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.

This enables us to prove (modulo the existence of a maximal slice) that if the stationary Killing field of a stationary black hole with bifurcate Killing horizon is normal to the horizon, and if the electrostatic potential asymptotically vanishes at infinity, then the black hole must be static. (This closes a significant gap in the black-hole uniqueness theorems.) Finally, by generalizing the type of argument used to predict the ‘‘sphaleron’’ solution of Yang-Mills-Higgs theory, we argue that the initial-data space for Einstein-Yang-Mills theory with a single asymptotic region should contain a countable infinity of saddle points of mass. Similarly in the case of an interior boundary, there should exist a countable infinity of saddle points of mass at fixed boundary area. We propose that this accounts for the existence and properties of the Bartnik-McKinnon and colored black-hole solutions. Similar arguments in the black-hole case indicate the presence of a countable infinity of extrema of mass at fixed area, electric charge, and angular momentum, thus suggesting the existence of colored generalizations of the charged Kerr solutions. A number of other conjectures concerning stationary solutions of the Einstein-Yang-Mills equations and related systems are formulated. Among these is the prediction of the existence of a countable infinity of new static solutions of the Yang-Mills-Higgs equations related to the sphaleron.

• Received 15 October 1991

DOI:https://doi.org/10.1103/PhysRevD.46.1453