TOPOLOGICAL BLACK HOLES OF (n+1)-DIMENSIONAL EINSTEIN–YANG–MILLS GRAVITY

Abstract

We present the topological solutions of Einstein gravity in the presence of a non-Abelian Yang–Mills field. In (n+1) dimensions, we consider the SO (n(n-1)/2-1, 1) semisimple group as the Yang–Mills gauge group, and introduce the black hole solutions with hyperbolic horizon. We argue that the four-dimensional solution is exactly the same as the four-dimensional solution of Einstein–Maxwell gravity, while the higher-dimensional solutions are new. We investigate the properties of the higher-dimensional solutions and find that these solutions in five dimensions have the same properties as the topological five-dimensional solution of Einstein–Maxwell (EM) theory although the metric function in five dimensions is different. But in six and higher dimensions, the topological solutions of EYM and EM gravities with non-negative mass have different properties. First, the singularity of EYM solution does not present a naked singularity and is spacelike, while the singularity of topological Reissner–Nordström solution is timelike. Second, there are no extreme six or higher-dimensional black holes in EYM gravity with non-negative mass, while these kinds of solutions exist in EM gravity. Furthermore, EYM theory has no static asymptotically de Sitter solution with non-negative mass, while EM gravity has.