Black holes with unusual topology

TL;DR

The paper demonstrates that asymptotically anti-de Sitter gravity admits black holes with non-spherical horizons, including higher-genus and toroidal topologies. It builds explicit metric Ansätze with constant-curvature horizon surfaces, derives horizon properties and a mass formula, and analyzes the Euclidean action to establish thermodynamic stability. By carefully choosing the reference background in the same topology class, the authors obtain a positive mass spectrum and entropy consistent with the Bekenstein-Hawking area law in the extremal limit, highlighting subtle issues in holographic interpretations. The work underscores the role of topology in AdS black hole thermodynamics and points to deeper connections with string theory and the dynamics of horizon topology.

Abstract

The Einstein's equations with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter space. Matter fields in anti-de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition. Hence there is no fundamental reason that black hole's horizons should have spherical topology. In anti-de Sitter space the Einstein's equations admit black hole solutions where the horizon can be a Riemann surface with genus $g$. The case $g=0$ is the asymptotically anti-de Sitter black hole first studied by Hawking-Page, which has spherical topology. The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs. The genus $g>1$ black hole has no other free parameters apart from the mass and the charge. All such black holes exhibits a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite.