Schwarzschild-anti de Sitter within an Isothermal Cavity: Thermodynamics, Phase Transitions and the Dirichlet Problem

TL;DR

This paper investigates the thermodynamics and phase transitions of Schwarzschild-AdS black holes within an isothermal cavity, deriving stability conditions, phase transition points, and exact solutions in five dimensions, extending and confirming previous theoretical results.

Contribution

It provides the first derivation of the area law for black-hole entropy in this setting and offers exact solutions in five dimensions, advancing understanding of black hole thermodynamics with boundary conditions.

Findings

Existence of a unique hot AdS solution and two or no black-hole solutions depending on temperature.

Confirmation that larger black holes are thermodynamically stable, smaller ones unstable.

Identification of a phase transition from hot AdS to larger black holes above a critical temperature.

Abstract

The thermodynamics of Schwarzschild black holes within an isothermal cavity and the associated Euclidean Dirichlet boundary-value problem are studied for four and higher dimensions in anti-de Sitter (AdS) space. For such boundary conditions classically there always exists a unique hot AdS solution and two or no Schwarzschild-AdS black-hole solutions depending on whether or not the temperature of the cavity-wall is above a minimum value, the latter being a function of the radius of the cavity. Assuming the standard area-law of black-hole entropy, it was known that larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable respectively. In this paper we present the first derivation of this by showing that the standard area law of black-hole entropy holds in the semi-classical approximation of the Euclidean path integral for such boundary conditions. We further show that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS within the cavity. The larger hole thus can be globally thermodynamically stable above this temperature. The phase transition can occur for a cavity of arbitrary radius above a (corresponding) critical temperature. In the infinite cavity limit this picture reduces to that considered by Hawking and Page. The case of five dimensions is found to be rather special since exact analytic expressions can be obtained for the masses of the two holes as functions of cavity radius and temperature thus solving exactly the Euclidean Dirichlet problem. This makes it possible to compute the on-shell Euclidean action as functions of them from which other quantities of interest can be evaluated exactly.