Analytical theory of Hawking radiation in dispersive media
Identifieur interne : 001369 ( Istex/Corpus ); précédent : 001368; suivant : 001370Analytical theory of Hawking radiation in dispersive media
Auteurs : Ulf Leonhardt ; Scott RobertsonSource :
- New Journal of Physics [ 1367-2630 ] ; 2012-05.
Abstract
Hawking's 1974 prediction that black holes radiate and evaporate has been hinting at a hidden connection between general relativity, quantum mechanics and thermodynamics. Recently, laboratory analogues of the event horizon have reached the level where tests of Hawking's idea are possible. In this paper we show how to go beyond Hawking's theory in such laboratory analogues in a way that is experimentally testable.
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DOI: 10.1088/1367-2630/14/5/053003
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Analytical theory of Hawking radiation in dispersive media</title><author><name sortKey="Leonhardt, Ulf" sort="Leonhardt, Ulf" uniqKey="Leonhardt U" first="Ulf" last="Leonhardt">Ulf Leonhardt</name><affiliation><mods:affiliation>School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK</mods:affiliation></affiliation><affiliation><mods:affiliation>Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria</mods:affiliation></affiliation><affiliation><mods:affiliation>Quantum Optics, Quantum Nanophysics, Quantum Information, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: ulf@st-andrews.ac.uk</mods:affiliation></affiliation></author><author><name sortKey="Robertson, Scott" sort="Robertson, Scott" uniqKey="Robertson S" first="Scott" last="Robertson">Scott Robertson</name><affiliation><mods:affiliation>Dipartimento di Fisica, Universit degli Studi di Pavia, Via Bassi 6, 27100 Pavia, Italy</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: scott.robertson@unipv.it</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:A0A7E0003B243B55F82A506D8750C879A8C9FB41</idno><date when="2012" year="2012">2012</date><idno type="doi">10.1088/1367-2630/14/5/053003</idno><idno type="url">https://api.istex.fr/document/A0A7E0003B243B55F82A506D8750C879A8C9FB41/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001369</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Analytical theory of Hawking radiation in dispersive media</title><author><name sortKey="Leonhardt, Ulf" sort="Leonhardt, Ulf" uniqKey="Leonhardt U" first="Ulf" last="Leonhardt">Ulf Leonhardt</name><affiliation><mods:affiliation>School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK</mods:affiliation></affiliation><affiliation><mods:affiliation>Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria</mods:affiliation></affiliation><affiliation><mods:affiliation>Quantum Optics, Quantum Nanophysics, Quantum Information, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: ulf@st-andrews.ac.uk</mods:affiliation></affiliation></author><author><name sortKey="Robertson, Scott" sort="Robertson, Scott" uniqKey="Robertson S" first="Scott" last="Robertson">Scott Robertson</name><affiliation><mods:affiliation>Dipartimento di Fisica, Universit degli Studi di Pavia, Via Bassi 6, 27100 Pavia, Italy</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: scott.robertson@unipv.it</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">New Journal of Physics</title><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2012-05">2012-05</date><biblScope unit="volume">14</biblScope><biblScope unit="issue">5</biblScope></imprint></series><idno type="istex">A0A7E0003B243B55F82A506D8750C879A8C9FB41</idno><idno type="DOI">10.1088/1367-2630/14/5/053003</idno><idno type="href">http://stacks.iop.org/NJP/14/053003</idno><idno type="ArticleID">nj416455</idno></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">Hawking's 1974 prediction that black holes radiate and evaporate has been hinting at a hidden connection between general relativity, quantum mechanics and thermodynamics. 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Phys.</abbrev-journal-title></journal-title-group><issn pub-type="epub">1367-2630</issn><publisher><publisher-name>IOP Publishing</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">nj416455</article-id><article-id pub-id-type="doi">10.1088/1367-2630/14/5/053003</article-id><article-id pub-id-type="manuscript">416455</article-id><article-categories><subj-group subj-group-type="article-type"><subject>Paper</subject></subj-group></article-categories><title-group><article-title>Analytical theory of Hawking radiation in dispersive media</article-title><alt-title alt-title-type="ascii">Analytical theory of Hawking radiation in dispersive media</alt-title><alt-title alt-title-type="short">Analytical theory of Hawking radiation in dispersive media</alt-title><alt-title alt-title-type="short-ascii">Analytical theory of Hawking radiation in dispersive media</alt-title></title-group><contrib-group content-type="all"><contrib contrib-type="author"><name><surname>Leonhardt</surname><given-names>Ulf</given-names></name><xref ref-type="aff" rid="nj416455af1">1</xref><xref ref-type="aff" rid="nj416455af2">2</xref><xref ref-type="aff" rid="nj416455af3">3</xref><xref ref-type="fn" rid="nj416455afn1">5</xref><xref ref-type="aff" rid="nj416455em1"></xref></contrib><contrib contrib-type="author"><name><surname>Robertson</surname><given-names>Scott</given-names></name><xref ref-type="aff" rid="nj416455af4">4</xref><xref ref-type="fn" rid="nj416455afn1">5</xref><xref ref-type="aff" rid="nj416455em2"></xref></contrib><aff id="nj416455af1"><label>1</label>School of Physics and Astronomy, <institution>University of St Andrews</institution>, North Haugh, St Andrews KY16 9SS, <country>UK</country></aff><aff id="nj416455af2"><label>2</label>Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, <country>Austria</country></aff><aff id="nj416455af3"><label>3</label>Quantum Optics, Quantum Nanophysics, Quantum Information, <institution>University of Vienna</institution>, Boltzmanngasse 5, A-1090 Vienna, <country>Austria</country></aff><aff id="nj416455af4"><label>4</label>Dipartimento di Fisica, <institution>Università degli Studi di Pavia</institution>, Via Bassi 6, 27100 Pavia, <country>Italy</country></aff><ext-link ext-link-type="email" id="nj416455em1">ulf@st-andrews.ac.uk</ext-link><ext-link ext-link-type="email" id="nj416455em2">scott.robertson@unipv.it</ext-link><author-comment content-type="short-author-list"><p>U Leonhardt and S Robertson</p></author-comment></contrib-group><author-notes><fn id="nj416455afn1"><label>5</label><p>Both authors contributed equally to this work.</p></fn></author-notes><pub-date pub-type="ppub"><month>5</month><year>2012</year></pub-date><pub-date pub-type="epub"><day>3</day><month>5</month><year>2012</year></pub-date><volume>14</volume><issue>5</issue><elocation-id content-type="artnum">053003</elocation-id><history><date date-type="received"><day>9</day><month>12</month><year>2011</year></date></history><permissions><copyright-statement>© IOP Publishing and Deutsche Physikalische Gesellschaft</copyright-statement><copyright-year>2012</copyright-year></permissions><self-uri xlink:href="http://stacks.iop.org/NJP/14/053003"></self-uri><abstract><title>Abstract</title><p>Hawking's 1974 prediction that black holes radiate and evaporate has been hinting at a hidden connection between general relativity, quantum mechanics and thermodynamics. Recently, laboratory analogues of the event horizon have reached the level where tests of Hawking's idea are possible. In this paper we show how to go beyond Hawking's theory in such laboratory analogues in a way that is experimentally testable.</p></abstract><counts><page-count count="15"></page-count></counts><custom-meta-group><custom-meta><meta-name>ccc</meta-name><meta-value>1367-2630/12/053003+15$33.00</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="nj416455s1"><label></label><p>Black-hole evaporation by Hawking radiation has been known to theoretical physics for nearly 40 years [<xref ref-type="bibr" rid="nj416455bib01">1</xref>, <xref ref-type="bibr" rid="nj416455bib02">2</xref>]. Yet it cannot claim to be understood, because Hawking radiation should originate from vacuum modes at frequencies well beyond the Planck scale [<xref ref-type="bibr" rid="nj416455bib03">3</xref>, <xref ref-type="bibr" rid="nj416455bib04">4</xref>]—well beyond the limit of our physical understanding. Unruh [<xref ref-type="bibr" rid="nj416455bib05">5</xref>] realized that the Hawking effect is not restricted to gravity, but arises for waves in moving media in analogues of the event horizon [<xref ref-type="bibr" rid="nj416455bib06">6</xref>–<xref ref-type="bibr" rid="nj416455bib08">8</xref>], allowing tests of Hawking's prediction [<xref ref-type="bibr" rid="nj416455bib01">1</xref>] by experiments [<xref ref-type="bibr" rid="nj416455bib09">9</xref>–<xref ref-type="bibr" rid="nj416455bib16">16</xref>]. Here the trans-Planckian problem does not appear, because of dispersion [<xref ref-type="bibr" rid="nj416455bib03">3</xref>] that tunes waves out of the horizon's grip. Despite the drastic alteration to wave propagation, Hawking radiation is thought to be preserved and to agree with Hawking's result in the low-frequency and low-temperature limit [<xref ref-type="bibr" rid="nj416455bib17">17</xref>–<xref ref-type="bibr" rid="nj416455bib29">29</xref>], although evidence from approximations [<xref ref-type="bibr" rid="nj416455bib24">24</xref>], analytical theory for a specific case [<xref ref-type="bibr" rid="nj416455bib26">26</xref>] and numerical simulations [<xref ref-type="bibr" rid="nj416455bib27">27</xref>, <xref ref-type="bibr" rid="nj416455bib28">28</xref>] already indicate that Hawking's theory is not universal. Based upon the work of others [<xref ref-type="bibr" rid="nj416455bib19">19</xref>, <xref ref-type="bibr" rid="nj416455bib23">23</xref>] we develop here an analytical theory of Hawking radiation in dispersive media. This theory predicts deviations from Hawking's formula that are observable in experimental tests with analogues of the event horizon [<xref ref-type="bibr" rid="nj416455bib06">6</xref>–<xref ref-type="bibr" rid="nj416455bib16">16</xref>].</p><p>Waves in moving media experience the analogue of the event horizon when the velocity of the medium exceeds the speed of the waves. Consider waves propagating with phase velocity <italic>c</italic> in a medium moving with local velocity <italic>u</italic>. Here <italic>c</italic> depends on the wavenumber <italic>k</italic> according to the dispersion relation in the material—in a co-moving frame—while <italic>u</italic> varies in space according to the velocity profile of the medium. We can always decompose a wave into its Fourier components <italic>φ</italic> with respect to the (angular) frequency <italic>ω</italic>. Near the horizon, we can assume a one-dimensional (1D) model [<xref ref-type="bibr" rid="nj416455bib04">4</xref>] where both <italic>φ</italic> and <italic>u</italic> depend on the spatial coordinate <italic>z</italic> only. The medium shall move to the left, i.e. with negative <italic>u</italic>, forming just one black-hole horizon, for simplicity. In a co-moving frame, the frequency is Doppler-shifted to <italic>ω</italic> − <italic>uk</italic> and equals <italic>ck</italic> for waves propagating against the current. We do not discuss waves propagating with the flow, as they are not affected by the horizon, and we assume that waves propagating with and against the current are decoupled. The wave propagation in moving media is thus described by the dispersion relation <disp-formula id="nj416455eqn1"><label>1</label><tex-math></tex-math><graphic xlink:href="nj416455eqn1.gif"></graphic></disp-formula> For isotropic media the phase velocity <italic>c</italic> is an even function of <italic>k</italic>, so <italic>c</italic>(<italic>k</italic>)<italic>k</italic> is an odd function. Figure <xref ref-type="fig" rid="nj416455fig1">1</xref> illustrates the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) and shows how to find solutions graphically. These solutions characterize the waves in the moving medium.</p><fig id="nj416455fig1" position="float"><label>Figure 1.</label><caption id="nj416455fc1"><p>Graphical solution of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) for waves in moving media. Waves may propagate with wavenumbers <italic>k</italic><sub><italic>j</italic></sub> given by the intersections of the dispersion curve <italic>ck</italic> (blue) with the straight lines (red) of the Doppler-shifted frequency <italic>ω</italic> − <italic>uk</italic> depending on the velocity <italic>u</italic> of the medium (<italic>u</italic> < 0). The group velocity <italic>v</italic><sub>g</sub> is given by the slope of the dispersion curve <italic>ck</italic>. For a wave with <italic>k</italic><sub><italic>u</italic>r</sub> the group velocity exceeds −<italic>u</italic>, the wave thus propagates to the right, away from the horizon. For <italic>k</italic><sub><italic>u</italic>l</sub> the wave is left moving with <italic>v</italic><sub>g</sub> + <italic>u</italic> < 0. Where <italic>k</italic><sub><italic>u</italic>r</sub> and <italic>k</italic><sub><italic>u</italic>l</sub> coincide the flow velocity reaches the group velocity, forming a group-velocity horizon. A third solution <italic>k</italic><sub><italic>u</italic></sub> may exist that corresponds to a Hawking wave with negative wavenumber that is left-moving. The fourth solution <italic>k</italic><sub><italic>v</italic></sub> describes waves propagating with the current that do not contribute to the Hawking effect. We used the dispersion relation <italic>c</italic><sup>2</sup> = <italic>c</italic><sup>2</sup><sub>0</sub>(1 − <italic>k</italic><sup>2</sup>/<italic>k</italic><sup>2</sup><sub>0</sub>) with the constants <italic>c</italic><sub>0</sub> and <italic>k</italic><sub>0</sub> and plotted <italic>k</italic> and <italic>ω</italic> in units of <italic>k</italic><sub>0</sub> and <italic>ω</italic><sub>0</sub> = <italic>c</italic><sub>0</sub><italic>k</italic><sub>0</sub>, respectively. This dispersion relation is said to be subluminal (or normal), because the phase velocity decreases with increasing <italic>k</italic>; it is the simplest subluminal dispersion relation. We could also chose superluminal (or anomalous) dispersion, for example <italic>c</italic><sup>2</sup> = <italic>c</italic><sup>2</sup><sub>0</sub>(1 + <italic>k</italic><sup>2</sup>/<italic>k</italic><sup>2</sup><sub>0</sub>), where <italic>c</italic> increases with <italic>k</italic>. It turns out that in this case we obtain similar Hawking temperatures.</p></caption><graphic id="nj416455f1_eps" content-type="print" xlink:href="nj416455f1_pr.eps"></graphic><graphic id="nj416455f1_online" content-type="online" xlink:href="nj416455f1_online.jpg"></graphic></fig><p>Consider now localized wave packets with position <italic>z</italic>, wavenumber <italic>k</italic> and carrier frequency <italic>ω</italic>. Figure <xref ref-type="fig" rid="nj416455fig2">2</xref> illustrates the typical trajectories of such wave packets. Their dynamics are governed by Hamilton's equations <disp-formula id="nj416455eqn2"><label>2</label><tex-math></tex-math><graphic xlink:href="nj416455eqn2.gif"></graphic></disp-formula> Here <italic>u</italic>' = d<italic>u</italic>/d<italic>z</italic> denotes the velocity gradient of the medium and <italic>v</italic><sub>g</sub> = d(<italic>ck</italic>)/d<italic>k</italic> the group velocity of the wave, which, in dispersive media, differs from the phase velocity <italic>c</italic>. The horizon is formed where the flow exceeds the wave velocity, but we need to identify which velocity we mean. Here we assume a <italic>group-velocity horizon</italic> where |<italic>u</italic>| reaches <italic>v</italic><sub>g</sub>. Yet Hawking radiation also relies on the possibility of exciting waves for which |<italic>u</italic>| exceeds <italic>c</italic>. Figure <xref ref-type="fig" rid="nj416455fig1">1</xref> shows [<xref ref-type="bibr" rid="nj416455bib19">19</xref>] that three types of waves may exist in a region where |<italic>u</italic>| < <italic>v</italic><sub>g</sub>: a right-moving wave <italic>φ</italic><sub><italic>u</italic>r</sub> that escapes from the horizon, a left-moving wave <italic>φ</italic><sub><italic>u</italic>l</sub> that drifts towards the horizon and a left-moving wave <italic>φ</italic><sub><italic>u</italic></sub> with negative wavenumber <italic>k</italic><sub><italic>u</italic></sub> [<xref ref-type="bibr" rid="nj416455bib10">10</xref>]. We see from the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) that <italic>k</italic> = (<italic>c</italic> + <italic>u</italic>)<sup>−1</sup><italic>ω</italic>, so <italic>k</italic> is negative when |<italic>u</italic>| > <italic>c</italic>; for <italic>φ</italic><sub><italic>u</italic></sub> the medium exceeds the phase velocity. (A fourth wave <italic>φ</italic><sub><italic>v</italic></sub> propagates with the flow and is ignored in our analysis.) On the other side of the horizon, where |<italic>u</italic>| > <italic>v</italic><sub>g</sub>, only the negative-<italic>k</italic> wave <italic>φ</italic><sub><italic>u</italic></sub> continues to propagate (apart from <italic>φ</italic><sub><italic>v</italic></sub> of course). In the astrophysical black hole, <italic>φ</italic><sub><italic>u</italic></sub> would be on its way to the singularity; it constitutes the Hawking partner of the escaping wave. Figure <xref ref-type="fig" rid="nj416455fig2">2</xref> indicates that incident waves with positive <italic>k</italic> are partially converted at the horizon into their Hawking partners with negative <italic>k</italic> and vice versa.</p><fig id="nj416455fig2" position="float"><label>Figure 2.</label><caption id="nj416455fc2"><p>Space–time diagrams of the trajectories of in-going and out-going modes near the horizon. In the diagram for the in-going mode <italic>φ</italic><sup>in</sup><sub><italic>u</italic>l</sub>, an incident wave with wavenumber <italic>k</italic><sub><italic>u</italic>l</sub> is reflected at the horizon and leaves with wavenumber <italic>k</italic><sub><italic>u</italic>r</sub>, while some of the incident amplitude is converted into a wave with negative <italic>k</italic><sub><italic>u</italic></sub> that crosses the horizon (dashed line). The other in-going mode <italic>φ</italic><sup>in</sup><sub><italic>u</italic></sub> is incident with negative <italic>k</italic><sub><italic>u</italic></sub> and continues across the horizon, sending off a <italic>k</italic><sub><italic>u</italic>r</sub> to the right (dashed line). The <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> and <italic>φ</italic><sup>out</sup><sub><italic>u</italic></sub> describe the out-going modes that leave with wavenumbers <italic>k</italic><sub><italic>u</italic>r</sub> and <italic>k</italic><sub><italic>u</italic></sub> and originate from <italic>φ</italic><sub><italic>u</italic>l</sub> and <italic>φ</italic><sub><italic>u</italic></sub> contributions. We used the velocity profile <italic>u</italic>(<italic>z</italic>) = (<italic>u</italic><sub>R</sub> + <italic>u</italic><sub>L</sub>)/2 + tanh(<italic>z</italic>/<italic>a</italic>)(<italic>u</italic><sub>R</sub> − <italic>u</italic><sub>L</sub>)/2 with constant asymptotic velocities <italic>u</italic><sub>R</sub> and <italic>u</italic><sub>L</sub> and constant scale <italic>a</italic> and solved Hamilton's equations (<xref ref-type="disp-formula" rid="nj416455eqn2">2</xref>) for the dispersion relation of figure <xref ref-type="fig" rid="nj416455fig1">1</xref> (<italic>u</italic><sub>R</sub> = −0.8<italic>c</italic><sub>0</sub>, <italic>u</italic><sub>L</sub> = −1.2<italic>c</italic><sub>0</sub>, <italic>a</italic> = 1/<italic>k</italic><sub>0</sub>, <italic>ω</italic> = 0.035<italic>ω</italic><sub>0</sub>); <italic>z</italic> and <italic>t</italic> are plotted in units of 1/<italic>k</italic><sub>0</sub> and 1/<italic>ω</italic><sub>0</sub>, respectively.</p></caption><graphic id="nj416455f2_eps" content-type="print" xlink:href="nj416455f2_pr.eps"></graphic><graphic id="nj416455f2_online" content-type="online" xlink:href="nj416455f2_online.jpg"></graphic></fig><p>Figure <xref ref-type="fig" rid="nj416455fig2">2</xref> also characterizes the in-going and out-going modes. As the mode conversion is linear, an out-going mode is the linear superposition of two in-going modes:
<disp-formula id="nj416455eqn3"><label>3</label><tex-math></tex-math><graphic xlink:href="nj416455eqn3.gif"></graphic></disp-formula> The modes are orthogonal and normalized to <italic>δ</italic>(<italic>ω</italic><sub>1</sub> − <italic>ω</italic><sub>2</sub>) in a sense made precise in the scalar product (A.2) defined in appendix <xref ref-type="sec" rid="nj416455app1">A</xref>. The negative-<italic>k</italic> mode <italic>φ</italic><sup>in</sup><sub><italic>u</italic></sub> turns out to have the negative norm −<italic>δ</italic>(<italic>ω</italic><sub>1</sub> − <italic>ω</italic><sub>2</sub>). From this follows an important conclusion that does not rely on the details of the scalar product (<italic>φ</italic><sub>1</sub>,<italic>φ</italic><sub>2</sub>) but only on the fact that (<italic>φ</italic><sub>1</sub>,<italic>φ</italic><sub>2</sub>) is bi-linear in <italic>φ</italic>*<sub>1</sub> and <italic>φ</italic><sub>2</sub>. Calculating the norm of the superposition (<xref ref-type="disp-formula" rid="nj416455eqn3">3</xref>) we obtain
<disp-formula id="nj416455eqn4"><label>4</label><tex-math></tex-math><graphic xlink:href="nj416455eqn4.gif"></graphic></disp-formula> which implies that the amplitude of the incident mode <italic>φ</italic><sup>in</sup><sub><italic>u</italic>l</sub> is multiplied by a factor |<italic>α</italic>| larger than 1; the horizon thus acts as an amplifier [<xref ref-type="bibr" rid="nj416455bib30">30</xref>, <xref ref-type="bibr" rid="nj416455bib31">31</xref>]. The negative-<italic>k</italic> mode <italic>φ</italic><sup>in</sup><sub><italic>u</italic></sub> represents the reservoir of the amplification noise [<xref ref-type="bibr" rid="nj416455bib30">30</xref>].</p><p>According to quantum mechanics [<xref ref-type="bibr" rid="nj416455bib32">32</xref>], amplifiers generate noise even when the reservoir is empty; they create real particles from the quantum vacuum in correlated pairs [<xref ref-type="bibr" rid="nj416455bib30">30</xref>]. In the case of a horizon, these particles constitute the Hawking radiation; the particles of the <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> are escaping while their Hawking partners of <italic>φ</italic><sup>out</sup><sub><italic>u</italic></sub> are swept away. Correlated particles are most strongly entangled; if we only consider the escaping radiation, averaged over the Hawking partners, the reduced quantum state of each escaping mode is therefore a state of maximal entropy [<xref ref-type="bibr" rid="nj416455bib33">33</xref>], a thermal state. Its temperature <italic>T</italic> is given by the Boltzmann law [<xref ref-type="bibr" rid="nj416455bib30">30</xref>] <disp-formula id="nj416455eqn5"><label>5</label><tex-math></tex-math><graphic xlink:href="nj416455eqn5.gif"></graphic></disp-formula> where ℏ denotes Planck's constant divided by 2<italic>π</italic> and <italic>k</italic><sub>B</sub> is Boltzmann's constant. In the following we calculate <italic>T</italic>. Here it is wise to characterize wave packets by their wavenumbers <italic>k</italic> instead of their positions <italic>z</italic>, because wavenumber is a better indicator of a horizon than position: there is a clear gap between positive and negative <italic>k</italic>, but not in <italic>z</italic>: negative-<italic>k</italic> waves may freely cross the horizon exploring the entire physical space (see <italic>φ</italic><sup>in</sup><sub><italic>u</italic>l</sub> in figure <xref ref-type="fig" rid="nj416455fig2">2</xref>). Therefore, we rather represent <italic>z</italic> as a function of <italic>k</italic>, identifying the present position of the wave packet by its present wavenumber. Mathematically, we simply solve the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) for <italic>z</italic> with <italic>k</italic> as variable.</p><p>The Hawking effect bridges positive and negative wavenumbers. Figure <xref ref-type="fig" rid="nj416455fig3">3</xref> visualizes this connection on the complex plane where we regard <italic>k</italic> as a complex variable. Assuming a slowly varying velocity profile, we derive in appendix <xref ref-type="sec" rid="nj416455app1">A</xref> a simple rule for obtaining the Hawking temperature (<xref ref-type="disp-formula" rid="nj416455eqn5">5</xref>) graphically, which goes as follows. The red lines of figure <xref ref-type="fig" rid="nj416455fig3">3</xref> indicate the physically allowed wavenumbers. The relative phase between the positive and negative <italic>k</italic> wave is given by the contour integral of <italic>z</italic> from positive to negative wavenumber, avoiding the branch cut between them. This phase is not a real number but contains an imaginary part. If we close the contour (black ellipse in figure <xref ref-type="fig" rid="nj416455fig3">3</xref>) the real part of the integral vanishes and twice the imaginary part remains. Now, the relative amplitude |<italic>α</italic>/<italic>β</italic>| between negative and positive component is the exponential of the imaginary part of the phase, and so |<italic>α</italic>/<italic>β</italic>|<sup>2</sup> must be the exponential of twice the imaginary part, i.e. of the closed contour integral (black ellipse in figure <xref ref-type="fig" rid="nj416455fig3">3</xref>). We thus obtain from formula (<xref ref-type="disp-formula" rid="nj416455eqn5">5</xref>) our result <disp-formula id="nj416455eqn6"><label>6</label><tex-math></tex-math><graphic xlink:href="nj416455eqn6.gif"></graphic></disp-formula> Note that we can also represent <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn1.gif"></inline-graphic></inline-formula> as <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn2.gif"></inline-graphic></inline-formula>, but the corresponding contour in the complex <italic>z</italic>-plane can be more complicated.</p><fig id="nj416455fig3" position="float"><label>Figure 3.</label><caption id="nj416455fc3"><p>Graphical representation of the Hawking temperature. The heart of the Hawking effect is the partial conversion of waves with positive wavenumbers <italic>k</italic> into waves with negative <italic>k</italic>. The relative amplitude of this mode conversion gives the Hawking temperature. For real <italic>k</italic> there is a gap between the physically allowed positive and negative values (figure <xref ref-type="fig" rid="nj416455fig1">1</xref>), but they are connected for complex <italic>k</italic>. Therefore it is wise to represent the solution <italic>z</italic>(<italic>k</italic>) of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) on the complex <italic>k</italic> plane; here we illustrate the imaginary part of <italic>z</italic> using different shades of blue (for the dispersion relation of figure <xref ref-type="fig" rid="nj416455fig1">1</xref> and the velocity profile in figure <xref ref-type="fig" rid="nj416455fig2">2</xref>). At the red lines Im(<italic>z</italic>) = 0, so <italic>z</italic> is real there; they are drawn along the values of <italic>k</italic> that appear during the real wave propagation. Lines with real <italic>z</italic>(<italic>k</italic>) are disconnected by branch cuts where the imaginary part of <italic>z</italic> abruptly changes (the shades of blue are discontinuous here). The solid blue line connects the wavenumbers (blue dots) that are physically allowed (figure <xref ref-type="fig" rid="nj416455fig1">1</xref>) for the out-going wave <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> on the far right side of the horizon (figure <xref ref-type="fig" rid="nj416455fig2">2</xref>); the dotted blue line corresponds to a point further to the left, beyond the group-velocity horizon, where the wave is exponentially damped. The contour integral along the blue curve gives the logarithm of the relative amplitude and the phase of the wave components of <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub>; the contour is freely deformable where <italic>z</italic>(<italic>k</italic>) is continuous. The closed-contour integral (black ellipse) gives the Hawking temperature according to formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>).</p></caption><graphic id="nj416455f3_eps" content-type="print" xlink:href="nj416455f3_pr.eps"></graphic><graphic id="nj416455f3_online" content-type="online" xlink:href="nj416455f3_online.jpg"></graphic></fig><p>Formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) is the main result of this paper. It extends Corley's analytical theory [<xref ref-type="bibr" rid="nj416455bib23">23</xref>] to arbitrary velocity profiles (beyond linearization at the horizon) and generalizes the interpretation of Hawking radiation as tunnelling [<xref ref-type="bibr" rid="nj416455bib34">34</xref>–<xref ref-type="bibr" rid="nj416455bib36">36</xref>] to dispersive media. We easily reproduce Hawking's formula in the regime of weak dispersion where <italic>v</italic><sub>g</sub> changes little with wavenumber. In this case, we see from Hamilton's equations (<xref ref-type="disp-formula" rid="nj416455eqn2">2</xref>) that d<italic>z</italic>/d<italic>t</italic> remains nearly zero, while the wavenumber <italic>k</italic> falls exponentially with the rate <italic>u</italic>' that remains nearly constant, because <italic>z</italic> does not change much. Differentiating the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) with respect to <italic>ω</italic> we get d<italic>z</italic>/d<italic>ω</italic> = 1/(<italic>u</italic>'<italic>k</italic>) and so <disp-formula id="nj416455eqn7"><label>7</label><tex-math></tex-math><graphic xlink:href="nj416455eqn7.gif"></graphic></disp-formula> which gives 2<italic>π</italic>/<italic>u</italic>' since <italic>u</italic>' remains constant over an exponentially large range of <italic>k</italic>. After integrating over <italic>ω</italic> we arrive at Hawking's formula [<xref ref-type="bibr" rid="nj416455bib01">1</xref>] in moving media [<xref ref-type="bibr" rid="nj416455bib05">5</xref>] <disp-formula id="nj416455eqn8"><label>8</label><tex-math></tex-math><graphic xlink:href="nj416455eqn8.gif"></graphic></disp-formula> For strong dispersion, however, we cannot regard <italic>u</italic>'(<italic>z</italic>) as almost constant in the contour integral (<xref ref-type="disp-formula" rid="nj416455eqn7">7</xref>). Figures <xref ref-type="fig" rid="nj416455fig4">4</xref> and <xref ref-type="fig" rid="nj416455fig5">5</xref> show that in this case <italic>T</italic> varies with frequency: the Hawking temperature (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>) is no longer universal and, as a consequence, the Hawking spectrum is non-Planckian. Formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) still predicts that <italic>T</italic> is inversely proportional to the scale <italic>a</italic> of the velocity profile. The Hawking temperature remains proportional to 1/<italic>a</italic> until for large velocity gradients our phase-integral method ceases to be valid, as numerical calculations show (figures <xref ref-type="fig" rid="nj416455fig4">4</xref> and <xref ref-type="fig" rid="nj416455fig5">5</xref>). For slowly-varying velocity gradients the contour integral (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) describes Hawking radiation very well, including deviations from Hawking's formula (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>) that have only been seen as an approximation [<xref ref-type="bibr" rid="nj416455bib24">24</xref>] or numerically [<xref ref-type="bibr" rid="nj416455bib27">27</xref>, <xref ref-type="bibr" rid="nj416455bib28">28</xref>] so far. Appendix <xref ref-type="sec" rid="nj416455app2">B</xref> describes three examples where our theory is exactly solvable, including a case closely related to the Schwarzschild black hole. The deviations from formula (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>) are remarkably small, except when the velocity profile is asymmetric (figure <xref ref-type="fig" rid="nj416455fig5">5</xref>). However, as they enter the mode-conversion rate (<xref ref-type="disp-formula" rid="nj416455eqn5">5</xref>) exponentially, they are observable in laboratory analogues within the validity range of our analytical method, see appendix <xref ref-type="sec" rid="nj416455app3">C</xref>. What lies beyond Hawking's theory is now at the horizon, coming into view.</p><fig id="nj416455fig4" position="float"><label>Figure 4.</label><caption id="nj416455fc4"><p>Variation of the Hawking temperature <italic>T</italic> with frequency <italic>ω</italic>. The analytical results (solid curves) based on formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) agree well with numerical calculations (dots), except close to the frequency where a group-velocity horizon ceases to exist and for steep velocity profiles. For <italic>ω</italic> → 0 the curves agree with Hawking's prediction (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>), but for larger <italic>ω</italic> they deviate. We employed the dispersion relation of figure <xref ref-type="fig" rid="nj416455fig1">1</xref> and the velocity profile used in figure <xref ref-type="fig" rid="nj416455fig2">2</xref> with three different length scales <italic>a</italic> in units of 1/<italic>k</italic><sub>0</sub>, and <italic>ω</italic> and <italic>T</italic> are given in units of <italic>ω</italic><sub>0</sub> and ℏ<italic>ω</italic><sub>0</sub>/<italic>k</italic><sub>B</sub>, respectively.</p></caption><graphic id="nj416455f4_eps" content-type="print" xlink:href="nj416455f4_pr.eps"></graphic><graphic id="nj416455f4_online" content-type="online" xlink:href="nj416455f4_online.jpg"></graphic></fig><fig id="nj416455fig5" position="float"><label>Figure 5.</label><caption id="nj416455fc5"><p>Variation of the Hawking temperature <italic>T</italic> with frequency <italic>ω</italic> for an asymmetric velocity profile. Here we assume the formula for the profile used in figure <xref ref-type="fig" rid="nj416455fig2">2</xref> with the parameters <italic>u</italic><sub>R</sub> = −0.7<italic>c</italic><sub>0</sub> and <italic>u</italic><sub>L</sub> = −1.1<italic>c</italic><sub>0</sub>; otherwise the situation is identical to figure <xref ref-type="fig" rid="nj416455fig4">4</xref>. We see that the deviations from Hawking's formula (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>) are more pronounced than for the symmetric profile assumed in figure <xref ref-type="fig" rid="nj416455fig4">4</xref>.</p></caption><graphic id="nj416455f5_eps" content-type="print" xlink:href="nj416455f5_pr.eps"></graphic><graphic id="nj416455f5_online" content-type="online" xlink:href="nj416455f5_online.jpg"></graphic></fig></sec></body><back><ack><title>Acknowledgments</title><p>We are grateful to Simon Horsley, Renaud Parentani and Thomas Philbin for valuable discussions. In particular we thank Renaud Parentani for pointing out that the negative-wavenumber component in the out-going mode gives directly <italic>β</italic>/<italic>α</italic>. Our work is supported by EPSRC and the Royal Society.</p></ack><app-group id="nj416455app"><app id="nj416455app1"><label>Appendix A.</label><p>In this appendix we describe the main method for obtaining our analytical results, the phase-integral method in <italic>k</italic>-space [<xref ref-type="bibr" rid="nj416455bib19">19</xref>]. Corley [<xref ref-type="bibr" rid="nj416455bib23">23</xref>] used phase integrals in position space, but horizons are better defined in <italic>k</italic>-space; Corley's method is restricted to linear velocity profiles, whereas our method works for arbitrary <italic>u</italic>(<italic>z</italic>).</p><p>We begin with an analysis of the classical wave equation in moving media in position space [<xref ref-type="bibr" rid="nj416455bib18">18</xref>]: <disp-formula id="nj416455eqnA.1"><label>A.1</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.1.gif"></graphic></disp-formula> For solutions <italic>φ</italic><sub>1</sub> and <italic>φ</italic><sub>2</sub> of equation (<xref ref-type="disp-formula" rid="nj416455eqnA.1">A.1</xref>) we define the scalar product [<xref ref-type="bibr" rid="nj416455bib05">5</xref>] <disp-formula id="nj416455eqnA.2"><label>A.2</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.2.gif"></graphic></disp-formula> that is a conserved quantity [<xref ref-type="bibr" rid="nj416455bib18">18</xref>] related to the number of particles associated with a given wave. In a region of uniform flow velocity the solutions are the plane waves <disp-formula id="nj416455eqnA.3"><label>A.3</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.3.gif"></graphic></disp-formula> for the roots <italic>k</italic><sub><italic>j</italic></sub> of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) shown in figure <xref ref-type="fig" rid="nj416455fig1">1</xref>. We have normalized the plane waves (<xref ref-type="disp-formula" rid="nj416455eqnA.3">A.3</xref>) to a delta function in <italic>ω</italic>. Note that <italic>φ</italic><sub><italic>u</italic>r</sub> and <italic>φ</italic><sub><italic>u</italic>l</sub> have positive norm, while the norm of <italic>φ</italic><sub><italic>u</italic></sub> is negative.</p><p>When <italic>u</italic>(<italic>z</italic>) varies, asymptotically plane waves (<xref ref-type="disp-formula" rid="nj416455eqnA.3">A.3</xref>) are partially converted into each other. Here it is advantageous [<xref ref-type="bibr" rid="nj416455bib19">19</xref>] to consider the spatial Fourier transform <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn3.gif"></inline-graphic></inline-formula>. We derive from equation (<xref ref-type="disp-formula" rid="nj416455eqnA.1">A.1</xref>) the wave equation in <italic>k</italic>-space: <disp-formula id="nj416455eqnA.4"><label>A.4</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.4.gif"></graphic></disp-formula> For slowly varying velocity profiles we approximate <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn4.gif"></inline-graphic></inline-formula> using a phase-integral method in
<italic>k</italic>-space. For this, we regard <italic>u</italic>(i∂<sub><italic>k</italic></sub>) as <italic>u</italic>(i<italic>ε</italic>∂<sub><italic>k</italic></sub>) where <italic>ε</italic> is a small parameter, we represent <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn5.gif"></inline-graphic></inline-formula> as <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn6.gif"></inline-graphic></inline-formula> and substitute this ansatz in equation (<xref ref-type="disp-formula" rid="nj416455eqnA.4">A.4</xref>). Sorting the result into powers of <italic>ε</italic> produces a coupled system of equations for the <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn7.gif"></inline-graphic></inline-formula> that we truncate at <italic>m</italic> = 1. Finally we remove <italic>ε</italic>, incorporating it in the scale of <italic>u</italic>(<italic>z</italic>), by formally setting <italic>ε</italic> = 1. In this way we obtain the square of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) with <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn8.gif"></inline-graphic></inline-formula> and <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn9.gif"></inline-graphic></inline-formula>, which gives in <italic>k</italic>-space <disp-formula id="nj416455eqnA.5"><label>A.5</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.5.gif"></graphic></disp-formula> To deduce <italic>φ</italic> in <italic>z</italic>-space, we evaluate the inverse Fourier transformation in saddle-point approximation: <disp-formula id="nj416455eqnA.6"><label>A.6</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.6.gif"></graphic></disp-formula> where, for Φ<sub>0</sub> = 1/(4<italic>π</italic><sup>2</sup>), the <italic>φ</italic><sub><italic>j</italic></sub> are the expressions (<xref ref-type="disp-formula" rid="nj416455eqnA.3">A.3</xref>) for the normalized plane waves with the solutions <italic>k</italic><sub><italic>j</italic></sub> of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>), but now <italic>z</italic> may vary. The phases and amplitudes of the components <italic>φ</italic><sub><italic>j</italic></sub> may depend on the contours of the phase integral in equation (<xref ref-type="disp-formula" rid="nj416455eqnA.6">A.6</xref>), although not on their form, but on their topology. Choosing topologically different contours [<xref ref-type="bibr" rid="nj416455bib23">23</xref>] we obtain the various incident and out-going waves of figure <xref ref-type="fig" rid="nj416455fig2">2</xref>.</p><p>Figure <xref ref-type="fig" rid="nj416455fig3">3</xref> shows the contour that distinguishes the out-going mode <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> of the escaping Hawking radiation defined in figure <xref ref-type="fig" rid="nj416455fig2">2</xref>. The contour is chosen by the following argument. On the left side of the group-velocity horizon, <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> decays exponentially; therefore the only physically allowed wavenumber is <italic>k</italic><sub><italic>u</italic>l</sub> with negative imaginary part, which in figure <xref ref-type="fig" rid="nj416455fig3">3</xref> lies on the lower half of the red ‘cross bow’. The contour for <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> (dotted curve) comes in from ∞ in a quadrant where <italic>z</italic>(<italic>k</italic>) has a negative imaginary part such that the phase integral in expression (<xref ref-type="disp-formula" rid="nj416455eqnA.5">A.5</xref>) vanishes there. On the right side of the horizon, we have the three saddle points <italic>k</italic><sub><italic>u</italic>l</sub>, <italic>k</italic><sub><italic>u</italic>r</sub> and <italic>k</italic><sub><italic>u</italic></sub> shown as blue dots. They correspond to the three partial waves (figure <xref ref-type="fig" rid="nj416455fig2">2</xref>) of <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub>. The contour for <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> for the right side of the horizon must be a deformation of the contour for the left side and pass through the physically relevant saddle points, which determines the contour shown in figure <xref ref-type="fig" rid="nj416455fig3">3</xref>.</p><p>The out-going mode <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> is the superposition (<xref ref-type="disp-formula" rid="nj416455eqn3">3</xref>) of the in-going modes <italic>φ</italic><sup>in</sup><sub><italic>u</italic>l</sub> and <italic>φ</italic><sup>in</sup><sub><italic>u</italic></sub>. On the right side of the horizon, the <italic>k</italic><sub><italic>u</italic>l</sub>-component comes from <italic>α</italic> <italic>φ</italic><sup>in</sup><sub><italic>u</italic>l</sub> and the <italic>k</italic><sub><italic>u</italic></sub>-component from <italic>β</italic> <italic>φ</italic><sup>in</sup><sub><italic>u</italic></sub>. The <italic>k</italic><sub><italic>u</italic>l</sub>-component can only differ from the <italic>k</italic><sub><italic>u</italic>r</sub>-component by a phase factor, because the <italic>z</italic>(<italic>k</italic>) integral between <italic>k</italic><sub><italic>u</italic>r</sub> and <italic>k</italic><sub><italic>u</italic>l</sub> is real on the real axis and hence real for any contour. We conclude that |<italic>β</italic>/<italic>α</italic>| corresponds to the relative weight of the <italic>k</italic><sub><italic>u</italic></sub>-component in <italic>φ</italic><sup>out</sup><sub><italic>u</italic>r</sub> that we can read off from our result (<xref ref-type="disp-formula" rid="nj416455eqnA.6">A.6</xref>). One advantage of this method is that we infer |<italic>β</italic>/<italic>α</italic>| from partial waves with high wavenumbers where the saddle-point approximation required for deriving expression (<xref ref-type="disp-formula" rid="nj416455eqnA.6">A.6</xref>) is sufficiently accurate. According to expression (<xref ref-type="disp-formula" rid="nj416455eqnA.6">A.6</xref>) the amplitude of the <italic>k</italic><sub><italic>u</italic></sub>-component is given by <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn10.gif"></inline-graphic></inline-formula> taking the lower half of the integration contour in figure <xref ref-type="fig" rid="nj416455fig3">3</xref>. Since the velocity profile is real for real <italic>z</italic>, its analytic continuation on the complex plane obeys <italic>u</italic>(<italic>z</italic>*) = <italic>u</italic>*(<italic>z</italic>), and the same must be true for <italic>z</italic>(<italic>k</italic>). Therefore, although we integrate from the negative to the positive wavenumber, we may close the contour on the upper half of the complex <italic>k</italic>-plane. The real part of the integral then vanishes, while the imaginary part is doubled. Exponentiating gives |<italic>β</italic>/<italic>α</italic>|<sup>2</sup> and thus the formula from which our result (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) follows: <disp-formula id="nj416455eqnA.7"><label>A.7</label><tex-math></tex-math><graphic xlink:href="nj416455eqnA.7.gif"></graphic></disp-formula></p></app><app id="nj416455app2"><label>Appendix B.</label><p>In this appendix we consider three simple examples where our formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) admits exact solutions for dispersion relations where <italic>c</italic>(<italic>k</italic>) is an analytic function in <italic>k</italic>: the linear velocity profile, the <italic>z</italic><sup>−1</sup> profile and the <italic>z</italic><sup>−1/2</sup> profile. The examples show how our theory is to be applied. Furthermore, the <italic>z</italic><sup>−1/2</sup> profile corresponds to a prominent case: the Schwarzschild black hole [<xref ref-type="bibr" rid="nj416455bib08">8</xref>].</p><p>Formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) requires <italic>z</italic> as a function of <italic>k</italic> for a given (angular) frequency <italic>ω</italic>. We obtain from the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>): <disp-formula id="nj416455eqnB.1"><label>B.1</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.1.gif"></graphic></disp-formula> To obtain <italic>z</italic>(<italic>k</italic>) we thus need to invert the function <italic>u</italic>(<italic>z</italic>) and substitute <italic>ω</italic>/<italic>k</italic> − <italic>c</italic>(<italic>k</italic>) for <italic>u</italic>. Consider first a linear velocity profile, <disp-formula id="nj416455eqnB.2"><label>B.2</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.2.gif"></graphic></disp-formula> where we get <disp-formula id="nj416455eqnB.3"><label>B.3</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.3.gif"></graphic></disp-formula> Assuming <italic>c</italic>(<italic>k</italic>) to be an analytic function, the only contribution to the contour integral (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) is the <italic>ω</italic>/(<italic>α</italic><italic>k</italic>) term, which gives <disp-formula id="nj416455eqnB.4"><label>B.4</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.4.gif"></graphic></disp-formula> Hawking's result (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>). We see that the Hawking radiation caused by a linear velocity profile does not depend on <italic>c</italic>(<italic>k</italic>): it is insensitive to dispersion. The wavenumber-dependance of <italic>c</italic> can only play a role in nonlinear profiles.</p><p>Consider a situation where <italic>u</italic> is inversely proportional to <italic>z</italic>. Although this is an artificial case, it has the advantage that the entire wave propagation is exactly solvable for analytic <italic>c</italic>(<italic>k</italic>) [<xref ref-type="bibr" rid="nj416455bib26">26</xref>]. Let us see whether our theory reproduces the known result [<xref ref-type="bibr" rid="nj416455bib26">26</xref>] for Hawking radiation. Assuming <disp-formula id="nj416455eqnB.5"><label>B.5</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.5.gif"></graphic></disp-formula> we obtain <disp-formula id="nj416455eqnB.6"><label>B.6</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.6.gif"></graphic></disp-formula> This expression has single poles at the zeros of the denominator that correspond to the solutions of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) for <italic>z</italic> → ∞ where the velocity (<xref ref-type="disp-formula" rid="nj416455eqnB.5">B.5</xref>) of the medium vanishes, i.e. <disp-formula id="nj416455eqnB.7"><label>B.7</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.7.gif"></graphic></disp-formula> As the integration contour (figure <xref ref-type="fig" rid="nj416455fig3">3</xref>) encloses the <italic>k</italic><sub><italic>u</italic>r</sub> of the outgoing wave, we take the <italic>k</italic><sub>∞</sub> that corresponds to <italic>k</italic><sub><italic>u</italic>r</sub> at ∞. We evaluate formula (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>) using Cauchy's residue theorem where we require expression (<xref ref-type="disp-formula" rid="nj416455eqnB.6">B.6</xref>) for <italic>k</italic> near the pole <italic>k</italic><sub>∞</sub>. Here <disp-formula id="nj416455eqnB.8"><label>B.8</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.8.gif"></graphic></disp-formula> where <italic>v</italic><sub>g</sub> denotes the group velocity. We obtain from the residue theorem
<disp-formula id="nj416455eqnB.9"><label>B.9</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.9.gif"></graphic></disp-formula> which is the previously known exact result for the 1/<italic>z</italic> profile [<xref ref-type="bibr" rid="nj416455bib26">26</xref>]. It shows that the dispersion influences the spectrum of Hawking radiation; it is no longer a Planck spectrum. We can cast expression (<xref ref-type="disp-formula" rid="nj416455eqnB.9">B.9</xref>) in a form where we immediately see how it reproduces Hawking's result for zero dispersion. Let us define the phase horizon <italic>z</italic><sub>∞</sub> as the position where the medium reaches the phase velocity at <italic>k</italic><sub>∞</sub>: <disp-formula id="nj416455eqnB.10"><label>B.10</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.10.gif"></graphic></disp-formula> For the profile (<xref ref-type="disp-formula" rid="nj416455eqnB.5">B.5</xref>) the velocity gradient at the phase horizon is <disp-formula id="nj416455eqnB.11"><label>B.11</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.11.gif"></graphic></disp-formula> and so we obtain from the result (<xref ref-type="disp-formula" rid="nj416455eqnB.9">B.9</xref>) <disp-formula id="nj416455eqnB.12"><label>B.12</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.12.gif"></graphic></disp-formula> In the absence of dispersion the group velocity agrees with the phase velocity, which gives Hawking's result (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>).</p><p>Finally, consider a velocity profile that closely corresponds to the Schwarzschild black hole. In Painlevé–Gullstrand coordinates the Schwarzschild solution appears like a medium moving with velocity profile <italic>u</italic> = −<italic>c</italic> (<italic>a</italic>/<italic>r</italic>)<sup>−1/2</sup> [<xref ref-type="bibr" rid="nj416455bib08">8</xref>], where <italic>r</italic> is the radius (the distance to the singularity), <italic>a</italic> the Schwarzschild radius and <italic>c</italic> the speed of light. Considering only the wave propagation in <italic>r</italic> direction and identifying <italic>r</italic> with <italic>z</italic> we assume <disp-formula id="nj416455eqnB.13"><label>B.13</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.13.gif"></graphic></disp-formula> In this case, <disp-formula id="nj416455eqnB.14"><label>B.14</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.14.gif"></graphic></disp-formula> Similar to the profile (<xref ref-type="disp-formula" rid="nj416455eqnB.5">B.5</xref>) the function <italic>z</italic>(<italic>k</italic>) has poles at <italic>k</italic><sub>∞</sub> that are solutions (<xref ref-type="disp-formula" rid="nj416455eqnB.7">B.7</xref>) of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) at <italic>z</italic> → ∞. As before, we take the <italic>k</italic><sub>∞</sub> that corresponds to <italic>k</italic><sub><italic>u</italic>r</sub> at ∞. However, due to the square in expression (<xref ref-type="disp-formula" rid="nj416455eqnB.14">B.14</xref>), <italic>z</italic>(<italic>k</italic>) has both a double and a single pole at <italic>k</italic><sub>∞</sub>. Only the single pole contributes to the contour integral (<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>). We extract the single pole and obtain from Cauchy's residue theorem <disp-formula id="nj416455eqnB.15"><label>B.15</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.15.gif"></graphic></disp-formula> As in the case of the <italic>z</italic><sup>−1</sup> profile we calculate the velocity gradient <italic>α</italic> at the phase horizon (<xref ref-type="disp-formula" rid="nj416455eqnB.10">B.10</xref>), where we get for the <italic>z</italic><sup>−1/2</sup> profile (<xref ref-type="disp-formula" rid="nj416455eqnB.13">B.13</xref>) <disp-formula id="nj416455eqnB.16"><label>B.16</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.16.gif"></graphic></disp-formula> In terms of <italic>α</italic> we obtain from expression (<xref ref-type="disp-formula" rid="nj416455eqnB.15">B.15</xref>) and the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqnB.7">B.7</xref>) the result <disp-formula id="nj416455eqnB.17"><label>B.17</label><tex-math></tex-math><graphic xlink:href="nj416455eqnB.17.gif"></graphic></disp-formula> This formula describes how the Hawking radiation of the black hole is modified if space were dispersive [<xref ref-type="bibr" rid="nj416455bib03">3</xref>]. For instance, if space were discrete at the Planck scale, this discreteness would first appear as a wavenumber dependance of the speed of light [<xref ref-type="bibr" rid="nj416455bib30">30</xref>], i.e. as dispersion. Our formula (<xref ref-type="disp-formula" rid="nj416455eqnB.17">B.17</xref>) would describe the non-Planckian spectrum of Hawking radiation due to the Planck scale.</p></app><app id="nj416455app3"><label>Appendix C.</label><p>In this appendix we apply our theory to an example of direct experimental relevance, the observation of stimulated Hawking radiation with water waves [<xref ref-type="bibr" rid="nj416455bib16">16</xref>]. We show that deviations from Hawking's prediction of a thermal spectrum are observable in the regime where our analytical theory works best.</p><p>Consider water waves in a channel of varying height <italic>h</italic> (figure <xref ref-type="fig" rid="nj416455figC1">C.1</xref>) where the water flows with a velocity profile <italic>u</italic>(<italic>z</italic>) that can be regulated by <italic>h</italic> and the initial speed of the water. Water waves—gravity waves—obey the dispersion relation [<xref ref-type="bibr" rid="nj416455bib37">37</xref>] <disp-formula id="nj416455eqnC.1"><label>C.1</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.1.gif"></graphic></disp-formula> where <italic>g</italic> denotes the gravitational acceleration of the Earth at the water surface. In the limit of long wavelengths, i.e. small wavenumbers <italic>k</italic>, the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqnC.1">C.1</xref>) reduces to <disp-formula id="nj416455eqnC.2"><label>C.2</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.2.gif"></graphic></disp-formula></p><fig id="nj416455figC1" position="float"><label>Figure C.1.</label><caption id="nj416455fcC1"><p>Aquatic analogue of the event horizon. Water waves experience an effective event horizon when the flow speed exceeds the speed of the waves. In practice [<xref ref-type="bibr" rid="nj416455bib16">16</xref>], this is a white-hole horizon. The white hole is simply the time-reverse of the black hole. In the aquatic analogue of the white hole the magnitude of the flow is rising at the horizon from sub- to super-wave speed (at the black-hole horizon the flow is falling from super- to sub-wave speed). An incident wave packet (top) is sent against the rising current. It propagates until the flow speed reaches the group velocity of the wave, whereupon it is converted into two wave packets (bottom), one with positive wavenumber (bottom, right) and one with negative wavenumber (bottom, left).</p></caption><graphic id="nj416455fC.1_eps" content-type="print" xlink:href="nj416455fC.1_pr.eps"></graphic><graphic id="nj416455fC.1_online" content-type="online" xlink:href="nj416455fC.1_online.jpg"></graphic></fig><p>The height <italic>h</italic> plays a triple role. Firstly, as we have seen, it determines the wave velocity <italic>c</italic><sub>0</sub> for long waves. Secondly, the dispersion of the wave velocity <italic>c</italic>—the deviation from <italic>c</italic><sub>0</sub>—is characterized by <italic>h</italic> as well, which corresponds to the trans-Planckian parameter in the water-wave analogue of the event horizon. Thirdly, the height determines the velocity profile, because, assuming the water to be incompressible and the flow to be steady, the velocity profile is given by <disp-formula id="nj416455eqnC.3"><label>C.3</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.3.gif"></graphic></disp-formula> where the constant <italic>q</italic> is the 2D flow rate per unit. The water shall flow to the left, hence the negative sign. Note that, for water waves in a channel of varying height, the phase velocity <italic>c</italic> depends on both <italic>k</italic><sup>2</sup> and <italic>z</italic>. We extend the theory of the main part of the paper to this case. For simplicity, we assume the profile (figure <xref ref-type="fig" rid="nj416455figC1">C.1</xref>) <disp-formula id="nj416455eqnC.4"><label>C.4</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.4.gif"></graphic></disp-formula> At the group-velocity horizon the two positive-wavenumber solutions of the dispersion relation (<xref ref-type="disp-formula" rid="nj416455eqn1">1</xref>) coincide, which happens at the maximum of (<italic>u</italic> + <italic>c</italic>)<italic>k</italic> seen as a function of <italic>k</italic>; the horizon thus appears at the position for which the curve [<italic>u</italic>(<italic>z</italic>) + <italic>c</italic>]<italic>k</italic> over the <italic>k</italic>-axis has its maximum at the value <italic>ω</italic>. With increasing <italic>k</italic> the horizon wanders out to increasing values of <italic>u</italic>. Therefore, a group-velocity horizon exists for all frequencies <italic>ω</italic> below the maximum <italic>ω</italic><sub>0</sub> of (<italic>u</italic> + <italic>c</italic>)<italic>k</italic> within maximal flow velocity, here <italic>u</italic><sub>L</sub> = −<italic>q</italic>/<italic>h</italic><sub>L</sub>.</p><p>The horizon converts incident waves with positive wavenumber <italic>k</italic> into waves with negative <italic>k</italic>, which is the classical analogue of Hawking radiation. According to our theory, the relative intensity |<italic>β</italic>|<sup>2</sup> of the negative-<italic>k</italic> waves is given by equations (<xref ref-type="disp-formula" rid="nj416455eqn4">4</xref>)–(<xref ref-type="disp-formula" rid="nj416455eqn6">6</xref>), or, in explicit form <disp-formula id="nj416455eqnC.5"><label>C.5</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.5.gif"></graphic></disp-formula> The contour integral connects positive with negative wavenumbers. We can choose any contour, as long as we do not cross branches, because the value of the integral is contour-independent. For our calculation we chose an ellipse between the positive and the negative <italic>k</italic> value for the asymptotic flow. Figure <xref ref-type="fig" rid="nj416455figC2">C.2</xref> compares |<italic>β</italic>|<sup>2</sup> with the analogue of Hawking's prediction, formula (<xref ref-type="disp-formula" rid="nj416455eqn8">8</xref>) in equations (<xref ref-type="disp-formula" rid="nj416455eqn4">4</xref>) and (<xref ref-type="disp-formula" rid="nj416455eqn5">5</xref>) that gives the Planck distribution <disp-formula id="nj416455eqnC.6"><label>C.6</label><tex-math></tex-math><graphic xlink:href="nj416455eqnC.6.gif"></graphic></disp-formula> where <italic>u</italic>' is taken at the horizon for the limit <italic>ω</italic> → 0. We see that the ratio |<italic>β</italic>|<sup>2</sup>/|<italic>β</italic><sub>0</sub>|<sup>2</sup> can deviate significantly from 1, in particular for slow variations of the flow velocity over a large length scale <italic>a</italic>, which is the regime where our theory is valid. Therefore, deviations of the Hawking spectrum from the Planck distribution (<xref ref-type="disp-formula" rid="nj416455eqnC.6">C.6</xref>) seem observable in laboratory analogues of the event horizon.</p><fig id="nj416455figC2" position="float"><label>Figure C.2.</label><caption id="nj416455fcC2"><p>Deviations of the Hawking spectrum (<xref ref-type="disp-formula" rid="nj416455eqnC.5">C.5</xref>) from the Planck distribution (<xref ref-type="disp-formula" rid="nj416455eqnC.6">C.6</xref>). We plot |<italic>β</italic>|<sup>2</sup>/|<italic>β</italic><sub>0</sub>|<sup>2</sup> as a function of <italic>ω</italic> (in units of <italic>c</italic><sub>0</sub>/<italic>h</italic><sub>0</sub>) for different scales <italic>a</italic> of the velocity profile (equations (<xref ref-type="disp-formula" rid="nj416455eqnC.3">C.3</xref>) and (<xref ref-type="disp-formula" rid="nj416455eqnC.4">C.4</xref>)) where <italic>u</italic><sub>L</sub> = −0.8<italic>c</italic><sub>0</sub> and <italic>u</italic><sub>R</sub> = −1.2<italic>c</italic><sub>0</sub>, with <inline-formula><tex-math></tex-math><inline-graphic xlink:href="nj416455ieqn11.gif"></inline-graphic></inline-formula> and 1/<italic>a</italic>∈{0.05,0.1,0.15,0.2,0.25} in units of 1/<italic>h</italic><sub>0</sub>.</p></caption><graphic id="nj416455fC.2_eps" content-type="print" xlink:href="nj416455fC.2_pr.eps"></graphic><graphic id="nj416455fC.2_online" content-type="online" xlink:href="nj416455fC.2_online.jpg"></graphic></fig></app></app-group><ref-list content-type="numerical"><title>References</title><ref id="nj416455bib01"><label>1</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hawking</surname><given-names>S W</given-names></name></person-group><year>1974</year><source>Nature</source><volume>248</volume><fpage>30</fpage><pub-id pub-id-type="doi">10.1038/248030a0</pub-id></element-citation></ref><ref id="nj416455bib02"><label>2</label><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Hawking</surname><given-names>S W</given-names></name></person-group><year>1975</year><source>Commun. 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Recently, laboratory analogues of the event horizon have reached the level where tests of Hawking's idea are possible. In this paper we show how to go beyond Hawking's theory in such laboratory analogues in a way that is experimentally testable.</abstract><note type="footnotes">Both authors contributed equally to this work.</note><relatedItem type="host"><titleInfo><title>New Journal of Physics</title></titleInfo><genre type="journal">journal</genre><identifier type="eISSN">1367-2630</identifier><identifier type="PublisherID">nj</identifier><part><date>2012</date><detail type="volume"><caption>vol.</caption><number>14</number></detail><detail type="issue"><caption>no.</caption><number>5</number></detail><extent unit="pages"><total>15</total></extent></part></relatedItem><identifier type="istex">A0A7E0003B243B55F82A506D8750C879A8C9FB41</identifier><identifier type="DOI">10.1088/1367-2630/14/5/053003</identifier><identifier type="href">http://stacks.iop.org/NJP/14/053003</identifier><identifier type="ArticleID">nj416455</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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