Thermal Fluctuations and Black Hole Entropy

Abstract

In this paper, we consider the effect of thermal fluctuations on the entropy of both neutral and charged black holes. We emphasize the distinction between fixed and fluctuating charge systems; using a canonical ensemble to describe the former and a grand canonical ensemble to study the latter. Our novel approach is based on the philosophy that the black hole quantum spectrum is an essential component in any such calculation. For definiteness, we employ a uniformly spaced area spectrum, which has been advocated by Bekenstein and others in the literature. The generic results are applied to some specific models; in particular, various limiting cases of an (arbitrary-dimensional) AdS-Reissner-Nordstrom black hole. We find that the leading-order quantum correction to the entropy can consistently be expressed as the logarithm of the classical quantity. For a small AdS curvature parameter and zero net charge, it is shown that, independent of the dimension, the logarithmic prefactor is +1/2 when the charge is fixed but +1 when the charge is fluctuating.We also demonstrate that, in the grand canonical framework, the fluctuations in the charge are large, $ΔQ\simΔA\sim S_{BH}^{1/2}$, even when $<Q> =0$. A further implication of this framework is that an asymptotically flat, non-extremal black hole can never achieve a state of thermal equilibrium.