Quantum-gravity fluctuations and the black-hole temperature

Abstract

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 byHere M is the mass of the Schwarzschild black hole. (We use gravitational units in which G = c = 1.) It should be stressed, however, that Hawking's 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

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