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

TL;DR

This paper explores the thermodynamics of rotating de Sitter black holes by analyzing boundaries at either the black hole or cosmological horizon, revealing their distinct properties and connections to anti-de Sitter cases.

Contribution

It introduces a framework for analyzing black hole thermodynamics with horizons as boundaries, clarifying their differences and relations to anti-de Sitter scenarios.

Findings

Energy and angular momentum are surface terms with opposite signs for the two horizons.

Thermodynamic parameters depend on the boundary chosen, either black hole or cosmological horizon.

Continuing the cosmological constant to negative values links de Sitter and anti-de Sitter thermodynamics.

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.