Einstein-Yang-Mills Isolated Horizons: Phase Space, Mechanics, Hair and Conjectures

TL;DR

This paper extends the isolated horizons framework to Einstein-Yang-Mills black holes, establishing laws, analyzing masses, and proposing conjectures about their uniqueness and stability within a Hamiltonian formalism.

Contribution

It generalizes the isolated horizons approach to non-Abelian gauge theories, deriving laws, formulas, and conjectures for hairy black holes in Einstein-Yang-Mills theory.

Findings

Established zeroth and first laws for non-Abelian horizons

Derived a canonical formula for the Horizon Mass of colored black holes

Proposed conjectures on the uniqueness and stability of Einstein-Yang-Mills black holes

Abstract

The concept of "Isolated Horizon" has been recently used to provide a full Hamiltonian treatment of black holes. It has been applied successfully to the cases of {\it non-rotating}, {\it non-distorted} black holes in Einstein Vacuum, Einstein-Maxwell and Einstein-Maxwell-Dilaton Theories. In this note, it is investigated the extent to which the framework can be generalized to the case of non-Abelian gauge theories where `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 non-linear 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 Horizons framework for Einstein-Yang-Mills theory in the same footing as the previously analyzed cases. Motivated by these considerations and using the fact that the Isolated Horizons 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.