Black holes at a finite distance: Quasi-local restricted phase space formalism

Abstract

We extend the restricted phase space formalism for spherically symmetric black hole solutions of Einstein-Maxwell theory to the quasi-local regime, with the static observers located at a finite radial distance. The first law and Euler relation for the RN and RN-AdS black holes are proved to hold, but only with the inclusion of an extra pair of thermodynamic variables, i.e. the pressure and the area of the codimension-2 hypersurface on which the observers reside. For the RN black holes, the quasi-local behavior is analyzed in detail. It turns out that the RN black holes in the quasi-local description behaves significantly different from itself in the asymptotic description, but is extremely similar to the RN-AdS black holes in the asymptotic description, e.g. allowing for isocharge temperature-entropy phase transitions and lack of isovoltage temperature-entropy phase transitions. In the neutral limit, the Hawking-Page-like transitions appear in the quasi-local description which is absent in the asymptotic description.