Extrema of mass, stationarity, and staticity, and solutions to the Einstein-Yang-Mills equations

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Abstract
A simple formula is derived for the variation of mass and other asymptotic conserved quantities in Einstein-Yang-Mills theory. For asymptotically flat initial data with a single asymptotic region and no interior boundary, it follows directly from our mass-variation formula that initial data for stationary solutions are extrema of mass at fixed electric charge. When generalized to include an interior boundary, this formula provides a simple derivation of a generalized form of the first law of black-hole mechanics. We also argue, but do not rigorously prove, that in the case of a single asymptotic region with no interior boundary stationarity is necessary for an extremum of mass at fixed charge; when an interior boundary is present, we argue that a necessary condition for an extremum of mass at fixed angular momentum, electric charge, and boundary area is that the solution be a stationary black hole, with the boundary serving as the bifurcation surface of the horizon. Then, by a completely different argument, we prove that if a foliation by maximal slices (i.e., slices with a vanishing trace of extrinsic curvature) exists, a necessary condition for an extremum of mass when no interior boundary is present is that the solution be static. A generalization of the argument to the case in which an interior boundary is present proves that a necessary condition for a solution of the Einstein-Yang-Mills equation to be an extremum of mass at fixed area of the boundary surface is that the solution be static.

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