Mass formulas for stationary Einstein-Yang-Mills black holes and a simple proof of two staticity theorems.

We derive two new integral mass formulas for stationary black holes in EinsteinYang-Mills theory. From these we derive a formula for JΩ − QV , from which it follows immediately that any stationary, nonrotating, uncharged black hole is static and has vanishing electric field on the static slices. In the Einstein-Maxwell case, we have, in addition, the “generalized Smarr mass formula”, for which we provide a new, simple derivation. When combined with the other two formulas, we obtain a simple proof that nonrotating Einstein-Maxwell black holes must be static and have vanishing magnetic field on the static slices. Our mass formulas also can be generalized to cases with other types of matter fields, and we describe the nature of these generalizations. In a recent paper by the authors [1], it was shown, among other things, that a solution of the Einstein-Yang-Mills (EYM) equations describing a stationary black hole with bifurcate Killing horizon and satisfying V Q = JΩ = 0 is necessarily static and has vanishing electric field on the static slices. A stronger result was obtained in the Einstein-Maxwell case: It was proven that a solution of the Einstein-Maxwell equations describing a stationary black hole with bifurcate Killing horizon and satisfying JΩ = 0 is necessarily static and has vanishing magnetic field on the static slices. These staticity theorems were obtained by deriving a generalized first law of black hole mechanics, using it to infer extremal properties of stationary black hole solutions, and then showing that these extremal properties could be violated unless the black hole is static. The theorems do not require the stationary Killing field to be globally timelike in the exterior region, i.e., “ergoregions” are permitted. Thus, in particular, the Einstein-Maxwell staticity theorem closed a gap in the black hole uniqueness theorems which had been open for nearly two decades (see [2]). The purpose of this paper is to derive some new “mass formulas” relating the asymptotically defined attributes of a stationary black hole in EYM theory and to use them