Temperature and entropy of a quantum black hole and conformal anomaly.

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(8πM2)−1 ) and entropy S = 4πM2 − σ logM 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. PACS number(s): 04.60.+n, 12.25.+e, 97.60.Lf, 11.10.Gh e-mail: [email protected] 1 Black hole thermodynamics is known to possess a number of puzzles like the meaning of black hole entropy, the information loss problem and the operation of the generalized second law [1]. The principal difficulty on the way to their resolution is the lack of a consistent theory of quantum gravity. Even investigation of quantum effects on the classical curved backgrounds sometime represents a technical problem where results can be obtained only approximately. This is a reason why exactly solvable two-dimensional models of black holes are of great interest at the present moment [2]. 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. To begin with, we remind that the energy E and entropy S of a canonical ensemble at the temperature β can be derived from the free energy F (β) as follows: E = ∂ ∂β (βF ) , S = β(E − F ) . (1) These quantities for a system being at the fixed temperature change until a system reaches a thermal equilibrium characterized by a minimum of F [3]. In this state the condition of extremum for F (δF )β = 0 (2) gives a relation between β and other parameters of the ensemble. Moreover, the first law of thermodynamics in its simplest form βδS = δE (3) turns out to be a consequence of (1) and (2). Now, returning to thermodynamics of black holes, an extremality condition of F , similar to (2), can be used to relate the temperature of the hole with its other parameters (mass, charge, etc.). To see this, we make use of the Gibbons-Hawking approach to gravitational thermodynamics [4]. In its framework the free energy in the semiclassical approximation is given by the Euclidean Einstein-Hilbert action Wcl with suitably 2 subtracted boundary terms 2 βF (β) = Wcl(β) = − 1 16π ( ∫