Questions of Stability near Black Hole Critical Points

In this letter, we discuss how thermal fluctuations can effect the stability of (generally) charged black holes when close to certain critical points. Our novel treatment utilizes the black hole area spectrum (which is, for definiteness, taken to be evenly spaced) and makes an important distinction between fixed and fluctuating charge systems (with these being modeled, respectively, as a canonical and grand canonical ensemble.) The discussion begins with a summary of a recent technical paper [1]. We then go on to consider the issue of stability when the system approaches the critical points of interest. These include the d-dimensional analogue of the Hawking-Page phase transition, a phase transition that is relevant to Reissner-Nordstrom black holes and various extremal-limiting cases. It has often been suggested that any viable theory of quantum gravity must somehow account for black hole thermodynamics [2]. In particular, a prospective fundamental theory would be expected to provide a statistical explanation for the Bekenstein-Hawking entropy [3,4]: SBH = 1 4G A, where A is the surface area of the black hole horizon. An important subplot is the leading-order quantum correction to this classical area law. Indeed, there has been substantial interest in deducing this correction, with many different techniques having been called upon for just this purpose (see [1] for a thorough list of references). A feature that is common to all of the relevant studies is a leading-order correction that arises at the logarithmic order in SBH . Nonetheless, there is significant disagreement on the value of the proportionality constant or logarithmic prefactor. This conflict over the prefactor can be partially resolved when one considers an often overlooked point: there are actually two distinct sources for this logarithmic correction. More precisely, one can anticipate both an uncertainty in the number of microstates describing a ∗E-mail: [email protected] †E-mail: [email protected] 1Here and throughout, all fundamental constants besides the gravitational coupling constant, G are set to unity.