Einstein-Yang-Mills isolated horizons: Phase space, mechanics, hair, and conjectures

The concept of an “isolated horizon” has been recently used to provide a full Hamiltonian treatment of black holes. It has been applied successfully to the cases of nonrotating, nondistorted black holes in the Einstein vacuum, Einstein-Maxwell, and Einstein-Maxwell-dilaton theories. In this paper, the extent to which the framework can be generalized to the case of non-Abelian gauge theories is investigated in which “hairy black holes” are known to exist. It is found that this extension is indeed possible, despite the fact that, in general, there is no “canonical normalization” yielding a preferred horizon mass. In particular the zeroth and first laws are established for all normalizations. Colored static spherically symmetric black hole solutions to the Einstein-Yang-Mills equations are considered from this perspective. A canonical formula for the horizon mass of such black holes is found. This analysis is used to obtain nontrivial relations between the masses of the colored black holes and the regular solitonic solutions in Einstein-Yang-Mills theory. A general testing bed for the instability of hairy black holes in general nonlinear theories is suggested. As an example, the embedded Abelian magnetic solutions are considered. It is shown that, within this framework, the total energy is also positive and thus the solutions are potentially unstable. Finally, it is discussed which elements would be needed to place the isolated horizon framework for Einstein-Yang-Mills theory on the same footing as the previously analyzed cases. Motivated by these considerations and using the fact that the isolated horizon framework seems to be the appropriate language to state uniqueness and completeness conjectures for the EYM equations, in terms of the horizon charges, two such conjectures are put forward.