Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons

TL;DR

The paper addresses whether Bekenstein–Hawking thermodynamics extends to four-dimensional AdS black holes with non-spherical horizons by replacing the S^2 with Σ_k (k=0,−1). It develops the global spacetime structure, computes the local Hawking temperature, and uses Brown–York quasilocal energy to formulate the first law with a finite boundary, showing the entropy obeys $S=\tfrac14 A_h$ for toroidal and higher-genus horizons. It then analyzes thermodynamic stability via heat capacities in finite boxes and constructs infinite-space ensembles (grand canonical and canonical) that are well-defined and dominated by a single classical solution, confirming stability and the persistence of the area law. The results extend topological black hole thermodynamics in AdS and have implications for holography and the role of horizon topology in gravitational thermodynamics.

Abstract

We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution.

Figures (5)

• Figure 1: The Penrose diagram for $M<M_{\rm crit}$. The straight line indicates an infinity and the wavy line a singularity.
• Figure 2: The Penrose diagram for $M=M_{\rm crit}$, if $Q\ne0$ or $k=-1$ or both. The point $p$ is an internal spacelike infinity, and the singularity consists of countably many connected components. The infinity, which is both spacelike and (future) null, consists of a single connected component. As the past of the infinity consists of all of the spacetime, the spacetime does not have an interpretation as a black hole.
• Figure 3: The Penrose diagram for $M>M_{\rm crit}$ if $Q\ne0$, and for $M_{\rm crit}< M < 0$ if $Q=0$ and $k=-1$. There is both an outer horizon and an inner horizon.
• Figure 4: The Penrose diagram for $M>0$ if $Q=0$, and for $M=0$ if $Q=0$ and $k=-1$. If $\Sigma_{-1}$ is simply connected, the singularity in the latter case is a coordinate one.
• Figure 5: The Penrose diagram for $M=0$, if $Q=0$ and $k=0$. If $\Sigma_0$ is simply connected, the singularity is a coordinate one.