Topological dilaton black holes

In the four-dimensional spacetime, when the two-sphere of black hole event horizons is replaced by a two-dimensional hypersurface with zero or negative constant curvature, the black hole is referred as to a topological black hole. In this paper we present some exact topological black hole solutions in the Einstein-Maxwell-dilaton theory with a Liouville-type dilaton potential. PACS numbers: 04.20.Jb, 04.20.Gz, 04.70.Dy Typeset using REVTEX 1 The topological structure of event horizon of black holes is an intriguing subject in black hole physics. It is generally believed that black holes in the four dimensional spacetime are always with a spherical topology. That is, the event horizon of black holes has the topology S. This was proved by Friedman, Schleich and Witt [1]. They suggested a “topological censorship theorem”, which states that in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve (nonspacelike curve) from J − to J + is homotopic to a topologically trivial curve from J − to J . That is, general relativity does not allow an observer to probe the topology of spacetime: Any topological structure collapses too quickly to allow light to traverse it. Later on , however, they found that nontrivial topologies could be observed passively [2]. The black holes with toroidal topology have indeed been found numerically in the gravitational collapse [3], although such a topological structure is temporal. When the asymptotic flatness and energy condition are given up, there are no fundamental reasons to forbid the existence of static or stationary black holes with nontrivial topologies. In particular, when the spacetime is asymptotically anti-de Sitter one, the matter field can be in stable equilibrium even if the potential energy is unbounded from below. In recent years, there has been a growing interest in these black holes with nontrivial topological structures (topological black holes) in the asymptotically anti-de Sitter space [4–13]. These investigations are mainly based on the Einstein (-Maxwell) theory with a negative cosmological constant. In general, one has the static solutions to Einstein-Maxwell equations with a cosmological constant ds = −(k − 2M r + Q r2 − 1 3 Λr)dt + (k − 2M r + Q r2 − 1 3 Λr)dr + rdΩk, (1) where dΩk is the line element of a two-dimensional hypersurface Σ with constant curvature, dΩk = 