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 Einstein-Yang-Mills theory. From these we derive a formula for scrJ\ensuremath{\Omega}-QV, from which it follows immediately that any stationary, nonrotating, uncharged black hole is static and has a 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 a 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.