Extremality in Reissner-Nordström black holes

Abstract

The Reissner-Nordström (RN) solution describes the most general static black hole of the Einstein-Maxwell system. The solution is characterized by its mass and electric charge and, depending on their relative values, it can possess two, one, or no horizons at all. After surveying the geometric and thermodynamic properties of RN black holes, we focus on the “extremal” case, corresponding to solutions with a single horizon and zero temperature. Near-extremal black holes present peculiar features which make them seemingly incompatible with classical gravity and thermodynamics. We comment on these features and explain how they can be resolved by making use of an effective two-dimensional theory of gravity which provides a quantum description of the black holes at ultra-low temperatures. We briefly comment on how supersymmetry dramatically alters the description, saving the day for the string-theoretical counting of extremal black hole microstates