ar X iv : g r - qc / 9 70 50 04 v 1 5 M ay 1 99 7 Black holes with unusual topology

The Einstein’s equations with a negative cosmological constant admit solutions which are asymtotically anti-de Sitter space. Matter fields in anti-de Sitter 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. Here it is shown that 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 size of the torus. The genus g > 1 black hole has no other free parameters apart from the mass. All such black holes exihibits a natural temperature which is identified as the period of its euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. But due to a peculiar character of some of its properties it is unknown, to the author, whether the black hole could form as a result of gravitational collapse. PACS numbers: 04.20.-q, 04.70.Bw, 04.70.Dy In general relativity it is widely believed that black holes formed by gravitazional collapse have spherical horizon [1]. The Einstein’s equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. These solutions have spherical horizon and obey thermodynamics laws like asymptotically flat black holes [2, 3]. In 2+1 dimensions there are the recently discovered globally anti-de Sitter black hole solutions [4], which have constant curvature everywhere, not just asymptotically at infinity. The horizon of a 2+1 solutions is a closed line, which leaves not much space for introducing non trivial topology. On the other hand, there do not seems to exist a reasonable, higher dimensional generalization of the BTZ black hole. Those recently found have horizons with spherical topology, but all the anti-de Sitter conserved charges are infinite [5]. We presents here a solution describing a family of black holes with non trivial horizon topology. The solution was inspired by trying to generalize to higher dimensions the BTZ construction in 2 + 1 dimensions. Although anti-de Sitter space does not seems to correspond to the world in which we live, its importance has been noticed in many occasions [6–11]. Two features are worth to mention. Firstly, anti-de Sitter and Weyl conformal gravity are the only type of gravity which have a consistent interaction with massless higher spin fields [12] and, secondly, consistent anti-de Sitter strings exist for any D 6= 26 (or D 6= 10) [13], provided the cosmological term has the critical value which is required by anomaly cancellation. After this brief digression, we begin by presenting the metric, and then we shall discuss the relevant geometric features. We compute the mass the euclidean action and use it to calculate the entropy of the black hole. We shall not discuss entirely its causal structure, nor we make it confident that the black hole could result from gravitational collapse of some, topologically non ∗e-mail: [email protected]