Quantum-gravity fluctuations and the black-hole temperature

The European Physical Journal C, May 2015

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.

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Quantum-gravity fluctuations and the black-hole temperature

Eur. Phys. J. C Quantum-gravity fluctuations and the black-hole temperature Shahar Hod 0 1 0 The Hadassah Institute , 91010 Jerusalem , Israel 1 The Ruppin Academic Center , 40250 Emeq Hefer , Israel 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 (semiclassical) black-hole spectrum. One of the most remarkable theoretical predictions of modern physics is Hawking's celebrated result that black holes are not completely black [1]. According to Hawking's semi-classical analysis, a black hole is quantum mechanically unstable-it emits continuous thermal radiation whose characteristic temperature is given by - Here M is the mass of the Schwarzschild black hole. (We use gravitational units in which G = c = 1.) It should be stressed, however, that Hawkings derivation of the continuous black-hole radiation spectrum is restricted to the semi-classical regime: the matter fields are treated quantum mechanically but the spacetime (and, in particular, A SBH = 4h . the black hole itself) are treated classically. One therefore expects to find important new features in the character of the black-hole radiation spectrum once quantum properties of the black hole itself are properly taken into account.1 It is therefore appropriate to regard the Hawking temperature (1) as the semi-classical (SC) temperature of the continuous black-hole radiation: The quantization of black holes was first proposed in the seminal work of Bekenstein [2, 3]. The original quantization procedure was based on the physical observation that the surface area of a black hole behaves as a classical adiabatic invariant [2, 3]. In the spirit of the Ehrenfest principle [4], any classical adiabatic invariant corresponds to a quantum entity with a discrete spectrum, Bekenstein suggested that the horizon area A of a quantum black hole should have a discrete spectrum of the form n = 1, 2, 3, . . . . Here is an unknown fudge factor which was introduced in [2, 3]. In order to determine the value of the coefficient , Mukhanov and Bekenstein [5, 6] have suggested, in the spirit of the BoltzmannEinstein formula in statistical physics [4], to relate gn exp[SBH(n)] to the number of blackhole micro-states that correspond to a particular external black-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] 1 This state of affairs is reminiscent of atomic spectroscopy: according to the classical laws of electrodynamics an atom should have a continuous emission spectrum, whereas quantum mechanics dictates a discrete line emission from the atom. The emission of a gravitational quantum from the black hole results in a change M = hR [see Eq. (7)] in the blackhole mass. Using the first-law of black-hole thermodynamics, A = 32 M 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 the quantized area spectrum: An = 4h ln 3 n; n = 1, 2, 3, . . . . It is worth emphasizing again that the black-hole area spectrum (9) is consistent both with the area-entropy thermodynamic relation (4) for black holes, with the Boltzmann Einstein formula (5) in statistical physics, and with the Bohr correspondence principle (7) [7]. One therefore concludes that, in a quantum theory of gravity, a Schwarzschild black hole has a discrete energy (mass) spectrum of the form:3 Mn = 2 Here we have used the relation A = 16 M2 for the Schwarzschild 3 See footnote 2. The statistical degeneracy [see Eqs. (3) and (4)] gn exp[SBH(n)] = exp of the nth black-hole area level has to be an integer for every integer n. This physical requirement dictates the relation [5, 6] for the fudge factor , where the unknown constant k should be an integer. Determining th (...truncated)


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Shahar Hod. Quantum-gravity fluctuations and the black-hole temperature, The European Physical Journal C, 2015, pp. 233, Volume 75, Issue 5, DOI: 10.1140/epjc/s10052-015-3465-y