Simplified derivation of the Hawking–Unruh temperature for an accelerated observer in vacuum Paul M. Alsing and Peter W. Milonni
Citation: American Journal of Physics 72, 1524 (2004); doi: 10.1119/1.1761064 View online: http://dx.doi.org/10.1119/1.1761064 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/72/12?ver=pdfcov Published by the American Association of Physics Teachers
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 Simplified derivation of the Hawking–Unruh temperature for an accelerated observer in vacuum Paul M. Alsinga) Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131 Peter W. Milonnib) Theoretical Division (T-DOT), Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ͑Received 27 January 2004; accepted 23 April 2004͒ A detector undergoing uniform acceleration a in a vacuum field responds as though it were immersed in thermal radiation of temperature Tϭបa/2kc. An intuitive derivation of this result is given for a scalar field in one spatial dimension. The approach is extended to the case where the field detected by the accelerated observer is a spin 1/2 Dirac field. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1761064͔
I. INTRODUCTION Dirac particles of spin 1/2 ͑fermions͒ brings in a whole host of new machinery, the least of which is the formulation of Hawking1 predicted that a black hole should radiate with a the Dirac equation in curvilinear coordinates ͑that is, in temperature Tϭបg/2kc, where g is the gravitational accel- curved space–time͒. This formulation quickly goes beyond eration at the surface of the black hole, k is Boltzmann’s the expertise of most nonexperts. However, in both the boson constant, and c is the speed of light ͑the Hawking effect͒. The and fermion cases, the beautiful and important results of radiation results from the effect of the strong gravitation on Hawking, Davies, and Unruh can be stated quite simply: for the vacuum field. Shortly thereafter it was shown separately a scalar field ͑bosons͒ a detector carried by an accelerated by Davies and Unruh that a uniformly accelerated detector observer detects a Bose–Einstein ͑BE͒ number distribution moving through the usual flat space–time vacuum of con- of particles at temperature T given by Eq. ͑1͒, while for a vential quantum field theory responds as though it were in a spin 1/2 field ͑fermions͒ a detector carried by an accelerated thermal field of temperature2–6 observer detects a Fermi–Dirac number distribution at the same temperature. បa The purpose of this paper is to present a simplified deri- Tϭ , ͑1͒ 2kc vation of Eq. ͑1͒ that is suitable for advanced undergraduate or beginning graduate students and elucidates the essential where a is the acceleration in the instantaneous rest frame of underlying physics of the flat space–time Davies–Unruh ef- the detector ͑the Davies–Unruh effect͒. We will refer to Eq. fect. Once the simplest features of a quantized vacuum field ͑1͒ as the Hawking–Unruh temperature because its applica- are accepted, Eq. ͑1͒ emerges as a consequence of time- bility to flat space–time accelerated detectors or to stationary dependent Doppler shifts in the field detected by the accel- detectors situated outside the horizon of a black hole7 de- erated observer. pends only on the interpretation of the source of the accel- In Sec. II the essential features of uniform acceleration are eration a. reviewed, and in Sec. III we use these results to obtain Eq. The results of Hawking, Davies, and Unruh suggest pro- ͑1͒ in an almost trivial way based on the Doppler effect. In found consequences for the merger of quantum field theory Sec. IV this simple approach to the derivation of the tem- and general relativity and sparked intense debates over unre- perature in Eq. ͑1͒ is developed in more detail. In Sec. V we solved questions that are still being actively investigated. If extend the previous calculations for scalar fields to spin 1/2 black holes are not really ‘‘black,’’ are naked singularities the Dirac fields. We close with a brief summary and discussion. ultimate fate of black holes, or will the long-sought fusion of quantum mechanics and general relativity into a coherent II. UNIFORM ACCELERATION theory of quantum gravity prevent such occurrences? If a quantum mechanical pure state is dropped into a black hole We will refer to an observer moving with constant velocity and pure thermal ͑uncorrelated͒ radiation results, how does in flat space–time as a Minkowski observer and refer to a 8 one explain the apparent nonunitary evolution of the pure Rindler observer as one who travels with uniform accelera- state to a mixed state? tion in the positive z direction with respect to the former. The intriguing consequence of quantum field theory for Uniform acceleration is defined as a constant acceleration a accelerated detectors indicated by Eq. ͑1͒ has not been de- in an instantaneous inertial frame in which the ͑Rindler͒ ob- rived in a physically intuitive way. Numerous explicit and server is at rest. The acceleration dv/dt of the Rindler ob- detailed calculations have appeared in the scientific literature server as measured in the lab frame ͑that is, by the over the last 30 years for a wide variety of space–times. Minkowski observer͒ is given in terms of a by a Lorentz However, even for the simplest calculation involving a scalar transformation formula which relates the acceleration in the field ͑bosons͒, the intricacies of field theory techniques, two frames9 coupled with a forest of special function properties, make dv v2 3/2 most derivations intractable for the nonspecialist. An inves- ϭaͩ 1Ϫ ͪ . ͑2͒ tigation of the flat space–time Davies–Unruh effect for dt c2
1524 Am. J. Phys. 72 ͑12͒, December 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 1524 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 ͑ ͒ ϭ ϭ ϱ ϱ If we integrate Eq. 2 and take v 0att 0, we have ⍀ a/c c ⍀ Ϫ i i( Kc/a)e ϭ (i c/a 1) i( Kc/a)y v(t)ϭat/ͱ1ϩa2t2/c2. The relation dtϭd/ͱ1Ϫv2/c2 be- ͵ d e e ͵ dy y e Ϫϱ a 0 tween the lab time, t, and the proper time, , for the accel- ϭ c i⍀c erated observer gives t( ) (c/a)sinh(a /c) if we take t( ϭ ⌫ͩ ͪ ϭ0)ϭ0. The velocity v of the accelerated observer as de- a a tected from the lab frame can be expressed in terms of the Ϫi⍀c/a Kc proper time as ϫͩ ͪ eϪ⍀c/2a, ͑8͒ a a 10 v͑͒ϭc tanhͩ ͪ . ͑3͒ where ⌫ is the gamma function. Then, because c ͉⌫(i⍀c/a)͉2ϭ/͓(⍀c/a)sinh(⍀c/a)͔,11 we obtain ϱ 2 A straightforward integration of Eq. ͑3͒ using v(t()) ⍀ a/c 2 c 1 ͵ i i( Kc/a)e ϭ ͑ ͒ ϭ ϭ ͯ d e e ͯ ⍀ 2⍀c/aϪ . 9 dz/dt dz( )/d •d /dt yields the well-known hyperbolic Ϫϱ a e 1 orbit of the accelerated, Rindler observer in the z direction:8 The time-dependent Doppler shift detected by the acceler- ប⍀/kT c a c2 a ated observer therefore leads to the Planck factor (e t͑͒ϭ sinhͩ ͪ , z͑͒ϭ coshͩ ͪ . ͑4͒ Ϫ1)Ϫ1 indicative of a Bose–Einstein distribution for scalar a c a c ͑boson͒ particles with Tϵបa/2kc, which is just Eq. ͑1͒. ϩ We will consider aϾ0, that is, the observer accelerates in the We obtain the same result for a wave propagating in the z direction. positive z direction. Note that the time-dependent phase can also be obtained directly by considering the standard nonaccelerated ϵ Ϯ Minkowski plane wave exp͓i Ϯ͔ exp͓i(Kz Kt)͔ and sub- ͑ ͒ ϭ Ϯ ϭ III. INDICATION OF THERMAL EFFECT OF stituting Eq. 4 : Ϯ( ) Kz( ) Kt( ) ( Kc/a) ϫ Ϯ ϭ 12,13 ACCELERATION exp( a /c) with K K /c. Consider a plane wave field of frequency K and wave vector K parallel or anti-parallel to the z direction along IV. A MORE FORMAL DERIVATION which the observer is accelerated. In the instantaneous rest Ј The derivation of the temperature in Eq. ͑1͒ leaves much frame of the observer, the frequency K of this field is given by the Lorentz transformation to be desired. We have restricted ourselves to a single field frequency K , whereas a quantum field in vacuum has com- a ponents at all frequencies. Moreover, we have noted the ap- ͫ 1Ϫtanhͩ ͪͬ pearance of the Planck factor, but have not actually com- Ϫ ͑͒ K K Kv c pared our result to that appropriate to an observer at rest in a Ј ͑͒ϭ ϭ K thermal field ͑that is, a field in which the average number of ͱ1Ϫv2͑͒/c2 a ͱ1Ϫtanh2ͩ ͪ particles is given by a BE distribution for bosons or a Fermi– c Dirac distribution for fermions at a fixed temperature T). To rectify these deficiencies, let us consider a massless ϭ Ϫa/c ͑ ͒ Ke 5 scalar field in one spatial dimension (z), quantized in a box 14 ϭϩ of volume V: for K K /c, that is, for plane wave propagation along ប 2 1/2 the z direction of the observer’s acceleration. For propaga- 2 c Ϫ ˆ ϭ ͩ ͪ ͓ i Ktϩ † i Kt͔ ͑ ͒ Ϫ ͚ aˆ Ke aˆ Ke . 10 tion in the z direction, K KV Ј ͑͒ϭ a/c ͑ ͒ Here KϭϮ /c, and aˆ and aˆ † are, respectively, the an- K Ke , 6 K K K ͓ † ͔ nihilation and creation operators for mode K ( aˆ K ,aKЈ where KϭϪ /c. Note that for small values of a, Ј ϭ␦ ϭ K K KKЈ , ͓aˆ K ,aKЈ͔ 0). We use ˆ to denote quantum me- Х ϯ ͑ ͒ K(1 a /c), the familiar Doppler shift. Equations 5 chanical operators. The expectation value ͗(dˆ /dt)2͘/4c2 ͑ ͒ and 6 involve time-dependent Doppler shifts detected by Ϫ1͚ ប ͓͗ † ͘ of the energy density of this field is V K K aˆ Kaˆ K the accelerated observer. ϩ ͔ Because of these Doppler shifts, the accelerated observer 1/2 . For simplicity, we consider the field at a particular ͑ ϭ sees waves with a time-dependent phase () point in space say, z 0), because spatial variations of the ϵ͐Ј Ј Јϭ field will be of no consequence for our purposes. K( )d ( Kc/a)exp(a /c). We suppose therefore ͑ ͒ † Ϫ For a bosonic thermal state the number operator aˆ Kaˆ K that, for a wave propagating in the z direction, for which ប /kT Ϫ1 has the expectation value (e K Ϫ1) . Consider the ͐ Ј (Ј)dЈϭ( c/a)exp(a/c), the observer sees a fre- K K Fourier transform operator quency spectrum S(⍀) proportional to 1 ϱ i⍀t ϱ 2 gˆ ͑⍀͒ϭ ͵ dt ˆ e ⍀ a/c 2 Ϫϱ ͯ͵ d ei ei( Kc/a)e ͯ . ͑7͒ Ϫϱ 2បc2 1/2 ϭ͚ ͩ ͪ aˆ ␦͑ Ϫ⍀͒ ͑⍀Ͼ0͒. ͑11͒ a/c K K If we change variables to yϭe , we have K KV
1525 Am. J. Phys., Vol. 72, No. 12, December 2004 Paul M. Alsing and Peter W. Milonni 1525 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 The expectation value ͗gˆ †(⍀)gˆ (⍀Ј)͘ in thermal equilibrium which is identical to the thermal result Eq. ͑13͒ if we define is therefore the temperature by Eq. ͑1͒. † ϭ␦ Note that the expectation value ͗aˆ Ka ͘ KKЈ involves 2បc2 KЈ ͗ †͑⍀͒ ͑⍀Ј͒͘ϭ ͩ ͪ ͗ † ͘␦͑⍀Ϫ⍀Ј͒ the creation and annihilation operators of the accelerated ob- gˆ gˆ ͚ aˆ Kaˆ K K KV server and is taken with respect to the accelerated observer’s ϫ␦͑ Ϫ⍀͒ vacuum, which is different from the vacuum detected by the K nonaccelerated observer. This point is discussed more fully 2បc2 1 in Sec. VI. ϭ ͩ ͪ ͚ ប V e K /kTϪ1 K K V. FERMI–DIRAC STATISTICS FOR DIRAC ϫ␦ ⍀Ϫ⍀ ͒␦ Ϫ⍀͒ ͑ ͒ ͑ Ј ͑ K . 12 PARTICLES We go to the limit where the volume of the quantization box We have considered a scalar field and derived the Planck ប⍀ Ϫ becomes very large, V→ϱ, so that we can replace the sum factor (e /kTϪ1) 1 indicative of Bose–Einstein ͑BE͒ sta- ͚ → ͐ 15 over K by an integral: K (V/2 ) dK. Thus tistics. We began with the standard plane-wave solutions of Ϯ the form exp͓i(Kz Kt)͔ for the nonaccelerated Minkowski ϱ 1 1 ͗ †͑⍀͒ ͑⍀Ј͒͘ϭប 2 ͵ observer and considered the time-dependent Doppler shifts gˆ gˆ c dK ប⍀/kTϪ Ϫϱ K e 1 as detected by the accelerated observer. For spin 1/2 Dirac particles we would expect an analogous derivation to obtain ϫ␦͉͑K͉cϪ⍀͒␦͑⍀Ϫ⍀Ј͒ (eប⍀/kTϩ1)Ϫ1 indicative of Fermi–Dirac ͑FD͒ statistics. 2បc/⍀ Mathematically, the essential point involves the replace- ϭ ␦͑⍀Ϫ⍀Ј͒ ͑ ͒ ment i⍀c/a→i⍀c/aϩ1/2 in the integrals in Eqs. ͑7͒–͑9͒,16 ប⍀/kTϪ . 13 e 1 and the relation ͉⌫(i⍀c/aϩ1/2)͉2ϭ/cosh(⍀c/a).11 Now consider an observer in uniform acceleration in the Physically, this replacment arises from the additional spinor quantized vacuum field, that is, the particle free vacuum ap- nature of the Dirac wave function over that of the scalar propriate for the accelerated Rindler observer. This observer plane wave. For a scalar field, only the phase had to be sees each field frequency Doppler-shifted according to Eqs. instantaneously Lorentz-transformed to the comoving frame ͑ ͒ ͑ ͒ ⍀ of the accelerated observer. For non-zero spin, the spinor 5 and 6 , and so for him/her the operator gˆ ( ) has the 17 form structure of the particles must also be transformed, or Fermi–Walker transported18 along a particle’s trajectory to ϱ ប 2 1/2 1 ⍀ 2 c ensure that it does not ‘‘rotate’’ as it travels along the accel- gˆ ͑⍀͒ϭ ͵ d ei ͚ ͩ ͪ erated trajectory. Ensuring this ‘‘nonrotating’’ condition in 2 Ϫϱ V K K the observer’s instantaneous rest frame leads to a time- Ϫ R ЈЈ Ј † R Ј Ј dependent Lorentz transformation of the Dirac bispinor of ϫ͓ i d K( )ϩ i d K( )͔ aˆ Ke aˆ Ke the form19 Sˆ ()ϭexp(␥0 ␥3 a/2c)ϭcosh(a/2c) 1 ϱ 2បc2 1/2 ϩ␥0 ␥3 ϫ ϭ ͵ i⍀ sinh(a /2c), where the 4 4 constant Dirac matrices d e ͚ ͩ ͪ 2 Ϫϱ K KV are given by
⑀ Ϫ⑀ a/c Ϫ ⑀ Ϫ⑀ /c 10 0 ϫ͓ i( K Kc/a)e K ϩ † i( K Kc/a)e K K ͔ z aˆ Ke aˆ Ke , ␥0ϭͩ ͪ , ␥3ϭͩ ͪ , ͑17͒ 0 Ϫ1 Ϫ 0 ͑14͒ z ϭ Ϫ ϫ and z diagonal(1, 1) is the usual 2 2 Pauli spin matrix where ⑀ ϭ͉K͉/K. Because aˆ ͉vacuum͘ϭ0, only the aˆ † K K K in the z direction. If Sˆ () acts on a spin up state ͉↑͘ terms in Eq. ͑14͒ contribute to the vacuum expectation value ϭ͓ ͔T 20 ˆ ͉↑ ϭ ͉↑ ͗gˆ †(⍀)gˆ (⍀Ј)͘. If we perform the integrals over as before 1,0,1,0 , S( ) ͘ exp(a /2c) ͘. Thus, for the spin † up Dirac particle we should replace the plane wave scalar and use ͗aˆ a ͘ϭ␦ , we obtain K KЈ KKЈ wave function exp͓i()͔ used in Eq. ͑7͒ by 21 c 2 2បc2 i⍀c 2 exp(a/2c)exp͓i()͔. This replacement leads to i⍀c/a ͗gˆ †͑⍀͒gˆ ͑⍀Ј͒͘ϭͩ ͪ ͩ ͪͯ⌫ͩ ͪͯ → ⍀ ϩ ͑ ͒ 2a V a i c/a 1/2 in Eq. 8 , and therefore the result ϱ 2 ⑀ ⍀Ϫ⍀Ј ⍀ a/c 2 c 1 1 c i K( )c/a ͯ͵ d ei ea /2cei( Kc/a)e ͯ ϭ . Ϫ ⍀ K 2⍀c/a ϫ c /a Ϫϱ a e ϩ1 e ͚ ͩ ͪ , K K K a ͑18͒ ͑15͒ If we compare Eq. ͑18͒ with Eq. ͑9͒, we note the crucial Ϫ where we have used the fact that the sum over K vanishes change of sign in the denominator from 1 for BE statistics ϩ unless ⍀ϭ⍀Ј. We show in the Appendix that the sum over to 1 for FD statistics. We also note that the prefactor in Eq. ͑ ͒ ⍀ K is (2Va/c2)␦(⍀Ϫ⍀Ј), so that 9 involves the dimensionless frequency c/a while in Eq. ͑ ͒ ͑ 18 the prefactor involves the factor Kc/a the argument of បc2 i⍀c 2 the exponential in the distribution function is still ប⍀/kT ͗gˆ †͑⍀͒gˆ ͑⍀Ј͒͘ϭ ͯ⌫ͩ ͪͯ eϪc⍀/a␦͑⍀Ϫ⍀Ј͒ a a with the same Hawking–Unruh temperature Tϭបa/2kc). This difference in the prefactor is no cause for concern, be- ប ⍀ 2 c/ cause in fact a single frequency detectable by a ϭ ␦͑⍀Ϫ⍀Ј͒, ͑16͒ K e2⍀c/aϪ1 Minkowski observer is actually spread over a continuous
1526 Am. J. Phys., Vol. 72, No. 12, December 2004 Paul M. Alsing and Peter W. Milonni 1526 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 range of frequencies ⍀ detectable by the accelerated Rindler portional to the Bose–Einstein distribution. We then consider ⍀ϭ 22 observer, with a peak centered at K . This fact allows the calculation of this correlation function again, but this ⍀ ͓ us to replace K by in the final result. This frequency time for an accelerated observer in his particle free Rindler 25 ͉ replacement is explicitly evidenced by the delta function vacuum state 0R͘, such that for a single mode, ␦ Ϫ⍀ ͑ ͒ ͑ ͒ ͗ ͉ † ͉ ͘ϭ ( K ) in Eqs. 11 – 16 in the comparison of the ther- 0R aˆ Raˆ R 0R 1. The new feature is that from his local sta- mal and accelerated correlation functions.͔ tionary perspective, the accelerated observer detects all For the spin up Dirac particle, the more formal field- Minkowski frequencies ͑arising from the the usual plane theoretic derivation of Sec. IV proceeds in exactly the same waves associated with Minkowski states͒ as time-dependent fashion, with the modification of the accelerated wave func- Doppler shifted frequencies. tion from exp͓i()͔→exp(a/2c)exp͓i()͔ and the use of The derivation presented here shows why quantum field † ϭ␦ fluctuations in the vacuum state are crucial for the thermal anti-commutators ͕aˆ K ,a ͖ KKЈ for the quantum me- KЈ ͗ † ⍀ ⍀Ј ͘ chanical creation and annihilation operators instead of the effect of acceleration: gˆ ( )gˆ ( ) is nonvanishing be- ͗ † ͘Þ commutators appropriate for scalar BE particles. For the cor- cause the vacuum expectation aˆ Kaˆ K 0. But there is more ͗ † ͘ relation function we find to it than that, because aˆ Kaˆ K also is nonvanishing for an ϭ 2បc/ observer with a 0. For such an observer, however, ͗ †͑⍀͒ ͑⍀Ј͒͘ϭ K ␦͑⍀Ϫ⍀Ј͒ ͑ ͒ gˆ gˆ 2⍀c/aϩ , 19 e 1 ϱ ϱ ⍀ R Ј Ј ⍀ϩ ͑ ͒ ͵ d ei ei d K( )ϭ ͵ d ei( K) the FD analogue of Eq. 16 . Ϫϱ Ϫϱ ϭ ␦͑⍀ϩ ͒ VI. SUMMARY AND DISCUSSION 2 K ϭ ͑ ͒ In the usual derivation of the Hawking-Unruh temperature 0 20 Eq. ͑1͒,2–6 one solves the wave ͑or Dirac͒ equation for the ⍀ field mode functions in the Rindler coordinates Eq. ͑4͒, and for scalar particles, because both and K are positive. In then quantizes them. Because the hyperbolic orbit of the ac- other words, the thermal effect of acceleration in our model celerated observer Eq. ͑4͒ is confined to the region of arises because of the nontrivial nature of the quantum Minkowski space–time zϾ0, zϾ͉t͉ bounded by the asymp- vacuum and the time-dependent Doppler shifts detected by totes tϭϮz called the right Rindler wedge ͓with mirror or- the accelerated observer. For Dirac particles, the essential bits confined to the left Rindler wedge defined by zϽ0, ͉z͉ new feature is the additional spinor structure of the wave Ͼ͉t͉ obtained from defining the accelerated observer’s coor- function over that of the scalar plane wave. To keep the spin dinates as t()ϭϪ(c/a)sinh(a/c) and z()ϭϪ(c2/ nonrotating in the comoving frame of the accelerated ob- server, the Dirac bispinor must be Fermi–Walker transported a)cosh(a /c)], it turns out that the vacuum detected by the along the accelerated trajectory, resulting in an additional accelerated observer in say, the right Rindler wedge is dif- ͑ time-dependent Lorentz transformation. Formally, this trans- ferent than the usual Minkowski vacuum defined for all z formation induces a shifting of i⍀c/a→i⍀c/aϩ1/2 in the and t) detected by the unaccelerated observer. The inequiva- calculation of relevant gamma function-like integrals, lead- ͑ lence of these vacua and hence the Minkowski versus Rin- ing to the FD Planck factor. 23͒ dler quantization procedures is due to the fact that the In the following we briefly discuss the relationship of our right and left Rindler wedges are causally disconnected from correlation function to those used in the usual literature on each other. Readers can easily convince themselves of the this subject and point out a not widely appreciated subtlety causal disconnectedness of the right and left Rindler wedges relating details of the spatial Rindler mode functions ͑which by drawing a Minkowski diagram in (z,t) coordinates and we have ignored in our model͒ to the statistics of the noise observing that light rays at Ϯ45° emanating from one wedge spectrum detected by the accelerated observer. do not penetrate the other wedge. Hence the Minkowski In our model, we have not motivated the use of the corre- vacuum that the accelerated observer moves through appears lation function ͗gˆ †(⍀)gˆ (⍀Ј)͘ aside from the fact that we to her/him as an excited state containing particles, and not as could calculate it for a nonaccelerated observer in a thermal the particle free vacuum appropriate for the right Rindler field and for a uniformly accelerated observer in vacuum and wedge. The Bose–Einstein distribution with the Hawking– compare the results. It is easy to show that a harmonic oscil- Unruh temperature T for scalar fields ͑Fermi–Dirac for Dirac ␥ lator with frequency 0 and dissipation coefficient , lin- fields͒ is usually derived by considering the expectation early coupled to the field Eq. ͑10͒, reaches a steady-state † value of the number operator aˆ Raˆ R for the accelerated ob- energy expectation value server ͑in the right Rindler wedge͒ in the unaccelerated ͉ ͘ ͗ ͉ † ͉ ͘ ϱ ϱ † Minkowski vacuum 0M , that is, 0M aˆ Raˆ R 0M ͗gˆ ͑⍀͒gˆ ͑⍀Ј͒͘ ϳ͓ ប⍀ Ϯ ͔Ϫ1 ͑ ͗E͘ϰ ͵ d⍀ ͵ d⍀Ј , ͑21͒ exp( /kT) 1 with the upper sign for scalar fields ͑⍀Ϫ Ϫi␥͒͑⍀ЈϪ ϩi␥͒ and lower sign for Dirac fields͒. The proportionality of the 0 0 0 0 particle number spectrum detected by the accelerated Rindler observer moving through the Minkowski vacuum to a ther- which offers some motivation for considering ͗gˆ †(⍀)gˆ (⍀Ј)͘. In fact, it can be shown that ͗E͘ mal spectrum is referred to as the thermalization theorem by ប Ϫ Takagi.5 ϭ͓e 0 /kTϪ1͔ 1, which shows again that our accelerated In this work we have taken a slightly different observer acquires the characteristics appropriate to being in a viewpoint.24 For a scalar field, we first consider an unaccel- thermal field at the temperature Tϭបa/2kc. erated Minkowski observer in a thermal state and find that In an extensive review of the Davies–Unruh effect, the expectation value of a field correlation function is pro- Takagi5 utilizes the quantum two-point correlation ͑Wight-
1527 Am. J. Phys., Vol. 72, No. 12, December 2004 Paul M. Alsing and Peter W. Milonni 1527 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 ͒ ϵ ͉ˆ ˆ † ͉ able by utilizing the expression aϭGM/r2 for the gravita- man function gW( , Ј) ͗0M ( ) ( Ј) 0M͘ to deter- mine the power spectrum of the vacuum noise detected by tional acceleration of a test mass at a distance r from a mass the accelerated observer for a scalar field, M, and determine T at the surface of the earth, the Sun, and a Schwarzschild black hole. ϱ ͑⍀͒ϵ ͵ Ϫi⍀Ϫs͉͉ ͑͒ ͑ ͒ S lim e gW , 22 ↓ ϱ s 0 ACKNOWLEDGMENTS which is very much in the spirit of our calculation in Sec. IV. Thanks go to H. Fearn, D. McMahon, G. J. Milburn E. Here the field operator ˆ () is expanded in terms of the Mottola, M. O. Scully, and M. Wolinsky for brief discussions Rindler mode functions and involves the creation and anni- relating to acceleration in vacuum. hilation operators for both the right and left Rindler wedges. Takagi shows the remarkable, although not widely known result, that for a scalar field in a Rindler space–time of di- APPENDIX: MODE SUM CALCULATION IN EQ. ⍀ ϳ ⍀ ͓ ប⍀ Ϫ Ϫ n͔ mension n, Sn( ) f n( )/ exp( /kT) ( 1) . For even- „15… dimensional space–times ͑for example, nϭ2 as considered in this work or the usual nϭ4) S (⍀) is proportional to the If we convert the sum over K to an integral, we have, for n Ͼ Bose–Einstein distribution function ͓exp(ប⍀/kT)Ϫ1͔Ϫ1, and K 0, ͑ ͓͒ ⍀ ⑀ ⍀Ϫ⍀Ј essentially reproduces our Eq. 16 up to powers of c/a, 1 c i K( )c/a V ϱ 1 ⍀ ͚ ͩ K ͪ ϭ ͵ contained in the function f n( )]. However, for odd n, dK ⍀ KϾ0 K a 2 0 K Sn( ) is proportional to the Fermi–Dirac distribution Ϫ ͓exp(ប⍀/kT)ϩ1͔ 1. For Dirac particles the opposite is true, i(⍀Ϫ⍀Ј)c/a Kc namely for even space–time dimensions S (⍀) is propor- ϫͩ ͪ . ͑A1͒ n a tional to the FD distribution and for odd space–time dimen- ⍀ ϭ ͑ ͒ sions Sn( ) is proportional to the BE distribution. This cu- We let x log( Kc/a) and write Eq. A1 as ⍀ rious fact arises from the dependence of Sn( ) on two V ϱ Va factors in its calculation. The first is the previously men- ͵ Ϫix(⍀Ϫ⍀Ј)c/aϭ ␦͑⍀Ϫ⍀Ј͒ ͑ ͒ dx e 2 . A2 tioned thermalization theorem, that is, the number spectrum 2 c Ϫϱ c ͑ ͒ of accelerated Rindler particles in the usual nonaccelerated The same result is obtained for the sum over KϽ0, so that Minkowski vacuum is proportional to the BE distribution the sum over all K is (2Va/c2)␦(⍀Ϫ⍀Ј). function for scalar fields and is proportional to the FD dis- tribution function for Dirac fields. The second factor that ͒ a Electronic mail: [email protected] ⍀ ͒ switches the form of Sn( ) from BE to FD depends on the b Electronic mail: [email protected] 5 detailed form of the Rindler mode functions. Although the 1S. W. Hawking, ‘‘Black hole explosions,’’ Nature ͑London͒ 248, 30–31 trajectory of the accelerated observer takes place in 1ϩ1 ͑1974͒; ‘‘Particle creation by black holes,’’ Commun. Math. Phys. 43, ͓ ͔ 199–220 ͑1975͒. dimensions the (z,t) plane , the quantum field exists in the 2 ⍀ W. G. Unruh, ‘‘Notes on black hole evaporation,’’ Phys. Rev. D 14, 870– full n-dimensional space–time, and thus Sn( ) ultimately 892 ͑1976͒. depends on the form of the mode functions in the full space– 3P. C. W. Davies, ‘‘Scalar production in Schwarzschild and Rindler met- time. In space–times of even dimensions the number spec- rics,’’ J. Phys. A 8, 609–616 ͑1975͒. trum of Rindler particles in the Minkowski vacuum and the 4The literature on this subject is vast. For some articles directly relevant for noise spectrum of the vacuum fluctuations ͑that is, the re- the work presented here see, for instance, P. Candelas and D. W. Sciama, ͒ ‘‘Irreversible thermodynamics of black holes,’’ Phys. Rev. Lett. 38, 1372– sponse of the accelerated particle detector both depend on 1375 ͑1977͒; T. H. Boyer, ‘‘Thermal effects of acceleration through ran- the same distribution function, and these two effects often dom classical radiation,’’ Phys. Rev. D 21, 2137–2148 ͑1980͒ and ‘‘Ther- are incorrectly equated. mal effects of acceleration for a classical dipole oscillator in classical In our simplified derivation we have bypassed this techni- electromagnetic zero-point radiation,’’ 29, 1089–1095 ͑1984͒;D.W. cality by performing our calculations in 1ϩ1 dimensions Sciama, P. Candelas, and D. Deutsch, ‘‘Quantum field theory, horizons, ͑ ͒ ͑that is, nϭ2). We have concentrated on the power spectrum and thermodynamics,’’ Adv. Phys. 30, 327–366 1981 . 5An extensive review is given by S. Takagi, ‘‘Vacuum noise and stress of vacuum fluctuations as seen by a particle detector carried induced by uniform acceleration,’’ Prog. Theor. Phys. 88, 1–142 ͑1986͒. by the accelerated observer. We have shown that in 1ϩ1 See Chap. 2 for a review of the Davies–Unruh effect. dimensions the spectrum of fluctuations is proportional to the 6N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space ͑Cam- bridge U.P., New York, 1982͒. Bose–Einstein distribution function for scalar fields and to 7 the Fermi–Dirac distribution for spin 1/2 fields, with the Note that in order for a detector to remain at a fixed location outside the ͑ ͒ horizon of a black hole, it must undergo constant acceleration just to Hawking–Unruh temperature defined by Eq. 1 . The depen- remain in place. dence of the noise spectrum on these distribution functions is 8W. Rindler, ‘‘Kruskal space and the uniformly accelerated observer,’’ Am. ultimately traced back to the time-dependent Doppler shifts J. Phys. 34, 1174–1178 ͑1966͒. as detected by the accelerated observer as he/she moves 9P. W. Milonni, The Quantum Vacuum ͑Academic, New York, 1994͒,pp. through the usual nonaccelerated Minkowski vacuum. 60–64. 10 We hope that the calculations exhibited here are suffi- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products ͑Academic, New York, 1980͒. We have used integrals 3.761.4 and 3.761.9, ciently straightforward to give an intuitive understanding of ͐ϱ Ϫ1 ϭ͓⌫ ͔ ͐ϱ Ϫ1 0 x sin(ax)dx ( )/a sin( /2) and 0 x cos(ax)dx the essential physical origin of the Hawking–Unruh tempera- ϭ͓⌫()/a͔cos(/2) respectively, in the combination of the second plus i ture experienced by a uniformly accelerated observer. times the first. Taken together, both integrals have a domain of definition Suggested Problem. Discuss when the Hawking–Unruh aϾ0, 0ϽRe()Ͻ1. In Eq. ͑8͒ we have ϭi⍀c/a with Re()ϭ0. The temperature from Eq. ͑1͒ would become physically detect- integrals can be regularized and thus remain valid in the limit Re()→0as
1528 Am. J. Phys., Vol. 72, No. 12, December 2004 Paul M. Alsing and Peter W. Milonni 1528 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45 can be seen by adding a small imaginary part Ϫi⑀a/c, ⑀Ͼ0 to the fre- This result can be understood as follows. If the observer was traveling at quency ⍀ so that →Јϵϩ⑀ and 0ϽRe(Ј)ϭ⑀Ͻ1 is strictly in the constant velocity v in the positive z direction, we would Lorentz transform ⑀→ ⌫ domain of definition of the integrals. In the limit of 0 we have ͓ ( the spinor in the usual special relativistic way via the operator Sˆ (v) ϩ⑀ ϩ⑀͔ i(ϩ⑀)/2→͓⌫ ͔ i/2 )/a e ( )/a e and thus obtain the same values ϭexp(␥0␥3/2) where tanh ϭv/c. See J. D. Bjorkin and S. D. Drell, Rela- Ј ϭ for the integrals as if we had just set Re( ) 0 initially. The equivalence tivistic Quantum Mechanics ͑McGraw–Hill, New York, 1964͒, pp. 28–30. of these two approaches to evaluate the integrals occurs because the stan- From Eq. ͑3͒ we have v/cϭtanh(a/c) so that ϭa/c, yielding the ⌫ ϭ͐ϱ Ϫz zϪ1 Ͼ dard integral form of the gamma function (z) 0 dt e t ,Re(z) 0 spinor Lorentz transformation to the instantaneous rest frame of the accel- can be analytically continued in the complex plane and in fact remains erated observer. → Þ well defined, in particular, for Re(z) 0, Im(z) 0, which is the case in Eq. 20For simplicity, we have chosen the spin up wave function as an eigenstate ͑8͒. See for for example, J. T. Cushing, Applied Analytical Mathematics ˆ for Physical Scientists ͑Wiley, New York, 1975͒, p. 343. of S( ) with eigenvalue exp(a /2c). The exact spatial dependence of the ͑ ͒ 11Reference 10, Sec. 8.332. accelerated Rindler spin up wave function is more complicated than this 12Related, although much more involved derivations of the Davies–Unruh simple form, although both have the same zero bispinor components. See effect based on a similar substitution can be found in L. Pringle, ‘‘Rindler W. Greiner, B. Mu¨ller, and J. Rafelski, Quantum Electrodynamics in observers, correlated states, boundary conditions, and the meaning of the Strong Fields ͑Springer, New York, 1985͒, Chap. 21.3, pp. 563–567; M. thermal spectrum,’’ Phys. Rev. D 39, 2178–2186 ͑1989͒; U. H. Gerlach, Soffel, B. Mu¨ller, and W. Greiner, ‘‘Dirac particles in Rindler spacetime,’’ ‘‘Minkowski Bessel modes,’’ ibid. 38, 514–521 ͑1988͒, gr-qc/9910097 Phys. Rev. D 22, 1935–1937 ͑1980͒. and ‘‘Quantum states of a field partitioned by an accelerated frame,’’ 40, 21The spinor Lorentz transformation Sˆ () does not mix spin components. 1037–1047 ͑1989͒. We can justify this substitution from the form Thus, for example, a spin up Minkowski state remains a spin up acceler- ϭ͐k(x)dx of the phase of a quantum mechanical particle in curved ated ͑Rindler͒ state. We can therefore drop the constant spinor ͉↑͘ from our space–time. See L. Stodolsky ‘‘Matter and light wave interferometry in calculations and retain the essential, new time-dependent modification ͑ ͒ gravitational fields,’’ Gen. Relativ. Gravit. 11, 391–405 1979 and P. M. exp(a/2c) to the plane wave for our Dirac ‘‘wave function.’’ Alsing, J. C. Evans, and K. K. Nandi, ‘‘The phase of a quantum mechani- 22Reference 5, Sec. 2, in particular Eqs. ͑2.7.4͒ and ͑2.8.8͒. ͑ ͒ cal particle in curved spacetime,’’ ibid. 33, 1459–1487 2001 , gr-qc/ 23S. A. Fulling, ‘‘Nonuniqueness of canonical field quantization in Riemann- 0010065. ian spacetime,’’ Phys. Rev. D 7, 2850–2862 ͑1973͒ and Aspects of Quan- 13A similar derivation in terms of Doppler shifts appears in the appendix of tum Field Theory in Curved Space-Time ͑Cambridge U.P., New York, H. Kolbenstvedt, ‘‘The principle of equivalence and quantum detectors,’’ 1989͒. Eur.J.Phys.12, 119–121 ͑1991͒. T. Padmanabhan and coauthors also 24This viewpoint is also taken in a different derivation of the Unruh–Davies have derived Eq. ͑9͒ by similarly considering the power spectrum of Dop- effect in Ref. 9. pler shifted plane waves as detected by the accelerated observer. See K. 25 ͉ ͘ Srinivasan, L. Sriramkumar, and T. Padmanabhan, ‘‘Plane waves viewed The unaccelerated Minkowski vacuum 0 M is unitarily related to the Rin- ͉ ͉ ͉ ϭ ˆ ͉ ͉ ˆ from an accelerated frame: Quantum physics in a classical setting,’’ Phys. dler vacuum 0R͘ 0L͘ via 0 M͘ S(r) 0R͘ 0L͘, where S(r) is the Rev. D 56, 6692–6694 ͑1997͒; T. Padmanabhan, ‘‘Gravity and the ther- ˆ ϭ ͓ Ϫ † † ͔ squeezing operator S(r) exp r(aˆRaˆL aˆRaˆL) . The subscripts R and L de- modynamics of horizons,’’ gr-qc/0311036. note the right (zϾ0,zϾ͉t͉) and left (zϽ0,͉z͉Ͼ͉t͉) Rindler wedges, re- 14 When we later convert a sum over K to an integral, we obtain the Lorentz- spectively, which are regions of Minkowski space–time bounded by the invariant measure dK/ as a consequence of the 1/ͱ in Eq. ͑10͒.The ϭϮ ͉ K K asymptotes t z. 0R͘ is the Fock state of zero particles in the right use of this invariant measure eliminates the need to explicitly transform Rindler wedge and ͉0 ͘ is the Fock state of zero particles in the left ͱ ͑ ͒ L the frequency term 1/ K in Eq. 14 , for instance. Rindler wedge. Note that the orbit of the accelerated Rindler observer 15 Because we use a one-dimensional model, the factor V/2 appears instead given by Eq. ͑4͒ is confined to the right Rindler wedge. Because the right 3 of the more familiar V/(2 ) . In other works, our volume V here is really and left Rindler wedges of Minkowski space–time are causally discon- just a length. nected from each other, the creation and annihilation operators aˆ † , aˆ and 16If we use the same gamma function integrals as in Ref. 10, we will have R R aˆ † , aˆ live in the right and left wedges, respectively, and mutually com- for the Dirac case ϭi⍀c/aϩ1/2, with Re()ϭ1/2 clearly in their do- L L mute with each other, that is, ͓aˆ ,aˆ † ͔ϭ0, etc. Because, physical states main of definition 0ϽRe()Ͻ1. R L that live in the right wedge have zero support in the left wedge ͑and vice 17S. Weinberg, Gravitation and Cosmology ͑Wiley, New York, 1972͒, pp. ͒ † 365–370. versa , they are described by functions solely of the operators aˆ R , aˆ R 18 ͉⌿ ͘ϭ † ͉ ͘ ͉ ͘ C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation ͑Freeman, San appropriate for the right wedge, that is, R f (aˆ R ,aˆ R) 0R 0L ϭ͉ ͉ ͉ Francisco, 1973͒, Chap. 6, pp. 163–176. R͘ 0L͘. It is in this sense that we can speak of 0R͘ as the vacuum 19 ͉ Reference 5, p. 101, and P. Candelas and D. Deutsch, ‘‘Fermion fields in for the right Rindler wedge, and similarly 0L͘ as the vacuum for the left accelerated states,’’ Proc. R. Soc. London, Ser. A 362, 251–262 ͑1978͒. Rindler wedge.
1529 Am. J. Phys., Vol. 72, No. 12, December 2004 Paul M. Alsing and Peter W. Milonni 1529 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.110.37.248 On: Fri, 28 Nov 2014 23:55:45