Euclidean de Sitter Black Holes and Microcanonical Equilibrium

TL;DR

This paper explores modifications to the gravitational path integral to interpret Euclidean Schwarzschild-de Sitter black holes as microcanonical equilibrium states, addressing issues with standard approaches and analyzing their thermodynamic stability.

Contribution

It introduces a microcanonical ensemble approach for Euclidean SdS black holes, overcoming limitations of the canonical ensemble in positive cosmological constant scenarios.

Findings

Microcanonical ensemble provides a well-defined partition function for SdS black holes.

States with black holes are thermodynamically unstable.

Discontinuities modeled as membranes facilitate energy exchange in closed systems.

Abstract

Schwarzschild-de Sitter (SdS) black holes do not admit a completely smooth Euclidean continuation. We discuss some modifications of the gravitational path integral that give Euclidean SdS a semiclassical equilibrium interpretation. First we consider "gravity in a cavity," defining the canonical ensemble in a box that excises one horizon. However, this standard approach does not work for positive cosmological constant: the solution of lowest free energy has a negative heat capacity, which is inconsistent if it is to provide the leading semiclassical contribution to a canonical partition function. Instead we modify the boundary conditions in the path integral to construct the microcanonical partition function, which appears to be well-defined. We then bring two ensembles into contact and remove the boundary, producing states of a larger microcanonical ensemble that contain, for example, both a black hole and a cosmological horizon at once. These systems are closed and have no boundary, but they must possess some form of mild metric discontinuity. We discuss the case where the discontinuity is equivalent to the insertion of a thin, rigid membrane, separating two systems that can exchange energy and are at local equilibrium. Equilibrium configurations obtained in this way are found to be thermodynamically unstable if they contain a black hole.