Mass Formulas for Stationary Einstein-Yang-Mills Black Holes and a Simple Proof of Two Staticity Theorems

TL;DR

A formula for JΩ-QV is derived, from which it follows immediately that any stationary, nonrotating, uncharged black hole is static and has a vanishing electric field on the static slices.

Abstract

We derive two new integral mass formulas for stationary black holes in Einstein-Yang-Mills theory. From these we derive a formula for $ \J Ω-Q V $, 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.