Temperature and entropy of a quantum black hole and conformal anomaly

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.

Quantum tunneling beyond semiclassical approximation

“…Interestingly the leading order correction is logarithmic in A or S BH which was found earlier in [22,23] by field theory calculations and later in [24,29] with β 1 = − 1 8π by quantum geometry method. The higher order corrections involve inverse powers of A or S BH .…”

“…Such a structure was obtained earlier [12] in radial null geodesic approach by explicitly taking into account the one loop back reaction effect. Also, as stated earlier, such a form follows from conformal field theory techniques [22] where α is given by (3.26).…”

“…We will now show that various choices of the coefficients β i in (3.19) correspond to higher order loop corrections to the surface gravity of the black hole, obtained by including back reaction effects [20,21] or by accounting for the trace anomaly [22]. To see this note that the standard relation between the surface gravity (K) and the Hawking temperature (T h ) is…”

“…Moreover, the coefficient α is related to the trace anomaly. Using conformal field theory techniques it was shown [22] that the one loop quantum correction to the surface gravity, for the Schwarzschild black hole, is given by (3.25) where,…”

Black hole entropy, log corrections and quantum ergosphere

“…Let us focus on the propagation of a classical shell in a Schwarzschild geometry. When no back reaction effects nor quantum 3 A similar logarithmic correction to the entropy-area law has also emerged from the calculation of one-loop effects of the (quantum) matter fields near the black hole [16]. 4 We now switch from k = = c = G = 1 units of the previous sections to k = = c = 1 to keep track of the Planck-scale suppressed terms.…”