Skyrme Black Holes in the Isolated Horizons Formalism

The isolated horizons formalism [1] has been used to formulate several conjectures regarding static black holes [2],[3]. In this article we study the static, spherically symmetric Einstein-Skyrme black hole solutions first found in [4],[5],[6]. The Einstein-Skyrme solutions have a number of features that, while generic to certain models involving non-abelian gauge fields [7], are not features of models that have been considered up to now in the context of the isolated horizons conjectures [8],[9]. Like a large class of models admitting solitons in flat space, Skyrme black holes have an upper limit on the radius of the black hole. This involves the merging of two different branches of solutions at some finite horizon radius. In addition they admit static solutions that are not spherically symmetric [10], thus extending the class of static solutions. Perhaps most importantly Skyrme black holes are of particular interest in the context of the no-hair theorem since they have been shown to be linearly stable [5],[11],[12] and thus demonstrate that exceptions to the no-hair theorem are not necessarily unstable. However, it has yet to be resolved whether Skyrme black holes are non-linearly stable and it is possible that the conjectures stemming from the isolated horizon formalism could shed some light on this unresolved issue. One of the advantages of the isolated horizons formalism is that it characterizes black hole solutions in terms of horizon values of their parameters rather than their values at infinity. Thus the formalism provides a well-suited framework in which to investigate so-called “hairy” black holes, where solutions that are indistinguishable at infinity have different parameter values at the horizon. This paper is organised as follows: Section one reviews the basics of the Skyrme model and the black hole solutions found previously [5],[6]. Section two reviews the essential elements of the isolated horizon formalism and the conjectures for static black holes that it gives rise to. Section three presents our results and a discussion of their relevance to the conjectures. Our metric convention is (−,+,+,+) and we use units where c = 1, although since we often want to compute masses in terms of spatial integrals we keep all factors of G. Otherwise our conventions are as appear in Wald [13].