rest_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invariant_mass" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Invariant_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Invariant mass</span> </div> </a> <ul id="toc-Invariant_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_energy–momentum_equation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_energy–momentum_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relativistic energy–momentum equation</span> </div> </a> <ul id="toc-Relativistic_energy–momentum_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_mass_of_composite_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_mass_of_composite_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The mass of composite systems</span> </div> </a> <ul id="toc-The_mass_of_composite_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservation_versus_invariance_of_mass_in_special_relativity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conservation_versus_invariance_of_mass_in_special_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Conservation versus invariance of mass in special relativity</span> </div> </a> <button aria-controls="toc-Conservation_versus_invariance_of_mass_in_special_relativity-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 Conservation versus invariance of mass in special relativity subsection</span> </button> <ul id="toc-Conservation_versus_invariance_of_mass_in_special_relativity-sublist" class="vector-toc-list"> <li id="toc-Closed_(meaning_totally_isolated)_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_(meaning_totally_isolated)_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Closed (meaning totally isolated) systems</span> </div> </a> <ul id="toc-Closed_(meaning_totally_isolated)_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_system_invariant_mass_vs._the_individual_rest_masses_of_parts_of_the_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_system_invariant_mass_vs._the_individual_rest_masses_of_parts_of_the_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>The system invariant mass vs. the individual rest masses of parts of the system</span> </div> </a> <ul id="toc-The_system_invariant_mass_vs._the_individual_rest_masses_of_parts_of_the_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History_of_the_relativistic_mass_concept" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_of_the_relativistic_mass_concept"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>History of the relativistic mass concept</span> </div> </a> <button aria-controls="toc-History_of_the_relativistic_mass_concept-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 History of the relativistic mass concept subsection</span> </button> <ul id="toc-History_of_the_relativistic_mass_concept-sublist" class="vector-toc-list"> <li id="toc-Transverse_and_longitudinal_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transverse_and_longitudinal_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Transverse and longitudinal mass</span> </div> </a> <ul id="toc-Transverse_and_longitudinal_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_mass_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_mass_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Relativistic mass</span> </div> </a> <ul id="toc-Relativistic_mass_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Popular_science_and_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Popular_science_and_textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Popular science and textbooks</span> </div> </a> <ul id="toc-Popular_science_and_textbooks-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">9</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">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Massa_en_la_relativitat_especial" title="Massa en la relativitat especial – Catalan" lang="ca" hreflang="ca" data-title="Massa en la relativitat especial" 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/Relativistick%C3%A1_hmotnost" title="Relativistická hmotnost – Czech" lang="cs" hreflang="cs" data-title="Relativistická hmotnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Masa_y_energ%C3%ADa_en_la_relatividad_especial" title="Masa y energía en la relatividad especial – Spanish" lang="es" hreflang="es" data-title="Masa y energía en la relatividad especial" 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/%D8%AC%D8%B1%D9%85_%D8%AF%D8%B1_%D9%86%D8%B3%D8%A8%DB%8C%D8%AA_%D8%AE%D8%A7%D8%B5" title="جرم در نسبیت خاص – Persian" lang="fa" hreflang="fa" data-title="جرم در نسبیت خاص" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Massa_relativistica" title="Massa relativistica – Italian" lang="it" hreflang="it" data-title="Massa relativistica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%BD%D0%B0%D0%B9%D1%8B_%D1%81%D0%B0%D0%BB%D1%8B%D1%81%D1%82%D1%8B%D1%80%D0%BC%D0%B0%D0%BB%D1%8B%D0%BB%D1%8B%D2%9B%D1%82%D0%B0%D2%93%D1%8B_%D0%BC%D0%B0%D1%81%D1%81%D0%B0" title="Арнайы салыстырмалылықтағы масса – Kazakh" lang="kk" hreflang="kk" data-title="Арнайы салыстырмалылықтағы масса" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BB%D0%B0%D1%82%D0%B8%D0%B2%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BC%D0%B0%D1%81%D0%B0" title="Релативистичка маса – Macedonian" lang="mk" hreflang="mk" data-title="Релативистичка маса" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Masa_relatywistyczna" title="Masa relatywistyczna – Polish" lang="pl" hreflang="pl" data-title="Masa relatywistyczna" 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/Massa_na_relatividade_especial" title="Massa na relatividade especial – Portuguese" lang="pt" hreflang="pt" data-title="Massa na relatividade especial" 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/Dependen%C8%9Ba_masei_de_vitez%C4%83" title="Dependența masei de viteză – Romanian" lang="ro" hreflang="ro" data-title="Dependența masei de viteză" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D1%81%D0%B0_%D0%B2_%D1%81%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B8_%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Масса в специальной теории относительности – Russian" lang="ru" hreflang="ru" data-title="Масса в специальной теории относительности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Relativisti%C4%8Dna_masa" title="Relativistična masa – Slovenian" lang="sl" hreflang="sl" data-title="Relativistična masa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Masa_po_teoriji_specijalnog_relativiteta" title="Masa po teoriji specijalnog relativiteta – Serbian" lang="sr" hreflang="sr" data-title="Masa po teoriji specijalnog relativiteta" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Relativisti%C4%8Dka_masa" title="Relativistička masa – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Relativistička masa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96zel_g%C3%B6relilikte_k%C3%BCtle" title="Özel görelilikte kütle – Turkish" lang="tr" hreflang="tr" data-title="Özel görelilikte kütle" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%E1%BB%91i_l%C6%B0%E1%BB%A3ng_trong_thuy%E1%BA%BFt_t%C6%B0%C6%A1ng_%C4%91%E1%BB%91i_h%E1%BA%B9p" title="Khối lượng trong thuyết tương đối hẹp – Vietnamese" lang="vi" hreflang="vi" data-title="Khối lượng trong thuyết tương đối hẹp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9D%9C%E8%B3%AA%E9%87%8F" title="靜質量 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="靜質量" 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0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">August 2023</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> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Meanings of mass in special relativity</div> <p>The word "<a href="/wiki/Mass" title="Mass">mass</a>" has two meanings in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>: <i><a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a></i> (also called rest mass) is an <a href="/wiki/Invariant_quantity" class="mw-redirect" title="Invariant quantity">invariant quantity</a> which is the same for all <a href="/wiki/Observer_(special_relativity)" title="Observer (special relativity)">observers</a> in all <a href="/wiki/Reference_frames" class="mw-redirect" title="Reference frames">reference frames</a>, while the <b>relativistic mass</b> is dependent on the velocity of the observer. According to the concept of <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>, invariant mass is equivalent to <i><a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">rest energy</a></i>, while relativistic mass is equivalent to <i><a href="/wiki/Relativistic_energy" class="mw-redirect" title="Relativistic energy">relativistic energy</a></i> (also called total energy). </p><p>The term "relativistic mass" tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity, in favor of referring to the body's relativistic energy.<sup id="cite_ref-roche_1-0" class="reference"><a href="#cite_note-roche-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In contrast, "invariant mass" is usually preferred over rest energy. The measurable inertia and the warping of spacetime by a body in a given frame of reference is determined by its relativistic mass, not merely its invariant mass. For example, photons have zero rest mass but contribute to the inertia (and weight in a gravitational field) of any system containing them. </p><p>The concept is generalized in <a href="/wiki/Mass_in_general_relativity" title="Mass in general relativity">mass in general relativity</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Rest_mass">Rest mass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=1" title="Edit section: Rest mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i>mass</i> in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The <i><a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a></i> is another name for the <i>rest mass</i> of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their <a href="/wiki/Center_of_momentum" class="mw-redirect" title="Center of momentum">center of momentum</a> frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by <i>c</i><sup>2</sup> (the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> squared). </p><p>The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time). </p><p>It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by <span class="texhtml"><i>c</i><sup>2</sup></span>) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed to all influences. (The technical term is <a href="/wiki/Isolated_system" title="Isolated system">isolated system</a> meaning that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.) </p> <div class="mw-heading mw-heading2"><h2 id="Relativistic_mass">Relativistic mass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=2" title="Edit section: Relativistic mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>relativistic mass</i> is the sum total quantity of energy in a body or system (divided by <span class="texhtml"><i>c</i><span style="padding-left:0.12em;"><sup>2</sup></span></span>). Thus, the mass in the <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">formula</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 E=m_{\text{rel}}c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=m_{\text{rel}}c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda4b4a68aafc6c72a394939997028c2ce717640" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.04ex; height:3.009ex;" alt="{\displaystyle E=m_{\text{rel}}c^{2}}"></span> is the relativistic mass. For a particle of non-zero rest mass <span class="texhtml mvar" style="font-style:italic;">m</span> moving at a speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> relative to the observer, one finds <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_{\text{rel}}={\frac {m}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{rel}}={\frac {m}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ff0ee8f9e06200820485c5655e689f0b871ca7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:18.031ex; height:8.843ex;" alt="{\displaystyle m_{\text{rel}}={\frac {m}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}"></span> </p><p>In the <a href="/wiki/Center_of_momentum" class="mw-redirect" title="Center of momentum">center of momentum</a> frame, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}"></span> and the relativistic mass equals the rest mass. In other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>. However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity. The relativistic mass is also the proportionality factor between velocity and momentum, <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 \mathbf {p} =m_{\text{rel}}\mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m_{\text{rel}}\mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0b9ebd6070cadfda7925c3101d73ec82f5daa4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.746ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m_{\text{rel}}\mathbf {v} .}"></span> </p><p><a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> remains valid in the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c301a874bc8fda093e2f02fbf54281bf996c4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.176ex; height:5.843ex;" alt="{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}}.}"></span> </p><p>When a body emits light of frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> and wavelength <span class="mwe-math-element"><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> as a <a href="/wiki/Photon" title="Photon">photon</a> of 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=h\nu =hc/\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mo>=</mo> <mi>h</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=h\nu =hc/\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913f72cac79789f8f236ec8adaace715e48da291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.407ex; height:2.843ex;" alt="{\displaystyle E=h\nu =hc/\lambda }"></span>, the mass of the body decreases 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 E/c^{2}=h/\lambda c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E/c^{2}=h/\lambda c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be49b2b7b5375ef4b6678f5a2d98486270aba79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.961ex; height:3.176ex;" alt="{\displaystyle E/c^{2}=h/\lambda c}"></span>,<sup id="cite_ref-inertia_2-0" class="reference"><a href="#cite_note-inertia-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which some<sup id="cite_ref-Sandin_3-0" class="reference"><a href="#cite_note-Sandin-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ketterle2020_4-0" class="reference"><a href="#cite_note-Ketterle2020-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> interpret as the relativistic mass of the emitted photon since it also <a href="/wiki/Photon#Physical_properties" title="Photon">fulfills</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 p=m_{\text{rel}}c=h/\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mi>c</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=m_{\text{rel}}c=h/\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd999a971e82782f9529dbbd600ca048b8a5fc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.424ex; height:2.843ex;" alt="{\displaystyle p=m_{\text{rel}}c=h/\lambda }"></span>. Although some authors present relativistic mass as a <i>fundamental</i> concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to space–time. There is disagreement over whether the concept is pedagogically useful.<sup id="cite_ref-okun_5-0" class="reference"><a href="#cite_note-okun-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sandin_3-1" class="reference"><a href="#cite_note-Sandin-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-okun2009_6-0" class="reference"><a href="#cite_note-okun2009-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> It explains simply and quantitatively why a body subject to a constant acceleration cannot reach the speed of light, and why the mass of a system emitting a photon decreases.<sup id="cite_ref-Sandin_3-2" class="reference"><a href="#cite_note-Sandin-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Relativistic_quantum_chemistry#Qualitative_treatment" title="Relativistic quantum chemistry">relativistic quantum chemistry</a>, relativistic mass is used to explain electron orbital contraction in heavy elements.<sup id="cite_ref-Pitzer_1979_7-0" class="reference"><a href="#cite_note-Pitzer_1979-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Norrby1991_8-0" class="reference"><a href="#cite_note-Norrby1991-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.<sup id="cite_ref-Vøyenli_9-0" class="reference"><a href="#cite_note-Vøyenli-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Relativistic mass is not referenced in nuclear and particle physics,<sup id="cite_ref-roche_1-1" class="reference"><a href="#cite_note-roche-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and a survey of introductory textbooks in 2005 showed that only 5 of 24 texts used the concept,<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> although it is still prevalent in popularizations. </p><p>If a stationary box contains many particles, its weight increases in its rest frame the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the box itself is moving (its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> is moving), there remains the question of whether the kinetic energy of the overall motion should be included in the mass of the system. The invariant mass is calculated excluding the kinetic energy of the system as a whole (calculated using the single velocity of the box, which is to say the velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass <i>plus</i> the kinetic energy of the system which is calculated from the velocity of the center of mass. </p> <div class="mw-heading mw-heading2"><h2 id="Relativistic_vs._rest_mass">Relativistic vs. rest mass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=3" title="Edit section: Relativistic vs. rest mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its <a href="/wiki/Center_of_momentum" class="mw-redirect" title="Center of momentum">center of momentum</a> frame). For example, if an electron in a <a href="/wiki/Cyclotron" title="Cyclotron">cyclotron</a> is moving in circles with a relativistic velocity, the mass of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed". If the electron is <i>stopped</i> and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different. </p><p>The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass also represents the measured mass when the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> is at rest for systems of many particles. This special frame where this occurs is also called the <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a>, and is defined as the <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a> in which the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the object is at rest (another way of stating this is that it is the frame in which the momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), only the center of mass of the system is required to be at rest, for the object's relativistic mass to be equal to its rest mass. </p><p>A so-called <a href="/wiki/Massless_particle" title="Massless particle">massless particle</a> (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless". However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference. </p> <div class="mw-heading mw-heading2"><h2 id="Invariant_mass">Invariant mass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=4" title="Edit section: Invariant mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a> is the ratio of <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> (the four-dimensional generalization of <a href="/wiki/Classical_three-dimensional_momentum" class="mw-redirect" title="Classical three-dimensional momentum">classical momentum</a>) to <a href="/wiki/Four-velocity" title="Four-velocity">four-velocity</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> <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 p^{\mu }=mv^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{\mu }=mv^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d76ba2f8a4875e09d9a68c38a135cc25e266e98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.972ex; height:2.676ex;" alt="{\displaystyle p^{\mu }=mv^{\mu }}"></span> and is also the ratio of <a href="/wiki/Four-acceleration" title="Four-acceleration">four-acceleration</a> to <a href="/wiki/Four-force" title="Four-force">four-force</a> when the rest mass is constant. The four-dimensional form of Newton's second law 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 F^{\mu }=mA^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\mu }=mA^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62417323412fe809f3682ec3ff41ecf79b9568ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.79ex; height:2.343ex;" alt="{\displaystyle F^{\mu }=mA^{\mu }.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Relativistic_energy–momentum_equation"><span id="Relativistic_energy.E2.80.93momentum_equation"></span>Relativistic energy–momentum equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=5" title="Edit section: Relativistic energy–momentum equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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/Mass_in_special_relativity" title="Special:EditPage/Mass in special relativity">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=%22Mass+in+special+relativity%22">"Mass in special relativity"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Mass+in+special+relativity%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Mass+in+special+relativity%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=%22Mass+in+special+relativity%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Mass+in+special+relativity%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Mass+in+special+relativity%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2016</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> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Invariant_and_additive_masses.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Invariant_and_additive_masses.svg/427px-Invariant_and_additive_masses.svg.png" decoding="async" width="427" height="427" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Invariant_and_additive_masses.svg/641px-Invariant_and_additive_masses.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Invariant_and_additive_masses.svg/854px-Invariant_and_additive_masses.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>Dependency between the rest mass and <i>E</i>, given in 4-momentum <span class="texhtml">(<i>p</i><sub>0</sub>, <i>p</i><sub>1</sub>)</span> coordinates, where <span class="texhtml"><i>p</i><sub>0</sub><i>c</i> = <i>E</i></span></figcaption></figure> <p>The relativistic expressions for <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">p</span> obey the relativistic <a href="/wiki/Energy%E2%80%93momentum_relation" title="Energy–momentum relation">energy–momentum relation</a>:<sup id="cite_ref-taylor_12-0" class="reference"><a href="#cite_note-taylor-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}-(pc)^{2}=\left(mc^{2}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}-(pc)^{2}=\left(mc^{2}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d8c73e4986e0801a17bf63d62361605c412f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.112ex; height:3.843ex;" alt="{\displaystyle E^{2}-(pc)^{2}=\left(mc^{2}\right)^{2}}"></span> where the <var>m</var> is the rest mass, or the invariant mass for systems, and <span class="texhtml mvar" style="font-style:italic;">E</span> is the total energy. </p><p>The equation is also valid for photons, which have <span class="texhtml"><var>m</var> = 0</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 E^{2}-(pc)^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}-(pc)^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22dd54c5a914674a2f823fbcd71814e9899a983f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.989ex; height:3.176ex;" alt="{\displaystyle E^{2}-(pc)^{2}=0}"></span> and therefore <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=pc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>p</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=pc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5498982f39427c7573448da10de26a7b3209e47a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.05ex; height:2.509ex;" alt="{\displaystyle E=pc}"></span> </p><p>A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always <span class="texhtml"><i>c</i></span>. </p><p>For an object at rest, the momentum <span class="texhtml mvar" style="font-style:italic;"><var>p</var></span> is zero, therefore <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=mc^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <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=mc^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f928e3ab3000919723fafe31e6212699e510744" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.622ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}.}"></span> Note that the formula is true only for particles or systems with zero momentum. </p><p>The rest mass is only proportional to the total energy in the rest frame of the object. </p><p>When the object is moving, the total energy is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\sqrt {\left(mc^{2}\right)^{2}+(pc)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</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> <mo stretchy="false">(</mo> <mi>p</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\sqrt {\left(mc^{2}\right)^{2}+(pc)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29057b220ec1578ecd29ca8b978bcf82fe2877fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:22.364ex; height:4.843ex;" alt="{\displaystyle E={\sqrt {\left(mc^{2}\right)^{2}+(pc)^{2}}}}"></span> </p><p>To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(c,{\vec {v}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(c,{\vec {v}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eabc9211c00b088e122637f5776c30857976a04c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.025ex; height:2.843ex;" alt="{\displaystyle \left(c,{\vec {v}}\right)}"></span>, is the only four-vector associated with the particle's motion, so that if there is a conserved four-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 \left(E,{\vec {p}}c\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(E,{\vec {p}}c\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/517f1c0b4b042da33e65a6811ae1a22dfed4ace8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.951ex; height:2.843ex;" alt="{\displaystyle \left(E,{\vec {p}}c\right)}"></span>, it must be proportional to this vector. This allows expressing the ratio of energy to momentum 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 pc=E{\frac {v}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>c</mi> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pc=E{\frac {v}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00337f08f5fd4dac43cf612f7dfa8ea7a07ece21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:9.75ex; height:4.676ex;" alt="{\displaystyle pc=E{\frac {v}{c}},}"></span> resulting in a relation between <span class="texhtml mvar" style="font-style:italic;"><var>E</var></span> and <span class="texhtml mvar" style="font-style:italic;"><var>v</var></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 E^{2}=\left(mc^{2}\right)^{2}+E^{2}{\frac {v^{2}}{c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</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> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}=\left(mc^{2}\right)^{2}+E^{2}{\frac {v^{2}}{c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e53ad8259608902c756c43f6c47586465077c8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.585ex; height:6.009ex;" alt="{\displaystyle E^{2}=\left(mc^{2}\right)^{2}+E^{2}{\frac {v^{2}}{c^{2}}},}"></span> </p><p>This results in <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={\frac {mc^{2}}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da837c5cfa4780288f4b9e0d0f28445643d56de7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:15.055ex; height:9.843ex;" alt="{\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {mv}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>v</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {mv}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22db4ed98c57f4725b5fcda491d699da934d8ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; margin-left: -0.089ex; width:15.185ex; height:8.843ex;" alt="{\displaystyle p={\frac {mv}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.}"></span> </p><p>these expressions can be written 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 {\begin{aligned}E_{0}&=mc^{2},\\E&=\gamma mc^{2},\\p&=mv\gamma ,\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> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> <mi>γ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{0}&=mc^{2},\\E&=\gamma mc^{2},\\p&=mv\gamma ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6caaff308e52278011d194d90233c32cf0458dd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:12.63ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}E_{0}&=mc^{2},\\E&=\gamma mc^{2},\\p&=mv\gamma ,\end{aligned}}}"></span> where the factor <span class="mwe-math-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 \gamma ={1}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma ={1}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ce0fd9c597d80408215922045e27c682bad6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.124ex; height:4.676ex;" alt="{\textstyle \gamma ={1}/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}"></span> </p><p>When working in <a href="/wiki/System_of_units" class="mw-redirect" title="System of units">units</a> where <span class="texhtml"><i>c</i> = 1</span>, known as the <a href="/wiki/Natural_unit_system" class="mw-redirect" title="Natural unit system">natural unit system</a>, all the relativistic equations are simplified and the quantities <a href="/wiki/Energy" title="Energy">energy</a>, <a href="/wiki/Momentum" title="Momentum">momentum</a>, and <a href="/wiki/Mass" title="Mass">mass</a> have the same natural dimension:<sup id="cite_ref-QFT_13-0" class="reference"><a href="#cite_note-QFT-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><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}=E^{2}-p^{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> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}=E^{2}-p^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d15b17ad5c20230ed17dc58a70b3ecac9842ac4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.752ex; height:3.009ex;" alt="{\displaystyle m^{2}=E^{2}-p^{2}.}"></span> </p><p>The equation is often written this way because the difference <span class="mwe-math-element"><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^{2}-p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}-p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81818e8f16c2aea700f6337f65073b54d8943101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.912ex; height:3.009ex;" alt="{\displaystyle E^{2}-p^{2}}"></span> is the relativistic length of the energy <a href="/wiki/4-momentum" class="mw-redirect" title="4-momentum">momentum four-vector</a>, a length which is associated with rest mass or invariant mass in systems. Where <span class="texhtml"><i>m</i> > 0</span> and <span class="texhtml"><i>p</i> = 0</span>, this equation again expresses the mass–energy equivalence <span class="texhtml"><i>E</i> = <i>m</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="The_mass_of_composite_systems">The mass of composite systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=6" title="Edit section: The mass of composite systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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/Mass_in_special_relativity" title="Special:EditPage/Mass in special relativity">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=%22Mass+in+special+relativity%22">"Mass in special relativity"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Mass+in+special+relativity%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Mass+in+special+relativity%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=%22Mass+in+special+relativity%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Mass+in+special+relativity%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Mass+in+special+relativity%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The rest mass of a composite system is not the sum of the rest masses of the parts, unless all the parts are at rest. The total mass of a composite system includes the kinetic energy and field energy in the system. </p><p>The total energy <span class="texhtml mvar" style="font-style:italic;">E</span> of a composite system can be determined by adding together the sum of the energies of its components. The total 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 {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy <span class="texhtml mvar" style="font-style:italic;">E</span> and the length (magnitude) <span class="texhtml mvar" style="font-style:italic;">p</span> of the total momentum vector <span class="mwe-math-element"><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 {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span>, the invariant mass is given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11afd531a58660e058b779fdbdcc52422fac5651" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.027ex; height:6.509ex;" alt="{\displaystyle m={\frac {\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}}"></span> </p><p>In the system of <a href="/wiki/Natural_units" title="Natural units">natural units</a> where <span class="texhtml"><i>c</i> = 1</span>, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following: <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}=\left(\sum E\right)^{2}-\left\|\sum {\vec {p}}\ \right\|^{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> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> </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^{2}=\left(\sum E\right)^{2}-\left\|\sum {\vec {p}}\ \right\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420e2a0b27c7ea35e5ccc1d27c49968178aac489" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.408ex; height:5.176ex;" alt="{\displaystyle m^{2}=\left(\sum E\right)^{2}-\left\|\sum {\vec {p}}\ \right\|^{2}}"></span> </p><p>Where, again, the particle momenta <span class="mwe-math-element"><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 {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> are first summed as vectors, and then the square of their resulting total magnitude (<a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy. </p><p>For such a system, in the special <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a> where momenta sum to zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where <span class="texhtml"><i>c</i> = 1</span>, is identical to it. This invariant mass for a system remains the same quantity in any inertial frame, although the system total energy and total momentum are functions of the particular inertial frame which is chosen, and will vary in such a way between inertial frames as to keep the invariant mass the same for all observers. Invariant mass thus functions for systems of particles in the same capacity as "rest mass" does for single particles. </p><p>Note that the invariant mass of an <a href="/wiki/Isolated_system" title="Isolated system">isolated system</a> (i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions. The concept of invariant mass is widely used in <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, because the invariant mass of a particle's decay products is equal to its <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a>. This is used to make measurements of the mass of particles like the <a href="/wiki/Z_particle" class="mw-redirect" title="Z particle">Z boson</a> or the <a href="/wiki/Top_quark" title="Top quark">top quark</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Conservation_versus_invariance_of_mass_in_special_relativity">Conservation versus invariance of mass in special relativity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=7" title="Edit section: Conservation versus invariance of mass in special relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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/Mass_in_special_relativity" title="Special:EditPage/Mass in special relativity">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=%22Mass+in+special+relativity%22">"Mass in special relativity"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Mass+in+special+relativity%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Mass+in+special+relativity%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=%22Mass+in+special+relativity%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Mass+in+special+relativity%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Mass+in+special+relativity%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Total energy is an additive conserved quantity (for single observers) in systems and in reactions between particles, but rest mass (in the sense of being a sum of particle rest masses) may not be conserved through an event in which rest masses of particles are converted to other types of energy, such as kinetic energy. Finding the sum of individual particle rest masses would require multiple observers, one for each particle rest inertial frame, and these observers ignore individual particle kinetic energy. Conservation laws require a single observer and a single inertial frame. </p><p>In general, for isolated systems and single observers, relativistic mass is conserved (each observer sees it constant over time), but is not invariant (that is, different observers see different values). Invariant mass, however, is both conserved <i>and</i> invariant (all single observers see the same value, which does not change over time). </p><p>The relativistic mass corresponds to the energy, so <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> automatically means that relativistic mass is conserved for any given observer and inertial frame. However, this quantity, like the total energy of a particle, is not invariant. This means that, even though it is conserved for any observer during a reaction, its absolute <i>value</i> will change with the frame of the observer, and for different observers in different frames. </p><p>By contrast, the rest mass and invariant masses of systems and particles are <em>both</em> conserved <em>and</em> also invariant. For example: A closed container of gas (closed to energy as well) has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the container (including the kinetic energy of the gas) only when it is measured in the <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a>. Just as is the case for single particles, the calculated "rest mass" of such a container of gas does not change when it is in motion, although its "relativistic mass" does change. </p><p>The container may even be subjected to a force which gives it an overall velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an overall velocity (that is, technically, a frame in which its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> has a velocity). In this case, its total relativistic mass and energy increase. However, in such a situation, although the container's total relativistic energy and total momentum increase, these energy and momentum increases subtract out in the <i>invariant mass</i> definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale. </p> <div class="mw-heading mw-heading3"><h3 id="Closed_(meaning_totally_isolated)_systems"><span id="Closed_.28meaning_totally_isolated.29_systems"></span>Closed (meaning totally isolated) systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=8" title="Edit section: Closed (meaning totally isolated) systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All conservation laws in special relativity (for energy, mass, and momentum) require isolated systems, meaning systems that are totally isolated, with no mass–energy allowed in or out, over time. If a system is isolated, then both total energy and total momentum in the system are conserved over time for any observer in any single inertial frame, though their <i>absolute values</i> will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does <i>not</i> change with different observers. This is also the familiar situation with single particles: all observers calculate <i>the same</i> particle rest mass (a special case of the invariant mass) no matter how they move (what inertial frame they choose), but different observers see different total energies and momenta for the same particle. </p><p>Conservation of invariant mass also requires the system to be enclosed so that no heat and radiation (and thus invariant mass) can escape. As in the example above, a physically enclosed or bound system does not need to be completely isolated from external forces for its mass to remain constant, because for bound systems these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the bound system, these two changes cancel, so that there is no change in the system's invariant mass. This is just the same result as with single particles: their calculated rest mass also remains constant no matter how fast they move, or how fast an observer sees them move. </p><p>On the other hand, for systems which are unbound, the "closure" of the system may be enforced by an idealized surface, inasmuch as no mass–energy can be allowed into or out of the test-volume over time, if conservation of system invariant mass is to hold during that time. If a force is allowed to act on (do work on) only one part of such an unbound system, this is equivalent to allowing energy into or out of the system, and the condition of "closure" to mass–energy (total isolation) is violated. In this case, conservation of invariant mass of the system also will no longer hold. Such a loss of rest mass in systems when energy is removed, according to <span class="texhtml"><i>E</i> = <i>mc</i><span style="padding-left:0.12em;"><sup>2</sup></span></span> where <span class="texhtml mvar" style="font-style:italic;">E</span> is the energy removed, and <span class="texhtml mvar" style="font-style:italic;">m</span> is the change in rest mass, reflect changes of mass associated with movement of energy, not "conversion" of mass to energy. </p> <div class="mw-heading mw-heading3"><h3 id="The_system_invariant_mass_vs._the_individual_rest_masses_of_parts_of_the_system">The system invariant mass vs. the individual rest masses of parts of the system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=9" title="Edit section: The system invariant mass vs. the individual rest masses of parts of the system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Again, in special relativity, the rest mass of a system is not required to be equal to the sum of the rest masses of the parts (a situation which would be analogous to gross mass-conservation in chemistry). For example, a massive particle can decay into photons which individually have no mass, but which (as a system) preserve the invariant mass of the particle which produced them. Also a box of moving non-interacting particles (e.g., photons, or an ideal gas) will have a larger invariant mass than the sum of the rest masses of the particles which compose it. This is because the total energy of all particles and fields in a system must be summed, and this quantity, as seen in the <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a>, and divided by <span class="texhtml"><i>c</i><span style="padding-left:0.12em;"><sup>2</sup></span></span>, is the system's invariant mass. </p><p>In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass. Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems. Thus, a system's invariant mass may change <i>only</i> because invariant mass is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is <i>not</i> allowed to escape (the system is closed and isolated), the energy will continue to contribute to the system rest mass, and the system mass will not change. Only if the energy is released to the environment will the mass be lost; this is because the associated mass has been allowed out of the system, where it contributes to the mass of the surroundings.<sup id="cite_ref-taylor_12-1" class="reference"><a href="#cite_note-taylor-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History_of_the_relativistic_mass_concept">History of the relativistic mass concept</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=10" title="Edit section: History of the relativistic mass concept"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Transverse_and_longitudinal_mass">Transverse and longitudinal mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=11" title="Edit section: Transverse and longitudinal mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Electromagnetic_mass" title="Electromagnetic mass">Electromagnetic mass</a></div> <p>Concepts that were similar to what nowadays is called "relativistic mass", were already developed before the advent of special relativity. For example, it was recognized by <a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a> in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out in more detail by <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> (1889) and <a href="/wiki/George_Frederick_Charles_Searle" title="George Frederick Charles Searle">George Frederick Charles Searle</a> (1897). So the electrostatic energy behaves as having some sort of electromagnetic mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m_{\text{em}}={\frac {4}{3}}E_{\text{em}}/c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>em</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>em</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m_{\text{em}}={\frac {4}{3}}E_{\text{em}}/c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfbb02fdb83c1c5604876f0e0936516eac1aa32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.398ex; height:3.676ex;" alt="{\textstyle m_{\text{em}}={\frac {4}{3}}E_{\text{em}}/c^{2}}"></span>, which can increase the normal mechanical mass of the bodies.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><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> </p><p>Then, it was pointed out by Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a> (1899, 1904) in the framework of <a href="/wiki/Lorentz_ether_theory" title="Lorentz ether theory">Lorentz ether theory</a>. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{L}}=\gamma ^{3}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{L}}=\gamma ^{3}m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b77a1de52cd8c29ca65d0d32569194a552a35c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.773ex; height:3.176ex;" alt="{\displaystyle m_{\text{L}}=\gamma ^{3}m}"></span> parallel to the direction of motion and the mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{T}}=\gamma m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{T}}=\gamma m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3cf909ca6cb7b245854251a280cf2a066392bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.176ex;" alt="{\displaystyle m_{\text{T}}=\gamma m}"></span> perpendicular to the direction of motion (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="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a28ea842c016e78a603843b6bbbb136e0418e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.418ex; height:3.343ex;" alt="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"></span> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>, <span class="texhtml"><i>v</i></span> is the relative velocity between the ether and the object, and <span class="texhtml"><i>c</i></span> is the speed of light). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass". <a href="/wiki/Max_Abraham" title="Max Abraham">Max Abraham</a> (1902) called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e350489c615dc73c70529c54f10c974126c060da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.3ex; height:2.009ex;" alt="{\displaystyle m_{\text{L}}}"></span> <i>longitudinal mass</i> 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_{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/617ae5b4b81a0e8db687ff1511244b0317ee9e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.459ex; height:2.009ex;" alt="{\displaystyle m_{\text{T}}}"></span> <i><a href="/wiki/Transverse_mass" title="Transverse mass">transverse mass</a></i> (although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><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/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> also initially used the concepts of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of Lorentz, but with a different <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/617ae5b4b81a0e8db687ff1511244b0317ee9e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.459ex; height:2.009ex;" alt="{\displaystyle m_{\text{T}}}"></span> by an unfortunate force definition, which was later corrected), and in another paper in 1906.<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><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> However, he later abandoned velocity dependent mass concepts (see quote at the end of <a href="#Relativistic_mass">next section</a>). </p><p>The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> moving in the <i>x</i> direction with velocity <i>v</i> and associated Lorentz factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> 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}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mtd> <mtd> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mtd> <mtd> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mtd> <mtd> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbbce44d5229059942eb858abfcdde4fa1a6bf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.329ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relativistic_mass_2">Relativistic mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=12" title="Edit section: Relativistic mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In special relativity, an object that has nonzero rest mass cannot travel at the speed of light. As the object approaches the speed of light, the object's energy and momentum increase without bound. </p><p>In the first years after 1905, following Lorentz and Einstein, the terms longitudinal and transverse mass were still in use. However, those expressions were replaced by the concept of <i>relativistic mass</i>, an expression which was first defined by <a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Gilbert N. Lewis</a> and <a href="/wiki/Richard_C._Tolman" title="Richard C. Tolman">Richard C. Tolman</a> in 1909.<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> They defined the total energy and mass of a body 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 m_{\text{rel}}={\frac {E}{c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{rel}}={\frac {E}{c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b127fb4c06049075895ba3dc4ee8c1e37faeabd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.747ex; height:5.509ex;" alt="{\displaystyle m_{\text{rel}}={\frac {E}{c^{2}}},}"></span> and of a body at rest <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}={\frac {E_{0}}{c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</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> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}={\frac {E_{0}}{c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1812dee1e1bad192e92ca4c0b1f2ba8fd5190517" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.446ex; height:5.676ex;" alt="{\displaystyle m_{0}={\frac {E_{0}}{c^{2}}},}"></span> with the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m_{\text{rel}}}{m_{0}}}=\gamma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{\text{rel}}}{m_{0}}}=\gamma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f66cb1b769a4767a7058918d6ca7cec6c057a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.948ex; height:5.009ex;" alt="{\displaystyle {\frac {m_{\text{rel}}}{m_{0}}}=\gamma .}"></span> </p><p>Tolman in 1912 further elaborated on this concept, and stated: "the expression <i>m</i><sub>0</sub>(1 − <i>v</i><span style="padding-left:0.12em;"><sup>2</sup></span>/<i>c</i><span style="padding-left:0.12em;"><sup>2</sup></span>)<sup>−1/2</sup> is best suited for the mass of a moving body."<sup id="cite_ref-RT_22-0" class="reference"><a href="#cite_note-RT-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In 1934, Tolman argued that the relativistic mass formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{rel}}=E/c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{rel}}=E/c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed7a1c1b633902dbdcd4c2a6dd7184e78c84793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.202ex; height:3.176ex;" alt="{\displaystyle m_{\text{rel}}=E/c^{2}}"></span> holds for all particles, including those moving at the speed of light, while the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{rel}}=\gamma m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{rel}}=\gamma m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258ce97d8282c110dc0d6c3ec623c7970a74eada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.56ex; height:2.176ex;" alt="{\displaystyle m_{\text{rel}}=\gamma m_{0}}"></span> only applies to a slower-than-light particle (a particle with a nonzero rest mass). Tolman remarked on this relation that "We have, moreover, of course the experimental verification of the expression in the case of moving electrons ... We shall hence have no hesitation in accepting the expression as correct in general for the mass of a moving particle."<sup id="cite_ref-RT34_25-0" class="reference"><a href="#cite_note-RT34-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>When the relative velocity is zero, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light <i>c</i>, the denominator of the right side approaches zero, and consequently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> approaches infinity. While <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> remains valid in the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e051b4ad701d1db0bdcce1f77258c8c99cbb1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.176ex; height:5.843ex;" alt="{\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}},}"></span> the derived 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 \mathbf {f} =m_{\text{rel}}\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =m_{\text{rel}}\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a846bc94459a73b680e2129a1920e053635eb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.556ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} =m_{\text{rel}}\mathbf {a} }"></span> is not valid because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{rel}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{rel}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6b2740000470bebfa88b87791ea1f9e88f89b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.009ex;" alt="{\displaystyle m_{\text{rel}}}"></span> 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 {d(m_{\text{rel}}\mathbf {v} )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d(m_{\text{rel}}\mathbf {v} )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31395c64ed9174654137532e9f31c21b55fd9574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.541ex; height:2.843ex;" alt="{\displaystyle {d(m_{\text{rel}}\mathbf {v} )}}"></span> is generally not a constant<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> (see the section above on transverse and longitudinal mass). </p><p>Even though Einstein initially used the expressions "longitudinal" and "transverse" mass in two papers (see <a href="#Transverse_and_longitudinal_mass">previous section</a>), in his first paper on <span class="mwe-math-element"><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=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f73dbd37a0cac34406ee89057fa1b36a1e6a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}"></span> (1905) he treated <span class="texhtml mvar" style="font-style:italic;">m</span> as what would now be called the <i>rest mass</i>.<sup id="cite_ref-inertia_2-1" class="reference"><a href="#cite_note-inertia-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Einstein never derived an equation for "relativistic mass", and in later years he expressed his dislike of the idea:<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>It is not good to introduce the concept of the mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M=m/{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M=m/{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8021134b2d3bf3f6cb1357018b624ad20b0971d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.476ex; height:3.343ex;" alt="{\textstyle M=m/{\sqrt {1-v^{2}/c^{2}}}}"></span> of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ <i>m</i>. Instead of introducing <i>M</i> it is better to mention the expression for the momentum and energy of a body in motion. </p><div class="templatequotecite">— <cite>Albert Einstein in letter to <a href="/wiki/Lincoln_Barnett" title="Lincoln Barnett">Lincoln Barnett</a>, 19 June 1948 (quote from <a href="/wiki/Lev_Okun" title="Lev Okun">L.B. Okun</a> (1989), p. 42<sup id="cite_ref-okun_5-1" class="reference"><a href="#cite_note-okun-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>)</cite></div></blockquote> <div class="mw-heading mw-heading3"><h3 id="Popular_science_and_textbooks">Popular science and textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=13" title="Edit section: Popular science and textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of relativistic mass is widely used in popular science writing and in high school and undergraduate textbooks. Authors such as Okun and A. B. Arons have argued against this as archaic and confusing, and not in accord with modern relativistic theory.<sup id="cite_ref-okun_5-2" class="reference"><a href="#cite_note-okun-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Arons_28-0" class="reference"><a href="#cite_note-Arons-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Arons wrote:<sup id="cite_ref-Arons_28-1" class="reference"><a href="#cite_note-Arons-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p>For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept. [See, for example, Okun (1989).<sup id="cite_ref-okun_5-3" class="reference"><a href="#cite_note-okun-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>]... The sound and rigorous approach to relativistic dynamics is through direct development of that expression for <i>momentum</i> that ensures conservation of momentum in all frames: <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 p={m_{0}v \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={m_{0}v \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dc89216307839fe2e053dc5da4c4de29922439" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-left: -0.089ex; width:13.985ex; height:7.509ex;" alt="{\displaystyle p={m_{0}v \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"></span> rather than through relativistic mass.</p></blockquote> <p>C. Alder takes a similarly dismissive stance on mass in relativity. Writing on said subject matter, he says that "its introduction into the theory of special relativity was much in the way of a historical accident", noting towards the widespread knowledge of <span class="texhtml"><i>E</i> = <i>mc</i><sup>2</sup></span> and how the public's interpretation of the equation has largely informed how it is taught in higher education.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> He instead supposes that the difference between rest and relativistic mass should be explicitly taught, so that students know why mass should be thought of as invariant "in most discussions of inertia". </p><p>Many contemporary authors such as Taylor and Wheeler avoid using the concept of relativistic mass altogether: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass – belonging to the magnitude of a 4-vector – to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself.<sup id="cite_ref-taylor_12-2" class="reference"><a href="#cite_note-taylor-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>While spacetime has the unbounded geometry of Minkowski space, the velocity-space is bounded by <span class="texhtml"><i>c</i></span> and has the geometry of <a href="/wiki/Beltrami%E2%80%93Klein_model" title="Beltrami–Klein model">hyperbolic geometry</a> where relativistic mass plays an analogous role to that of Newtonian mass in the barycentric coordinates of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> The connection of velocity to hyperbolic geometry enables the 3-velocity-dependent relativistic mass to be related to the 4-velocity Minkowski formalism.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </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=Mass_in_special_relativity&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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energy and momentum</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=Mass_in_special_relativity&action=edit&section=15" 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-roche-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-roche_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-roche_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free 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Vøyenli (1976), "The classical and relativistic concepts of mass", <i><a href="/wiki/Foundations_of_Physics" title="Foundations of Physics">Foundations of Physics</a></i>, <b>6</b> (1): 115–124, <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/1976FoPh....6..115E">1976FoPh....6..115E</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%2FBF00708670">10.1007/BF00708670</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:120139174">120139174</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=The+classical+and+relativistic+concepts+of+mass&rft.volume=6&rft.issue=1&rft.pages=115-124&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120139174%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00708670&rft_id=info%3Abibcode%2F1976FoPh....6..115E&rft.au=E.+Eriksen&rft.au=K.+V%C3%B8yenli&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Oas, "On the Abuse and Use of Relativistic Mass," 2005, <a rel="nofollow" class="external free" href="http://arxiv.org/abs/physics/0504110">http://arxiv.org/abs/physics/0504110</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGlinn2004" class="citation cs2">McGlinn, William D. (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PoDYLk6Ugd8C"><i>Introduction to relativity</i></a>, JHU Press, p. 43, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-7047-7" title="Special:BookSources/978-0-8018-7047-7"><bdi>978-0-8018-7047-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+relativity&rft.pages=43&rft.pub=JHU+Press&rft.date=2004&rft.isbn=978-0-8018-7047-7&rft.aulast=McGlinn&rft.aufirst=William+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPoDYLk6Ugd8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PoDYLk6Ugd8C&pg=PA43">Extract of page 43</a></span> </li> <li id="cite_note-taylor-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-taylor_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-taylor_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-taylor_12-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE._F._TaylorJ._A._Wheeler1992" class="citation cs2">E. 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(1909), <span class="cs1-ws-icon" title="s:The Principle of Relativity, and Non-Newtonian Mechanics"><a class="external text" href="https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics">"The Principle of Relativity, and Non-Newtonian Mechanics" </a></span>, <i>Proceedings of the American Academy of Arts and Sciences</i>, <b>44</b> (25): 709–726, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F20022495">10.2307/20022495</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20022495">20022495</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=The+Principle+of+Relativity%2C+and+Non-Newtonian+Mechanics&rft.volume=44&rft.issue=25&rft.pages=709-726&rft.date=1909&rft_id=info%3Adoi%2F10.2307%2F20022495&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20022495%23id-name%3DJSTOR&rft.au=Lewis%2C+Gilbert+N.&rft.au=Tolman%2C+Richard+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-RT-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-RT_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._Tolman1911" class="citation cs2">R. Tolman (1911), <span class="cs1-ws-icon" title="s:Derivation of Fifth Fundamental Equation"><a class="external text" href="https://en.wikisource.org/wiki/Derivation_of_Fifth_Fundamental_Equation">"Note on the Derivation from the Principle of Relativity of the Fifth Fundamental Equation of the Maxwell–Lorentz Theory" </a></span>, <i>Philosophical Magazine</i>, <b>21</b> (123): 296–301, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440308637034">10.1080/14786440308637034</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=Note+on+the+Derivation+from+the+Principle+of+Relativity+of+the+Fifth+Fundamental+Equation+of+the+Maxwell%E2%80%93Lorentz+Theory&rft.volume=21&rft.issue=123&rft.pages=296-301&rft.date=1911&rft_id=info%3Adoi%2F10.1080%2F14786440308637034&rft.au=R.+Tolman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._Tolman1911" class="citation cs2">R. Tolman (1911), <span class="cs1-ws-icon" title="s:The Direction of Force and Acceleration"><a class="external text" href="https://en.wikisource.org/wiki/The_Direction_of_Force_and_Acceleration">"Non-Newtonian Mechanics :— The Direction of Force and Acceleration." </a></span>, <i>Philosophical Magazine</i>, <b>22</b> (129): 458–463, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440908637142">10.1080/14786440908637142</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=Non-Newtonian+Mechanics+%3A%E2%80%94+The+Direction+of+Force+and+Acceleration.&rft.volume=22&rft.issue=129&rft.pages=458-463&rft.date=1911&rft_id=info%3Adoi%2F10.1080%2F14786440908637142&rft.au=R.+Tolman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></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="CITEREFR._Tolman1912" class="citation cs2">R. Tolman (1912), <span class="cs1-ws-icon" title="s:The Mass of a Moving Body"><a class="external text" href="https://en.wikisource.org/wiki/The_Mass_of_a_Moving_Body">"Non-Newtonian Mechanics. 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Tolman (1934), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1ZOgD9qlWtsC&pg=PR3"><i>Relativity, Thermodynamics, and Cosmology</i></a>, Oxford: <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65383-9" title="Special:BookSources/978-0-486-65383-9"><bdi>978-0-486-65383-9</bdi></a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/34032023">34032023</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity%2C+Thermodynamics%2C+and+Cosmology&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1934&rft_id=info%3Alccn%2F34032023&rft.isbn=978-0-486-65383-9&rft.au=R.C.+Tolman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1ZOgD9qlWtsC%26pg%3DPR3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span> Reissued (1987), New York: <a href="/wiki/Dover" title="Dover">Dover</a>, <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/0-486-65383-8" title="Special:BookSources/0-486-65383-8">0-486-65383-8</a>.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhilip_GibbsJim_Carr" class="citation web cs1">Philip Gibbs; Jim Carr. <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html">"What is relativistic mass?"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2011-09-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+is+relativistic+mass%3F&rft.au=Philip+Gibbs&rft.au=Jim+Carr&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FRelativity%2FSR%2Fmass.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEugene_Hecht2009" class="citation journal cs1">Eugene Hecht (19 August 2009). "Einstein Never Approved of Relativistic Mass". <i>The Physics Teacher</i>. <b>47</b> (6): 336–341. <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/2009PhTea..47..336H">2009PhTea..47..336H</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.5072">10.1.1.205.5072</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3204111">10.1119/1.3204111</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+Teacher&rft.atitle=Einstein+Never+Approved+of+Relativistic+Mass&rft.volume=47&rft.issue=6&rft.pages=336-341&rft.date=2009-08-19&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.205.5072%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1119%2F1.3204111&rft_id=info%3Abibcode%2F2009PhTea..47..336H&rft.au=Eugene+Hecht&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-Arons-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-Arons_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Arons_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA.B._Arons1990" class="citation cs2">A.B. Arons (1990), <i>A Guide to Introductory Physics Teaching</i>, p. 263</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+Introductory+Physics+Teaching&rft.pages=263&rft.date=1990&rft.au=A.B.+Arons&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span> Also in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><i>Teaching Introductory Physics</i>, 2001, p. 308</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teaching+Introductory+Physics&rft.pages=308&rft.date=2001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdler1986" class="citation journal cs1">Adler, Carl (September 30, 1986). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210506164309/http://sites.fas.harvard.edu/~phys191r/References/b5/Adler1987.pdf">"Does mass really depend on velocity, dad?"</a> <span class="cs1-format">(PDF)</span>. <i>American Journal of Physics</i>. <b>55</b> (8): 739–743. <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/1987AmJPh..55..739A">1987AmJPh..55..739A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.15314">10.1119/1.15314</a>. Archived from <a rel="nofollow" class="external text" href="https://sites.fas.harvard.edu/~phys191r/References/b5/Adler1987.pdf">the original</a> <span class="cs1-format">(PDF)</span> on May 6, 2021<span class="reference-accessdate">. Retrieved <span class="nowrap">December 12,</span> 2017</span> – via HUIT Sites Hosting.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Does+mass+really+depend+on+velocity%2C+dad%3F&rft.volume=55&rft.issue=8&rft.pages=739-743&rft.date=1986-09-30&rft_id=info%3Adoi%2F10.1119%2F1.15314&rft_id=info%3Abibcode%2F1987AmJPh..55..739A&rft.aulast=Adler&rft.aufirst=Carl&rft_id=https%3A%2F%2Fsites.fas.harvard.edu%2F~phys191r%2FReferences%2Fb5%2FAdler1987.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUngar2010" class="citation book cs1">Ungar, Abraham A. (2010). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/663096629"><i>Hyperbolic Triangle Centers: The Special Relativistic Approach</i></a>. Dordrecht: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-8636-5" title="Special:BookSources/978-90-481-8636-5"><bdi>978-90-481-8636-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/663096629">663096629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperbolic+Triangle+Centers%3A+The+Special+Relativistic+Approach&rft.place=Dordrecht&rft.pub=Springer&rft.date=2010&rft_id=info%3Aoclcnum%2F663096629&rft.isbn=978-90-481-8636-5&rft.aulast=Ungar&rft.aufirst=Abraham+A.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F663096629&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/communications-in-mathematical-analysis/volume-10/issue-1/When-Relativistic-Mass-Meets-Hyperbolic-Geometry/cma/1305810734.full">When Relativistic Mass Meets Hyperbolic Geometry</a>, Abraham A. Ungar, Commun. Math. Anal. Volume 10, Number 1 (2011), 30–56.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mass_in_special_relativity&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilagadze2008" class="citation journal cs2">Silagadze, Z. K. (2008), "Relativity without tears", <i><a href="/wiki/Acta_Physica_Polonica_B" class="mw-redirect" title="Acta Physica Polonica B">Acta Physica Polonica B</a></i>, <b>39</b> (4): 811–885, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0708.0929">0708.0929</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008AcPPB..39..811S">2008AcPPB..39..811S</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Physica+Polonica+B&rft.atitle=Relativity+without+tears&rft.volume=39&rft.issue=4&rft.pages=811-885&rft.date=2008&rft_id=info%3Aarxiv%2F0708.0929&rft_id=info%3Abibcode%2F2008AcPPB..39..811S&rft.aulast=Silagadze&rft.aufirst=Z.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOas2005" class="citation arxiv cs2">Oas, Gary (2005), "On the Abuse and Use of Relativistic Mass", <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0504110">physics/0504110</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+the+Abuse+and+Use+of+Relativistic+Mass&rft.date=2005&rft_id=info%3Aarxiv%2Fphysics%2F0504110&rft.aulast=Oas&rft.aufirst=Gary&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/">Usenet Physics FAQ</a> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html">"Does mass change with velocity?"</a> by Philip Gibbs et al., 2002, retrieved August 10, 2006</li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html">"What is the mass of a photon?"</a> by Matt Austern et al., 1998, retrieved June 27, 2007</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMax_Jammer1997" class="citation book cs2"><a href="/wiki/Max_Jammer" title="Max Jammer">Max Jammer</a> (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lYvz0_8aGsMC&q=%22inertia+of+a+body+depend+upon%22&pg=PA177"><i>Concepts of Mass in Classical and Modern Physics</i></a>, Courier Dover Publications, pp. 177–178, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-29998-3" title="Special:BookSources/978-0-486-29998-3"><bdi>978-0-486-29998-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+of+Mass+in+Classical+and+Modern+Physics&rft.pages=177-178&rft.pub=Courier+Dover+Publications&rft.date=1997&rft.isbn=978-0-486-29998-3&rft.au=Max+Jammer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlYvz0_8aGsMC%26q%3D%2522inertia%2Bof%2Ba%2Bbody%2Bdepend%2Bupon%2522%26pg%3DPA177&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMass+in+special+relativity" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.relativitycalculator.com/mass_variable.shtml">Mass as a Variable Quantity</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150716015502/http://www.relativitycalculator.com/mass_variable.shtml">Archived</a> 2015-07-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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.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:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a class="mw-selflink selflink">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a href="/wiki/Friedmann_equations" title="Friedmann equations">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mass_in_special_relativity&oldid=1221052050">https://en.wikipedia.org/w/index.php?title=Mass_in_special_relativity&oldid=1221052050</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:Special_relativity" title="Category:Special relativity">Special relativity</a></li><li><a href="/wiki/Category:Mass" title="Category:Mass">Mass</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:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_August_2023" title="Category:Articles lacking in-text citations from August 2023">Articles lacking in-text citations from August 2023</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_February_2016" title="Category:Articles needing additional references from February 2016">Articles needing additional references from February 2016</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:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 April 2024, at 15:25<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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