Quantum-gravity fluctuations and the black-hole temperature

Shahar Hod

doi:10.1140/epjc/s10052-015-3465-y

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 (semi-classical) black-hole spectrum.

Highlights

M is the mass of the Schwarzschild black hole. (We use gravitational units in which G = c = 1.)
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

Summary

Introduction

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 It is well known that a Schwarzschild black hole is characterized by a discrete spectrum of gravitational resonances [8,9,10] with the fundamental asymptotic frequency [7,11] 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].

Results

Conclusion