New Hairy Black Holes with Negative Cosmological Constant

Black hole solutions with nonspherical event horizon topology are shown to exist in an EinsteinYang-Mills theory with negative cosmological constant. The main characteristics of the solutions are presented and differences with respect to the spherically symmetric case are studied. The stability of these configurations is also addressed. PACS: 04.40; 04.70.-s; 98.80.E Although anti-de Sitter (AdS) spacetime does not seem to correspond to the world in which we live [1], its importance has been noticed in many occasions. As shown by Hawking and Page, the presence of a negative comological constant Λ makes it possible for a black hole to reach stable thermal equilibrium with a heat bath [2]. The black holes discussed by Hawking and Page have a spherically symmetric event horizon. The topological structure of the event horizon of a black hole is an intriguing subject in black hole physics. When asymptotic flatness and the energy conditions are given up, there are no fundamental reasons to forbid the existence of black holes with nontrivial topologies. In particular, for a negative cosmological constant, black holes for which the topology of the horizon is an arbitrary genus Riemann surface have been considered by many authors (see e.g. [3]-[15]). The thermodynamics of these solutions has been discussed in [8]-[11] while higher dimensional generalizations were obtained in [12]-[14]. The theorems about spherical horizon topology do not apply here because the negative cosmological constant can be interpreted as a negative vacuum energy density. They generalize the known solutions replacing the round two-sphere by a two-dimensional space Σ of negative or vanishing curvature. These black holes embedded in ’locally AdS’ background spacetimes (background locally isometric to spacetimes of constant negative curvature) have been seminal to recent developments in black hole physics. All these investigations are mainly based on Einstein(-Maxwell) theory (note however the inclusion of a dilaton field in [15]). It may be of interest to generalize these solutions for a nonabelian matter content. Here we restrict attention to SU(2) Einstein-Yang-Mills (EYM) theory and investigate black hole solutions with locally flat or hyperbolic horizons, which asymptotically approach a locally AdS spacetime. The case of a spherically symmetric horizon has been discussed in [16, 17] with some surprising results. For example, there are finite energy solutions for several intervals of the shooting parameter (the value of gauge function at event horizon), rather than discrete values. Solutions exist for all values of Λ < 0. There are also stable solutions in which the gauge field has no zeros, and the asymptotic value of the gauge potential is not fixed. corresponding author: Albert-Ludwigs-Universität Freiburg, Fakultät für Physik, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany email: [email protected] Telephone: +49-761/203-7630 Fax: +49-761/203-5967