Gravitational Thermodynamics of Schwarzschild-de Sitter Space

TL;DR

Teitelboim analyzes the thermodynamics of Schwarzschild–de Sitter space by treating the Euclidean geometry as an extremum of two distinct action principles, corresponding to fixing variables at the cosmological horizon or at the black-hole horizon. The on-shell action is shown to equal $I_{ ext{on-shell}}=rac{1}{4}A_H$, yielding entropies $S_{+}= m{ pi} r_{+}^{2}$ and $S_{++}= m{ pi} r_{++}^{2}$, while a semiclassical first-law analysis gives internal energies $U_{+}=+m$ and $U_{++}=-m$, indicating a thermodynamic instability under black hole formation. The framework relies on horizon boundary conditions and conical singularities, demonstrating that the two horizons cannot be simultaneously in thermal equilibrium and revealing a negative specific heat $C<0$ for both configurations. The study discusses possible dynamical outcomes, such as the Nariai limit or persistent non-equilibrium energy exchange, highlighting the nontrivial coupling between horizons and the unresolved question of true equilibrium in de Sitter thermodynamics.

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.