Topological Black Holes of (n+1)-dimensional Einstein-Yang-Mills Gravity

TL;DR

This paper explores topological black hole solutions in higher-dimensional Einstein-Yang-Mills gravity, revealing differences from Einstein-Maxwell solutions in dimensions six and above, including singularity structure and existence of certain black holes.

Contribution

It introduces new higher-dimensional solutions in Einstein-Yang-Mills gravity and compares their properties with Einstein-Maxwell solutions, highlighting key differences in dimensions six and higher.

Findings

Higher-dimensional EYM solutions have spacelike singularities, unlike timelike in EM.

No extreme black holes or de Sitter solutions in EYM for dimensions ≥6.

4D EYM solutions match Einstein-Maxwell solutions exactly.

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 4-dimensional solution is exactly the same as the 4-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 5 dimensions have the same properties as the topological 5-dimensional solution of Einstein-Maxwell (EM) theory although the metric function in 5 dimensions is different. But in 6 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-Nordstrom solution is timelike. Second, there are no extreme 6 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.