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href="#Unruh_radiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Unruh radiation</span> </div> </a> <ul id="toc-Unruh_radiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experimental_observation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Experimental_observation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Experimental observation</span> </div> </a> <ul id="toc-Experimental_observation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 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id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Unruh effect</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 18 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-18" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%A3%D8%AB%D9%8A%D8%B1_%D8%A3%D9%88%D9%86%D8%B1%D9%88%D9%87" title="تأثير أونروه – Arabic" lang="ar" hreflang="ar" data-title="تأثير أونروه" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Efecte_Unruh" title="Efecte Unruh – Catalan" lang="ca" hreflang="ca" data-title="Efecte Unruh" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Unruh-Effekt" title="Unruh-Effekt – German" lang="de" hreflang="de" data-title="Unruh-Effekt" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CE%B1%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF_%CE%9F%CF%8D%CE%BD%CF%81%CE%BF%CF%85%CF%87" title="Φαινόμενο Ούνρουχ – Greek" lang="el" hreflang="el" data-title="Φαινόμενο Ούνρουχ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Efecto_Unruh" title="Efecto Unruh – Spanish" lang="es" hreflang="es" data-title="Efecto Unruh" 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%A7%D8%AB%D8%B1_%D8%A2%D9%86%D8%B1%D9%88" title="اثر آنرو – Persian" lang="fa" hreflang="fa" data-title="اثر آنرو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Effet_Unruh" title="Effet Unruh – French" lang="fr" hreflang="fr" data-title="Effet Unruh" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%B8%EB%A3%A8_%ED%9A%A8%EA%B3%BC" title="언루 효과 – Korean" lang="ko" hreflang="ko" data-title="언루 효과" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_Unruh" title="Effetto Unruh – Italian" lang="it" hreflang="it" data-title="Effetto Unruh" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A6%E3%83%B3%E3%83%AB%E3%83%BC%E5%8A%B9%E6%9E%9C" title="ウンルー効果 – Japanese" lang="ja" hreflang="ja" data-title="ウンルー効果" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Unruh-effekt" title="Unruh-effekt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Unruh-effekt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Efekt_Unruha" title="Efekt Unruha – Polish" lang="pl" hreflang="pl" data-title="Efekt Unruha" 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/Efeito_Unruh" title="Efeito Unruh – Portuguese" lang="pt" hreflang="pt" data-title="Efeito Unruh" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D1%84%D1%84%D0%B5%D0%BA%D1%82_%D0%A3%D0%BD%D1%80%D1%83" title="Эффект Унру – Russian" lang="ru" hreflang="ru" data-title="Эффект Унру" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Unruh-effekten" title="Unruh-effekten – Swedish" lang="sv" hreflang="sv" data-title="Unruh-effekten" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A3%D0%BD%D1%80%D1%83_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82%D1%8B" title="Унру эффекты – Tatar" lang="tt" hreflang="tt" data-title="Унру 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item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button 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effect</b>) is a theoretical prediction in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> that an observer who is uniformly accelerating through empty space will perceive a <a href="/wiki/Thermal_reservoir" title="Thermal reservoir">thermal bath</a>. This means that even in the absence of any external heat sources, an accelerating observer will detect particles and experience a temperature. In contrast, an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial observer</a> in the same region of spacetime would observe no temperature.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In other words, the background appears to be warm from an accelerating <a href="/wiki/Reference_frame" class="mw-redirect" title="Reference frame">reference frame</a>. In layman's terms, an accelerating thermometer in empty space (like one being waved around), without any other contribution to its temperature, will record a non-zero temperature, just from its acceleration. Heuristically, for a uniformly accelerating observer, the <a href="/wiki/Ground_state" title="Ground state">ground state</a> of an inertial observer is seen as a mixed state in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> with a non-zero temperature bath. </p><p>The Unruh effect was first described by <a href="/wiki/Stephen_A._Fulling" title="Stephen A. Fulling">Stephen Fulling</a> in 1973, <a href="/wiki/Paul_Davies" title="Paul Davies">Paul Davies</a> in 1975 and <a href="/wiki/W._G._Unruh" title="W. G. Unruh">W. G. Unruh</a> in 1976.<sup id="cite_ref-fdu_2-0" class="reference"><a href="#cite_note-fdu-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-davies_1975_3-0" class="reference"><a href="#cite_note-davies_1975-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-unruh_1976_4-0" class="reference"><a href="#cite_note-unruh_1976-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are disputed. There is also some doubt about whether the Unruh effect implies the existence of <a href="#Unruh_radiation">Unruh radiation</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Temperature_equation"><span class="anchor" id="Unruh_temperature"></span>Temperature equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=1" title="Edit section: Temperature equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Unruh temperature</b>, sometimes called the Davies–Unruh temperature,<sup id="cite_ref-takagi_1986_5-0" class="reference"><a href="#cite_note-takagi_1986-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> was derived separately by Paul Davies<sup id="cite_ref-davies_1975_3-1" class="reference"><a href="#cite_note-davies_1975-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and William Unruh<sup id="cite_ref-unruh_1976_4-1" class="reference"><a href="#cite_note-unruh_1976-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and is the effective temperature experienced by a uniformly accelerating detector in a <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum field</a>. It is given by<sup id="cite_ref-DUMB_6-0" class="reference"><a href="#cite_note-DUMB-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}}\approx 4.06\times 10^{-21}\,\mathrm {K{\cdot }s^{2}{\cdot }m^{-1}} \times a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>c</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>4.06</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>21</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>×</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}}\approx 4.06\times 10^{-21}\,\mathrm {K{\cdot }s^{2}{\cdot }m^{-1}} \times a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2061742defffa2db0fbe9d99d4a324ec68576e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.677ex; height:5.843ex;" alt="{\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}}\approx 4.06\times 10^{-21}\,\mathrm {K{\cdot }s^{2}{\cdot }m^{-1}} \times a,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ħ</span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, <span class="texhtml mvar" style="font-style:italic;">a</span> is the proper uniform acceleration, <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, and <span class="texhtml"><i>k</i><sub>B</sub></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>. Thus, for example, a <a href="/wiki/Proper_acceleration" title="Proper acceleration">proper acceleration</a> of <span class="nowrap"><span data-sort-value="7020247000000000000♠"></span>2.47<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>20</sup> m⋅s<sup>−2</sup></span> corresponds approximately to a temperature of <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 K</span>. Conversely, an acceleration of <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 m⋅s<sup>−2</sup></span> corresponds to a temperature of <span class="nowrap"><span data-sort-value="6979405999999999999♠"></span>4.06<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−21</sup> K</span>. </p><p>The Unruh temperature has the same form as the <a href="/wiki/Hawking_temperature" class="mw-redirect" title="Hawking temperature">Hawking temperature</a> <span class="texhtml"><i>T</i><sub>H</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħg</i></span><span class="sr-only">/</span><span class="den">2π<i>ck</i><sub>B</sub></span></span>⁠</span></span> with <span class="texhtml mvar" style="font-style:italic;">g</span> denoting the surface gravity of a <a href="/wiki/Black_hole" title="Black hole">black hole</a>, which was derived by <a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a> in 1974.<sup id="cite_ref-Hawking_1974_7-0" class="reference"><a href="#cite_note-Hawking_1974-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In the light of the <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a>, it is, therefore, sometimes called the Hawking–Unruh temperature.<sup id="cite_ref-SIMPLE_8-0" class="reference"><a href="#cite_note-SIMPLE-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Solving the Unruh temperature for the uniform acceleration, it can be expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {2\pi ck_{\mathrm {B} }}{\hbar }}T=2\pi a_{\mathrm {P} }{\frac {T}{T_{\mathrm {P} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>c</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {2\pi ck_{\mathrm {B} }}{\hbar }}T=2\pi a_{\mathrm {P} }{\frac {T}{T_{\mathrm {P} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b5c0dd49b39471f5e2c981962f103a25f225ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.628ex; height:5.676ex;" alt="{\displaystyle a={\frac {2\pi ck_{\mathrm {B} }}{\hbar }}T=2\pi a_{\mathrm {P} }{\frac {T}{T_{\mathrm {P} }}}}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {P} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {P} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73dbb53a6e26e47c841e4a9df18a31ae7b596522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.581ex; height:2.009ex;" alt="{\displaystyle a_{\mathrm {P} }}"></span> is <a href="/wiki/Planck_units#Derived_units" title="Planck units">Planck acceleration</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {P} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {P} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11eafc04f483146555c10e620a70bf2f990415ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {P} }}"></span> is <a href="/wiki/Planck_units#History_and_definition" title="Planck units">Planck temperature</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=2" title="Edit section: Explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unruh demonstrated theoretically that the notion of <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> depends on the path of the observer through <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles in thermal equilibrium—a warm gas.<sup id="cite_ref-Bertlmann_9-0" class="reference"><a href="#cite_note-Bertlmann-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The Unruh effect would only appear to an accelerating observer. And although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word <i>vacuum</i> is interpreted in the following specific way. In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the concept of "<a href="/wiki/Vacuum" title="Vacuum">vacuum</a>" is not the same as "empty space": <a href="/wiki/Space" title="Space">Space</a> is filled with the quantized fields that make up the <a href="/wiki/Universe" title="Universe">universe</a>. Vacuum is simply the lowest <i>possible</i> <a href="/wiki/Energy" title="Energy">energy</a> state of these fields. </p><p>The energy states of any quantized field are defined by the <a href="/wiki/Hamiltonian_(quantum_theory)" class="mw-redirect" title="Hamiltonian (quantum theory)">Hamiltonian</a>, based on local conditions, including the time coordinate. According to <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system. Hence, the observers will see different quantum states and thus different vacua. </p><p>In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field <a href="/wiki/Canonical_commutation_relations" class="mw-redirect" title="Canonical commutation relations">canonical commutation relations</a>. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices. </p><p>An accelerating observer will perceive an apparent event horizon forming (see <a href="/wiki/Rindler_spacetime" class="mw-redirect" title="Rindler spacetime">Rindler spacetime</a>). The existence of Unruh radiation could be linked to this apparent <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a>, putting it in the same conceptual framework as <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a>. On the other hand, the theory of the Unruh effect explains that the definition of what constitutes a "particle" depends on the state of motion of the observer. </p><p>The <a href="/wiki/Free_field" title="Free field">free field</a> needs to be decomposed into positive and <a href="/wiki/Negative_frequency" title="Negative frequency">negative frequency</a> components before defining the <a href="/wiki/Creation_operator" class="mw-redirect" title="Creation operator">creation</a> and <a href="/wiki/Annihilation_operator" class="mw-redirect" title="Annihilation operator">annihilation operators</a>. This can only be done in spacetimes with a <a href="/wiki/Timelike" class="mw-redirect" title="Timelike">timelike</a> <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vector</a> field. This decomposition happens to be different in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian</a> and <a href="/wiki/Rindler_coordinates" title="Rindler coordinates">Rindler coordinates</a> (although the two are related by a <a href="/wiki/Bogoliubov_transformation" title="Bogoliubov transformation">Bogoliubov transformation</a>). This explains why the "particle numbers", which are defined in terms of the creation and annihilation operators, are different in both coordinates. </p><p>The Rindler spacetime has a horizon, and locally any non-extremal black hole horizon is Rindler. So the Rindler spacetime gives the local properties of <a href="/wiki/Black_hole" title="Black hole">black holes</a> and <a href="/wiki/Observable_universe#Horizons" title="Observable universe">cosmological horizons</a>. It is possible to rearrange the metric restricted to these regions to obtain the Rindler metric.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The Unruh effect would then be the near-horizon form of <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a>. </p><p>The Unruh effect is also expected to be present in <a href="/wiki/De_Sitter_space" title="De Sitter space">de Sitter space</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> </p><p>It is worth stressing that the Unruh effect only says that, according to uniformly-accelerated observers, the vacuum state is a thermal state specified by its temperature, and one should resist reading too much into the thermal state or bath. Different thermal states or baths at the same temperature need not be equal, for they depend on the Hamiltonian describing the system. In particular, the thermal bath seen by accelerated observers in the vacuum state of a quantum field is not the same as a thermal state of the same field at the same temperature according to inertial observers. Furthermore, uniformly accelerated observers, static with respect to each other, can have different proper accelerations <span class="texhtml mvar" style="font-style:italic;">a</span> (depending on their separation), which is a direct consequence of relativistic red-shift effects. This makes the Unruh temperature spatially inhomogeneous across the uniformly accelerated frame.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Calculations">Calculations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=3" title="Edit section: Calculations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, an observer moving with uniform <a href="/wiki/Proper_acceleration" title="Proper acceleration">proper acceleration</a> <span class="texhtml mvar" style="font-style:italic;">a</span> through <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a> is conveniently described with <a href="/wiki/Rindler_coordinates" title="Rindler coordinates">Rindler coordinates</a>, which are related to the standard (<a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian</a>) <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski coordinates</a> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=\rho \cosh(\sigma )\\t&=\rho \sinh(\sigma ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=\rho \cosh(\sigma )\\t&=\rho \sinh(\sigma ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db39141a41ed8148bd0f40527fe312358f953af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.703ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}x&=\rho \cosh(\sigma )\\t&=\rho \sinh(\sigma ).\end{aligned}}}"></span></dd></dl> <p>The <a href="/wiki/Line_element" title="Line element">line element</a> in Rindler coordinates, i.e. <a href="/wiki/Rindler_space" class="mw-redirect" title="Rindler space">Rindler space</a> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s^{2}=-\rho ^{2}\,\mathrm {d} \sigma ^{2}+\mathrm {d} \rho ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=-\rho ^{2}\,\mathrm {d} \sigma ^{2}+\mathrm {d} \rho ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4bb5d600c635024497f43f826880ec1592abce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.7ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} s^{2}=-\rho ^{2}\,\mathrm {d} \sigma ^{2}+\mathrm {d} \rho ^{2},}"></span></dd></dl> <p>where <span class="texhtml"><i>ρ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span>, and where <span class="texhtml mvar" style="font-style:italic;">σ</span> is related to the observer's proper time <span class="texhtml mvar" style="font-style:italic;">τ</span> by <span class="texhtml"><i>σ</i> = <i>aτ</i></span> (here <span class="texhtml"><i>c</i> = 1</span>). </p><p>An observer moving with fixed <span class="texhtml mvar" style="font-style:italic;">ρ</span> traces out a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> in Minkowski space, therefore this type of motion is called <a href="/wiki/Hyperbolic_motion_(relativity)" title="Hyperbolic motion (relativity)">hyperbolic motion</a>. The coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is related to the Schwarzschild spherical coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae68e7c41a953a56ab0e85f3c5a3d712fe533d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.341ex; height:2.009ex;" alt="{\displaystyle r_{S}}"></span> by the relation<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\int _{r_{S}}^{r}{\frac {dr^{\prime }}{\sqrt {1-r_{S}/r^{\prime }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\int _{r_{S}}^{r}{\frac {dr^{\prime }}{\sqrt {1-r_{S}/r^{\prime }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf2ebbf017fb21d9727d349ea56e05f268ab95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.172ex; height:6.843ex;" alt="{\displaystyle \rho =\int _{r_{S}}^{r}{\frac {dr^{\prime }}{\sqrt {1-r_{S}/r^{\prime }}}}.}"></span></dd></dl> <p>An observer moving along a path of constant <span class="texhtml mvar" style="font-style:italic;">ρ</span> is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function of <span class="texhtml mvar" style="font-style:italic;">σ</span>. These modes are constantly <a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Doppler shifted</a> relative to ordinary Minkowski time as the detector accelerates, and they change in frequency by enormous factors, even after only a short proper time. </p><p>Translation in <span class="texhtml mvar" style="font-style:italic;">σ</span> is a symmetry of Minkowski space: it can be shown that it corresponds to a <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">boost</a> in <i>x</i>, <i>t</i> coordinate around the origin. Any time translation in quantum mechanics is generated by the Hamiltonian operator. For a detector coupled to modes with a definite frequency in <span class="texhtml mvar" style="font-style:italic;">σ</span>, we can treat <span class="texhtml mvar" style="font-style:italic;">σ</span> as "time" and the boost operator is then the corresponding Hamiltonian. In Euclidean field theory, where the minus sign in front of the time in the Rindler metric is changed to a plus sign by multiplying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> to the Rindler time, i.e. a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> or imaginary time, the Rindler metric is turned into a polar-coordinate-like metric. Therefore any rotations must close themselves after 2<span class="texhtml mvar" style="font-style:italic;">π</span> in a Euclidean metric to avoid being singular. So </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2\pi iH}=Id.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>H</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2\pi iH}=Id.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb022ca77a282ef17d008d4c509d30a66bc19013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.239ex; height:2.676ex;" alt="{\displaystyle e^{2\pi iH}=Id.}"></span></dd></dl> <p>A path integral with real time coordinate is dual to a thermal partition function, related by a <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a>. The periodicity <span class="mwe-math-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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> of imaginary time corresponds to a temperature of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =1/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1e21c8b3cc0966416fbb167dc0f257e83fd65e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.392ex; height:2.843ex;" alt="{\displaystyle \beta =1/T}"></span> in <a href="/wiki/Thermal_quantum_field_theory" title="Thermal quantum field theory">thermal quantum field theory</a>. Note that the path integral for this Hamiltonian is closed with period 2<span class="texhtml mvar" style="font-style:italic;">π</span>. This means that the <span class="texhtml mvar" style="font-style:italic;">H</span> modes are thermally occupied with temperature <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2<span class="texhtml mvar" style="font-style:italic;">π</span></span></span>⁠</span>. This is not an actual temperature, because <span class="texhtml mvar" style="font-style:italic;">H</span> is dimensionless. It is conjugate to the timelike polar angle <span class="texhtml mvar" style="font-style:italic;">σ</span>, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency <span class="texhtml mvar" style="font-style:italic;">f</span> in <span class="texhtml mvar" style="font-style:italic;">σ</span> at position <span class="texhtml mvar" style="font-style:italic;">ρ</span> has a frequency which is determined by the square root of the (absolute value of the) metric at <span class="texhtml mvar" style="font-style:italic;">ρ</span>, the <a href="/wiki/Redshift" title="Redshift">redshift</a> factor. This can be seen by transforming the time coordinate of a Rindler observer at fixed <span class="texhtml mvar" style="font-style:italic;">ρ</span> to an inertial, co-moving observer observing a <a href="/wiki/Proper_time" title="Proper time">proper time</a>. From the Rindler-line-element given above, this is just <span class="texhtml mvar" style="font-style:italic;">ρ</span>. The actual inverse temperature at this point is therefore </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =2\pi \rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =2\pi \rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f3f524c174d2f2fb8ecfdf0457f840d5c8297e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.774ex; height:2.676ex;" alt="{\displaystyle \beta =2\pi \rho .}"></span></dd></dl> <p>It can be shown that the acceleration of a trajectory at constant <span class="texhtml mvar" style="font-style:italic;">ρ</span> in Rindler coordinates is equal to <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>ρ</i></span></span>⁠</span></span>, so the actual inverse temperature observed is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\frac {2\pi }{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {2\pi }{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc7c015fc0f92a4ba60097bc4b029cef23fa528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.408ex; height:5.176ex;" alt="{\displaystyle \beta ={\frac {2\pi }{a}}.}"></span></dd></dl> <p>Restoring units yields </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{B}}T={\frac {\hbar a}{2\pi c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}T={\frac {\hbar a}{2\pi c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e5ba90b7737945fe63d49d0339f71ea1974e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.326ex; height:5.343ex;" alt="{\displaystyle k_{\text{B}}T={\frac {\hbar a}{2\pi c}}.}"></span></dd></dl> <p>The <a href="/wiki/Temperature" title="Temperature">temperature</a> of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration of <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Standard_gravity" title="Standard gravity">g</a></span> = <span class="nowrap"><span data-sort-value="7000981000000000000♠"></span>9.81 m·s<sup>−2</sup></span>, is only <span class="nowrap"><span data-sort-value="6980399999999999999♠"></span>4<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−20</sup> K</span>. For an experimental test of the Unruh effect it is planned to use accelerations up to <span class="nowrap"><span data-sort-value="7026100000000000000♠"></span>10<sup>26</sup> m·s<sup>−2</sup></span>, which would give a temperature of about <span class="nowrap"><span data-sort-value="7005400000000000000♠"></span>400<span style="margin-left:.25em;">000</span> K</span>.<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>The Rindler derivation of the Unruh effect is unsatisfactory to some<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (August 2020)">who?</span></a></i>]</sup>, since the detector's path is <a href="/wiki/Superdeterminism" title="Superdeterminism">super-deterministic</a>. Unruh later developed the <a href="/w/index.php?title=Unruh%E2%80%93DeWitt_particle_detector&action=edit&redlink=1" class="new" title="Unruh–DeWitt particle detector (page does not exist)">Unruh–DeWitt particle detector</a> model to circumvent this objection. </p> <div class="mw-heading mw-heading2"><h2 id="Other_implications">Other implications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=4" title="Edit section: Other implications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Unruh effect would also cause the decay rate of accelerating particles to differ from inertial particles. Stable particles like the electron could have nonzero transition rates to higher mass states when accelerating at a high enough rate.<sup id="cite_ref-muel_16-0" class="reference"><a href="#cite_note-muel-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-van_17-0" class="reference"><a href="#cite_note-van-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-suz_18-0" class="reference"><a href="#cite_note-suz-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Unruh_radiation">Unruh radiation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=5" title="Edit section: Unruh radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Unruh's prediction that an accelerating detector would see a thermal bath is not controversial, the interpretation of the transitions in the detector in the non-accelerating frame is.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2022)">citation needed</span></a></i>]</sup> It is widely, although not universally, believed that each transition in the detector is accompanied by the emission of a particle, and that this particle will propagate to infinity and be seen as <b>Unruh radiation</b>. </p><p>The existence of Unruh radiation is not universally accepted. Smolyaninov claims that it has already been observed,<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> while O'Connell and Ford claim that it is not emitted at all.<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> While these skeptics accept that an accelerating object thermalizes at the Unruh temperature, they do not believe that this leads to the emission of photons, arguing that the emission and absorption rates of the accelerating particle are balanced. </p> <div class="mw-heading mw-heading2"><h2 id="Experimental_observation">Experimental observation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=6" title="Edit section: Experimental observation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Researchers claim experiments that successfully detected the <a href="/wiki/Sokolov%E2%80%93Ternov_effect" title="Sokolov–Ternov effect">Sokolov–Ternov effect</a><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> may also detect the Unruh effect under certain conditions.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Theoretical work in 2011 suggests that accelerating detectors could be used for the direct detection of the Unruh effect with current technology.<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> </p><p>The Unruh effect may have been observed for the first time in 2019 in the high energy <a href="/wiki/Channelling_(physics)" title="Channelling (physics)">channeling radiation</a> explored by the <a href="/wiki/NA63_experiment" title="NA63 experiment">NA63 experiment</a> at CERN.<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> <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=Unruh_effect&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Casimir_effect#Dynamical_Casimir_effect" title="Casimir effect">Dynamical Casimir effect</a></li> <li><a href="/wiki/Cosmic_Background_Radiation" class="mw-redirect" title="Cosmic Background Radiation">Cosmic Background Radiation</a></li> <li><a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a></li> <li><a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">Black hole thermodynamics</a></li> <li><a href="/wiki/Pair_production" title="Pair production">Pair production</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Superradiance" title="Superradiance">Superradiance</a></li> <li><a href="/wiki/Virtual_particle" title="Virtual particle">Virtual particle</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=Unruh_effect&action=edit&section=8" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMatsas2002" class="citation journal cs1">Matsas, George (2002). "The Fulling-Davies-Unruh Effect is Mandatory: The Proton's Testimony". <i>International Journal of Modern Physics D</i>. <b>11</b> (10): 1573–1577. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0205078">gr-qc/0205078</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/10.1142%2FS0218271802002918">10.1142/S0218271802002918</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:16555072">16555072</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+D&rft.atitle=The+Fulling-Davies-Unruh+Effect+is+Mandatory%3A+The+Proton%27s+Testimony&rft.volume=11&rft.issue=10&rft.pages=1573-1577&rft.date=2002&rft_id=info%3Aarxiv%2Fgr-qc%2F0205078&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16555072%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0218271802002918&rft.aulast=Matsas&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-fdu-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-fdu_2-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFulling1973" class="citation journal cs1">Fulling, S. 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"Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>7</b> (10): 2850–2862. <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/1973PhRvD...7.2850F">1973PhRvD...7.2850F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.7.2850">10.1103/PhysRevD.7.2850</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Nonuniqueness+of+Canonical+Field+Quantization+in+Riemannian+Space-Time&rft.volume=7&rft.issue=10&rft.pages=2850-2862&rft.date=1973&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.7.2850&rft_id=info%3Abibcode%2F1973PhRvD...7.2850F&rft.aulast=Fulling&rft.aufirst=S.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-davies_1975-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-davies_1975_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-davies_1975_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavies1975" class="citation journal cs1">Davies, P. C. W. (1975). "Scalar production in Schwarzschild and Rindler metrics". <i><a href="/wiki/Journal_of_Physics_A" title="Journal of Physics A">Journal of Physics A</a></i>. <b>8</b> (4): 609–616. <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/1975JPhA....8..609D">1975JPhA....8..609D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F8%2F4%2F022">10.1088/0305-4470/8/4/022</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+A&rft.atitle=Scalar+production+in+Schwarzschild+and+Rindler+metrics&rft.volume=8&rft.issue=4&rft.pages=609-616&rft.date=1975&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F8%2F4%2F022&rft_id=info%3Abibcode%2F1975JPhA....8..609D&rft.aulast=Davies&rft.aufirst=P.+C.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-unruh_1976-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-unruh_1976_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-unruh_1976_4-1"><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="CITEREFUnruh1976" class="citation journal cs1">Unruh, W. G. (1976). "Notes on black-hole evaporation". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>14</b> (4): 870–892. <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/1976PhRvD..14..870U">1976PhRvD..14..870U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.14.870">10.1103/PhysRevD.14.870</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Notes+on+black-hole+evaporation&rft.volume=14&rft.issue=4&rft.pages=870-892&rft.date=1976&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.14.870&rft_id=info%3Abibcode%2F1976PhRvD..14..870U&rft.aulast=Unruh&rft.aufirst=W.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-takagi_1986-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-takagi_1986_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakagi1986" class="citation journal cs1">Takagi, Shin (1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FPTP.88.1">"Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimensions"</a>. <i>Progress of Theoretical Physics Supplement</i> (88): 1–142. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FPTP.88.1">10.1143/PTP.88.1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+of+Theoretical+Physics+Supplement&rft.atitle=Vacuum+Noise+and+Stress+Induced+by+Uniform+Acceleration%3A+Hawking-Unruh+Effect+in+Rindler+Manifold+of+Arbitrary+Dimensions&rft.issue=88&rft.pages=1-142&rft.date=1986&rft_id=info%3Adoi%2F10.1143%2FPTP.88.1&rft.aulast=Takagi&rft.aufirst=Shin&rft_id=https%3A%2F%2Fdoi.org%2F10.1143%252FPTP.88.1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-DUMB-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-DUMB_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUnruh2001" class="citation book cs1">Unruh, W. 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"Is there Unruh radiation?". <i><a href="/wiki/Physics_Letters_A" class="mw-redirect" title="Physics Letters A">Physics Letters A</a></i>. <b>350</b> (1–2): 17–26. <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/quant-ph/0509151">quant-ph/0509151</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006PhLA..350...17F">2006PhLA..350...17F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physleta.2005.09.068">10.1016/j.physleta.2005.09.068</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:119352739">119352739</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+A&rft.atitle=Is+there+Unruh+radiation%3F&rft.volume=350&rft.issue=1%E2%80%932&rft.pages=17-26&rft.date=2005&rft_id=info%3Aarxiv%2Fquant-ph%2F0509151&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119352739%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physleta.2005.09.068&rft_id=info%3Abibcode%2F2006PhLA..350...17F&rft.aulast=Ford&rft.aufirst=G.+W.&rft.au=O%27Connell%2C+R.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBellLeinaas1983" class="citation journal cs1">Bell, J. 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T.; Singleton, D. (2007). "On the physical meaning of the Unruh effect". <i><a href="/wiki/JETP_Letters" class="mw-redirect" title="JETP Letters">JETP Letters</a></i>. <b>86</b> (9): 615–619. <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/0705.2525">0705.2525</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JETPL..86..615A">2008JETPL..86..615A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS0021364007210138">10.1134/S0021364007210138</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:14895426">14895426</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=JETP+Letters&rft.atitle=On+the+physical+meaning+of+the+Unruh+effect&rft.volume=86&rft.issue=9&rft.pages=615-619&rft.date=2007&rft_id=info%3Aarxiv%2F0705.2525&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14895426%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1134%2FS0021364007210138&rft_id=info%3Abibcode%2F2008JETPL..86..615A&rft.aulast=Akhmedov&rft.aufirst=E.+T.&rft.au=Singleton%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" 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="CITEREFMartín_MartínezFuentesMann2011" class="citation journal cs1">Martín Martínez, E.; Fuentes, I.; Mann, R. B. (2011). "Using Berry's Phase to Detect the Unruh Effect at Lower Accelerations". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>107</b> (13): 131301. <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/1012.2208">1012.2208</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhRvL.107m1301M">2011PhRvL.107m1301M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.107.131301">10.1103/PhysRevLett.107.131301</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22026837">22026837</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:21024756">21024756</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Using+Berry%27s+Phase+to+Detect+the+Unruh+Effect+at+Lower+Accelerations&rft.volume=107&rft.issue=13&rft.pages=131301&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21024756%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2011PhRvL.107m1301M&rft_id=info%3Aarxiv%2F1012.2208&rft_id=info%3Apmid%2F22026837&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.107.131301&rft.aulast=Mart%C3%ADn+Mart%C3%ADnez&rft.aufirst=E.&rft.au=Fuentes%2C+I.&rft.au=Mann%2C+R.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" 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="CITEREFLynchCohenHadadKaminer2021" class="citation journal cs1">Lynch, M. H.; Cohen, E.; Hadad, Y.; Kaminer, I. (2021). "Experimental Observation of Acceleration-Induced Thermality". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>104</b> (4): 025015. <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/1903.00043">1903.00043</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PhRvD.104b5015L">2021PhRvD.104b5015L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.104.025015">10.1103/PhysRevD.104.025015</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:214623535">214623535</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Experimental+Observation+of+Acceleration-Induced+Thermality&rft.volume=104&rft.issue=4&rft.pages=025015&rft.date=2021&rft_id=info%3Aarxiv%2F1903.00043&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A214623535%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.104.025015&rft_id=info%3Abibcode%2F2021PhRvD.104b5015L&rft.aulast=Lynch&rft.aufirst=M.+H.&rft.au=Cohen%2C+E.&rft.au=Hadad%2C+Y.&rft.au=Kaminer%2C+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThorne1995" class="citation book cs1">Thorne, K. P. (1995). "Black holes evaporate". <a href="/wiki/Black_Holes_and_Time_Warps" title="Black Holes and Time Warps"><i>Black Holes and Time Warps</i></a> (Reprint ed.). <a href="/wiki/W._W._Norton_%26_Company" title="W. W. Norton & Company">W. W. Norton & Company</a>. Box 12.5, p. 444. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-31276-3" title="Special:BookSources/0-393-31276-3"><bdi>0-393-31276-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Black+holes+evaporate&rft.btitle=Black+Holes+and+Time+Warps&rft.pages=Box+12.5%2C+p.+444&rft.edition=Reprint&rft.pub=W.+W.+Norton+%26+Company&rft.date=1995&rft.isbn=0-393-31276-3&rft.aulast=Thorne&rft.aufirst=K.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald1994" class="citation book cs1">Wald, R. M. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Iud7eyDxT1AC&q=quantum+inauthor:wald"><i>Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics</i></a>. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. Ch. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-87027-8" title="Special:BookSources/0-226-87027-8"><bdi>0-226-87027-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+in+Curved+Spacetime+and+Black+Hole+Thermodynamics&rft.pages=Ch.+5&rft.pub=University+of+Chicago+Press&rft.date=1994&rft.isbn=0-226-87027-8&rft.aulast=Wald&rft.aufirst=R.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIud7eyDxT1AC%26q%3Dquantum%2Binauthor%3Awald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrispinoHiguchiMatsas2008" class="citation journal cs1">Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (2008). "The Unruh effect and its applications". <i><a href="/wiki/Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a></i>. <b>80</b> (3): 787–838. <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/0710.5373">0710.5373</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008RvMP...80..787C">2008RvMP...80..787C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.80.787">10.1103/RevModPhys.80.787</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:119223632">119223632</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=The+Unruh+effect+and+its+applications&rft.volume=80&rft.issue=3&rft.pages=787-838&rft.date=2008&rft_id=info%3Aarxiv%2F0710.5373&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119223632%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.80.787&rft_id=info%3Abibcode%2F2008RvMP...80..787C&rft.aulast=Crispino&rft.aufirst=L.+C.+B.&rft.au=Higuchi%2C+A.&rft.au=Matsas%2C+G.+E.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unruh_effect&action=edit&section=10" 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="CITEREFStephen_Fulling_and_George_Matsas" class="citation web cs1"><a href="/wiki/Stephen_Fulling" class="mw-redirect" title="Stephen Fulling">Stephen Fulling</a> and George Matsas (ed.). <a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Unruh_effect">"Unruh effect"</a>. <i><a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Scholarpedia&rft.atitle=Unruh+effect&rft_id=http%3A%2F%2Fwww.scholarpedia.org%2Farticle%2FUnruh_effect&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnruh+effect" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output 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