Temperature and Entropy of a Quantum Black Hole and Conformal Anomaly

TL;DR

The aim of this paper is to show how the one-loop corrections to the temperature and entropy of the 4-dimensional Schwarzschild black hole with massless quantum fields can be derived explicitly in a simple thermodynamical treatment based on the scaling properties of the theory.

Abstract

Attention is paid to the fact that temperature of a classical black hole can be derived from the extremality condition of its free energy with respect to variation of the mass of a hole. For a quantum Schwarzschild black hole evaporating massless particles the same condition is shown to result in the following one-loop temperature $T=(8πM)^{-1} (1+σ(8πM^2)^{-1})$ and entropy $S = 4πM^2 - σ\log M$ expressed in terms of the effective mass $M$ of a hole together with its radiation and the integral of the conformal anomaly $σ$ that depends on the field species. Thus, in the given case quantum corrections to $T$ and $S$ turn out to be completely provided by the anomaly. When it is absent ($σ=0$), which happens in a number of supersymmetric models, the one-loop expressions of $T$ and $S$ preserve the classical form. On the other hand, if the anomaly is negative ($σ<0$) an evaporating quantum hole seems to cease to heat up when its mass reaches the Planck scales.