de Sitter Black Holes with Either of the Two Horizons as a Boundary

TL;DR

This work analyzes Kerr--de Sitter black holes with a positive cosmological constant by formulating two horizon-boundary action principles in which either the cosmological or the black hole horizon serves as the boundary. It derives energy $U$ and angular momentum $J$ as boundary surface terms, with explicit expressions that change sign when swapping horizon roles and defines the angular velocity (chemical potential) $\Omega$ in terms of horizon data, yielding the entropy $S = 4\pi A$. The paper also connects the de Sitter case to its anti--de Sitter counterpart via analytic continuation $\Lambda \to -\Lambda$, and discusses the implications for the thermodynamics and ensemble structure of horizons in spacetimes without spatial infinity.

Abstract

The action and the thermodynamics of a rotating black hole in the presence of a positive cosmological constant are analyzed. Since there is no spatial infinity, one must bring in, instead, a platform where the parameters characterizing the thermodynamic ensemble are specified. In the present treatment the platform in question is taken to be one of the two horizons, which is considered as a boundary. If the boundary is taken to be the cosmological horizon one deals with the action and thermodynamics of the black hole horizon. Conversely, if one takes the black hole horizon as the boundary, one deals with the action and thermodynamics of the cosmological horizon. The two systems are different. Their energy and angular momenta are equal in magnitude but have opposite sign. In either case, the energy and the angular momentum are obtained as surface terms on the boundary, according to the standard Hamiltonian procedure. The temperature and the rotational chemical potential are also expressed in terms of magnitudes on the boundary. If, in the resulting expressions, one continues the cosmological constant to negative values, the black hole thermodynamic parameters defined on the cosmological horizon coincide with those calculated at spatial infinity in the asymptotically anti-de Sitter case.