Bekenstein has put forward the idea that, in a quantum theory of gravity, a
black hole should have a discrete energy spectrum with concomitant discrete
line emission. The quantized black-hole radiation spectrum is expected to be
very different from Hawking's semi-classical prediction of a thermal black-hole
radiation spectrum. One naturally wonders: Is it possible to reconcile the {\it
discrete} quantum spectrum suggested by Bekenstein with the {\it continuous}
semi-classical spectrum suggested by Hawking ? In order to address this
fundamental question, in this essay we shall consider the zero-point
quantum-gravity fluctuations of the black-hole spacetime. In a quantum theory
of gravity, these spacetime fluctuations are closely related to the
characteristic gravitational resonances of the corresponding black-hole
spacetime. Assuming that the energy of the black-hole radiation stems from
these zero-point quantum-gravity fluctuations of the black-hole spacetime, we
derive the effective temperature of the quantized black-hole radiation
spectrum. Remarkably, it is shown that this characteristic temperature of the
{\it discrete} (quantized) black-hole radiation agrees with the well-known
Hawking temperature of the {\it continuous} (semi-classical) black-hole
spectrum.Comment: 6 page

Hod, Shahar

English

arXiv

Bekenstein has put forward the idea that, in a quantum theory of gravity, a black hole should have a discrete energy spectrum with concomitant discrete line emission. The quantized black-hole radiation spectrum is expected to be very different from Hawking’s semi-classical prediction of a thermal black-hole radiation spectrum. One naturally wonders: Is it possible to reconcile the discrete quantum spectrum suggested by Bekenstein with the continuous semi-classical spectrum suggested by Hawking? In order to address this fundamental question, in this essay we shall consider the zero-point quantum-gravity fluctuations of the black-hole spacetime. In a quantum theory of gravity, these spacetime fluctuations are closely related to the characteristic gravitational resonances of the corresponding black-hole spacetime. Assuming that the energy of the black-hole radiation stems from these zero-point quantum-gravity fluctuations of the black-hole spacetime, we derive the effective temperature of the quantized black-hole radiation spectrum. Remarkably, it is shown that this characteristic temperature of the discrete (quantized) black-hole radiation agrees with the well-known Hawking temperature of the continuous (semi-classical) black-hole spectrum

Shahar Hod

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Quantum-gravity fluctuations and the black-hole temperature

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Eur. Phys. J. C (2015) 75:233DOI 10.1140/epjc/s10052-015-3465-yLetterQuantum-gravity fluctuations and the black-hole temperatureShahar Hod1,2,a1 The Ruppin Academic Center, 40250 Emeq Hefer, Israel2 The Hadassah Institute, 91010 Jerusalem, IsraelReceived: 3 January 2015 / Accepted: 18 May 2015 / Published online: 29 May 2015© The Author(s) 2015. This article is published with open access at Springerlink.comAbstract Bekenstein has put forward the idea that, in aquantum theory of gravity, a black hole should have a discreteenergy spectrum with concomitant discrete line emission.The quantized black-hole radiation spectrum is expectedto be very different from Hawking’s semi-classical predic-tion of a thermal black-hole radiation spectrum. One natu-rally wonders: Is it possible to reconcile the discrete quan-tum spectrum suggested by Bekenstein with the continuoussemi-classical spectrum suggested by Hawking? In order toaddress this fundamental question, in this essay we shallconsider the zero-point quantum-gravity fluctuations of theblack-hole spacetime. In a quantum theory of gravity, thesespacetime fluctuations are closely related to the characteris-tic gravitational resonances of the corresponding black-holespacetime. Assuming that the energy of the black-hole radi-ation stems from these zero-point quantum-gravity fluctu-ations of the black-hole spacetime, we derive the effectivetemperature of the quantized black-hole radiation spectrum.Remarkably, it is shown that this characteristic temperature ofthe discrete (quantized) black-hole radiation agrees with thewell-known Hawking temperature of the continuous (semi-classical) black-hole spectrum.One of the most remarkable theoretical predictions ofmodern physics is Hawking’s celebrated result that blackholes are not completely black [1]. According to Hawking’ssemi-classical analysis, a black hole is quantum mechani-cally unstable–it emits continuous thermal radiation whosecharacteristic temperature is given byTH = h¯8πM. (1)Here M is the mass of the Schwarzschild black hole. (We usegravitational units in which G = c = 1.)It should be stressed, however, that Hawking’s derivationof the continuous black-hole radiation spectrum is restrictedto the semi-classical regime: the matter fields are treatedquantum mechanically but the spacetime (and, in particular,a e-mail: shaharhod@gmail.comthe black hole itself) are treated classically. One thereforeexpects to find important new features in the character ofthe black-hole radiation spectrum once quantum propertiesof the black hole itself are properly taken into account.1 Itis therefore appropriate to regard the Hawking temperature(1) as the semi-classical (SC) temperature of the continuousblack-hole radiation:TSC ≡ TH = h¯8πM. (2)The quantization of black holes was first proposed in theseminal work of Bekenstein [2,3]. The original quantizationprocedure was based on the physical observation that thesurface area of a black hole behaves as a classical adiabaticinvariant [2,3]. In the spirit of the Ehrenfest principle [4],any classical adiabatic invariant corresponds to a quantumentity with a discrete spectrum, Bekenstein suggested thatthe horizon area A of a quantum black hole should have adiscrete spectrum of the formAn = γ h¯ · n; n = 1, 2, 3, . . . . (3)Here γ is an unknown “fudge” factor which was introducedin [2,3].In order to determine the value of the coefficient γ ,Mukhanov and Bekenstein [5,6] have suggested, in the spiritof the Boltzmann–Einstein formula in statistical physics[4], to relate gn ≡ exp[SBH(n)] to the number of black-hole micro-states that correspond to a particular externalblack-hole macro-state. Here SBH is the black-hole entropy,which is related to its surface area A by the thermodynamic–geometric relation [1,2]SBH = A4h¯. (4)1 This state of affairs is reminiscent of atomic spectroscopy: accordingto the classical laws of electrodynamics an atom should have a continu-ous emission spectrum, whereas quantum mechanics dictates a discreteline emission from the atom.123233 Page 2 of 3 Eur. Phys. J. C (2015) 75 :233The statistical degeneracy [see Eqs. (3) and (4)]gn ≡ exp[SBH(n)] = exp(14γ · n)(5)of the nth black-hole area level has to be an integer for everyinteger n. This physical requirement dictates the relation [5,6]γ = 4 ln k (6)for the fudge factor γ , where the unknown constant k shouldbe an integer.Determining the specific value of the integer k requiresadditional physical input. This physical information emergesby applying the Bohr correspondence principle [4] to thediscrete resonance spectrum of the black hole [7]. Accord-ing to the Bohr correspondence principle, transition frequen-cies at large quantum numbers should equal classical oscilla-tion frequencies. Namely, the asymptotic energy differenceMn+1 − Mn between the (n + 1)th and the nth black-holequantum levels should be given by the characteristic classicaloscillation frequency of the black hole:Mn+1 − Mn = h¯ωBH. (7)It is well known that a Schwarzschild black hole is char-acterized by a discrete spectrum of gravitational resonances[8–10] with the fundamental asymptotic frequency [7,11]MωR = ln 38π. (8)The emission of a gravitational quantum from the black holeresults in a change M = h¯ωR [see Eq. (7)] in the black-hole mass. Using the first-law of black-hole thermodynam-ics, A = 32πMM ,2 one finds the fundamental changeA = 4 ln 3· h¯ in the Schwarzschild black-hole surface area.Taking cognizance of Eqs. (3) and (8), one finally obtains thequantized area spectrum:An = 4h¯ ln 3 · n; n = 1, 2, 3, . . . . (9)It is worth emphasizing again that the black-hole area spec-trum (9) is consistent both with the area-entropy thermo-dynamic relation (4) for black holes, with the Boltzmann–Einstein formula (5) in statistical physics, and with the Bohrcorrespondence principle (7) [7].One therefore concludes that, in a quantum theory of grav-ity, a Schwarzschild black hole has a discrete energy (mass)spectrum of the form:3Mn =√h¯ ln 34π· n1/2; n = 1, 2, 3, . . . , (10)2 Here we have used the relation A = 16πM2 for the Schwarzschildblack hole.3 See footnote 2.with concomitant discrete line emission [5–7]. In particular,the radiation emitted by the quantized black hole consists ofgravitational quanta whose frequencies are integer multiplesof the fundamental black-hole frequencyω0 ≡ (Mn+1 − Mn)/h¯ = ln 38πM. (11)The quantized (discrete) black-hole radiation spectrum isobviously different from Hawking’s semi-classical predic-tion of a thermal (continuous) spectrum.One naturally wonders: Is it possible to reconcile the dis-crete quantum spectrum predicted by Bekenstein with thecontinuous semi-classical spectrum predicted by Hawking?In order to address this fundamental question, we first pointout that, in a quantum theory of gravity, the black-hole space-time is expected to possess a set of zero-point quantum-gravity fluctuations. It has been suggested [12,13] that thesezero-point fluctuations of the black-hole spacetime (and, inparticular, the quantum-gravity fluctuations of the black-holehorizon) may enable quanta to tunnel out of the black hole.We shall now conjecture that these black-hole spacetimefluctuations are characterized by the fundamental resonancefrequency ω0 [see Eq. (11)] of the black-hole spacetime.In particular, we propose a physical picture in which thequantum-gravity fluctuations of the black-hole spacetimemay enable quanta with the appropriate frequencies (theones which are in resonance with the fluctuating horizon:ω0, 2ω0, 3ω0, . . .) to tunnel out of the quantum black hole.According to this physical picture, the energy of the black-hole radiation stems from these zero-point quantum-gravityfluctuations of the black-hole spacetime. The characteristicquantum temperature, TQ, of the discrete black-hole radia-tion spectrum may be defined by equating the mean ther-mal energy of the radiating fields (including their zero-pointquantum energy) with the corresponding energy of an emit-ted quantum with the characteristic black-hole resonance fre-quency ω0. Namely,12h¯ω0 + h¯ω0eh¯ω0/TQ − 1 = h¯ω0. (12)Substituting the fundamental black-hole resonance ω0 =ln 3/8πM [see Eq. (11)] into Eq. (12), one finds that thecharacteristic temperature of the quantized (Q) black-holeradiation spectrum is given byTQ = h¯8πM. (13)Remarkably, we find here that the characteristic temper-ature (13) of the discrete (quantized) black-hole radiationspectrum exactly matches the semi-classical temperature (2)of the continuous black-hole radiation:TQ = TSC. (14)123Eur. Phys. J. C (2015) 75 :233 Page 3 of 3 233Summary One of the most important theoretical predic-tions of modern physics is Hawking’s semi-classical4 resultthat black holes are not completely black [1]. In particu-lar, according to Hawking’s analysis, a Schwarzschild blackhole is expected to emit continuous thermal radiation whosecharacteristic semi-classical temperature is given by Eq. (2).However, Bekenstein [2,3] has put forward the idea that, ina quantum theory of gravity,5 a quantum black hole shouldhave a discrete mass spectrum. As a consequence, a quantumblack hole is expected to be characterized by a discrete lineemission.In the present essay we have proposed a physical mecha-nism which relates in a natural way the two seemingly dif-ferent spectra: the discrete black-hole quantum spectrum aspredicted by Bekenstein and the semi-classical continuousspectrum as predicted by Hawking. The proposed model isbased on the fact that, in a quantum theory of gravity, theblack-hole spacetime is expected to possess a set of zero-point quantum-gravity fluctuations which are characterizedby the fundamental black-hole resonance frequency (11).These quantum-gravity fluctuations of the black-hole hori-zon may enable quanta with the appropriate frequencies (thefrequencies ω0, 2ω0, 3ω0, . . . which are in resonance withthe fluctuating horizon) to tunnel out of the quantum blackhole.The resulting quantum-gravity black-hole radiation spec-trum is characterized by a discrete line emission as predictedby Bekenstein. Remarkably, we have shown here that thisquantized (discrete) black-hole radiation spectrum is charac-terized by an effective quantum temperature TQ [see Eq. (13)]4 It is worth emphasizing again that, in Hawking’s original analysis[1], the matter fields are treated quantum mechanically but the blackhole itself is treated as a classical entity. Thus, Hawking’s analysis isrestricted to the semi-classical regime.5 In a quantum theory of gravity, the spacetime itself (and, in particular,the black-hole energy spectrum) should be treated as a quantum entity.which agrees with the well-known semi-classical Hawkingtemperature TH [see Eq. (2)] of the continuous black-holeradiation spectrum.Acknowledgments This research is supported by the Carmel ScienceFoundation. I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, andAlona B. Tea for stimulating discussions.OpenAccess This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.References1. S.W. Hawking, Commun. Math. Phys. 43, 199 (1975)2. J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)3. J.D. Bekenstein, Lett. Nuovo Cimento 11, 467 (1974)4. M. Born, Atomic Physics (Blackie, London, 1969)5. V. Mukhanov, JETP Lett. 44, 63 (1986)6. J.D. Bekenstein, V.F. Mukhanov, Phys. Lett. B 360, 7 (1995)7. S. Hod, Phys. Rev. Lett. 81, 4293 (1998)8. H.P. Nollert, Class. Quantum Gravity 16, R159 (1999)9. E.W. Leaver, Proc. R. Soc. A 402, 285 (1985)10. H.P. Nollert, Phys. Rev. D 47, 5253 (1993)11. See S. Hod, Class. Quantum Gravity 23, L23 (2006) [for the phys-ical motivation of focusing on the asymptotic black-hole reso-nances]12. J.W. York, Phys. Rev. D 28, 2929 (1983)13. J.B. Hartle, S.W. Hawking, Phys. Rev. D 13, 2188 (1975)123

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