Gravitational Thermodynamics of Schwarzschild-de Sitter Space

TL;DR

This paper investigates the thermodynamics of Schwarzschild-de Sitter space from two perspectives, analyzing how fixing different horizons affects the thermodynamic parameters and revealing potential instability of de Sitter space.

Contribution

It provides a detailed analysis of the thermodynamics of Schwarzschild-de Sitter space under different boundary conditions, calculating the internal energy in each case.

Findings

Internal energy U=+m when fixing parameters at the cosmological horizon.

Internal energy U=-m when fixing parameters at the black hole horizon.

De Sitter space may be thermodynamically unstable under black hole formation.

Abstract

The Euclidean Schwarzschild-de Sitter geometry may be considered as an extremum of two different action principles. If the thermodynamical parameters are held fixed at the cosmological horizon, one deals with the gravitational thermodynamical effects of the black hole but ignores those of the cosmological horizon. Conversely, if the macroscopical variables are held fixed at the black hole horizon, it is only the cosmological horizon thermodynamics which is dealt with. Both cases are analyzed. In particular, the internal energy U is calculated in the semiclassical approximation as a function of the mass parameter m of Schwarzschild de Sitter space. In the first case one finds U=+m, while in the second one gets U=-m. This suggests that de Sitter space is thermodynamically unstable under black hole formation.