On all possible static spherically symmetric EYM solitons and black holes

TL;DR

This paper proves the local existence and uniqueness of static spherically symmetric Einstein-Yang-Mills solutions for various gauge groups, analyzing both regular and irregular cases, and discusses the complexity of finding global solutions.

Contribution

It establishes the local existence and uniqueness of solutions for a broad class of Einstein-Yang-Mills equations, including irregular cases, and explores the parameters and methods for global solution search.

Findings

Local solutions exist near singularities, horizons, and infinity.

Regular solutions can be characterized by specific parameters.

The set of global solutions, especially irregular ones, remains largely unexplored.

Abstract

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over space-time whose structure group is a compact semisimple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of g and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang-Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the center, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that su(2) is a subalgebra of any compact semisimple Lie algebra. But the set of less trivial global solutions remains to be explored.