Schwarzschild-anti-de Sitter black holes within isothermal cavity: Thermodynamics, phase transitions, and the Dirichlet problem

The thermodynamics of Schwarzschild black holes within spherical isothermal cavities in anti-de Sitter (AdS) space is studied for arbitrary dimensions in the semiclassical approximation of the Euclidean path integral of quantum gravity. For such boundary conditions, known classical solutions are a hot AdS and two or no Schwarzschild-AdS depending on whether or not the wall-temperature of the cavity is above or below a minimum value. Earlier work in four dimensions with such boundary conditions showed that the larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable, respectively. The standard area-law of entropy was known to hold too. We derive the area-law for arbitrary dimensions and show that qualitative behavior of local stability remains the same. Then using a careful analysis of the associated Dirichlet boundary-value problem we address global issues. We find that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS. The larger hole thus can be globally thermodynamically stable. We find that the smaller the cavity the higher the critical temperature for phase transition is and it always remains above the minimum temperature needed for the classical existence of the holes in that cavity. In the infinite limit of cavity this picture reduces to that considered by Hawking and Page. All these hold for arbitrary dimensions, however the case of five dimensions turns out to be special in that the Dirichlet problem can be solved exactly giving exact analytic expressions for the black-hole masses as functions of boundary variables (cavity-radius and temperature). This makes it possible to compute the on-shell Euclidean action as a function of boundary variables too from which other quantities of interest can be evaluated. In particular, we obtain the minimum temperature (for the holes to exist classically) and the critical temperature (for phase transition) as functions of the cavity-radius in five dimensions.