A Yang-Mills field on the extremal Reissner-Nordstr\"om black hole

TL;DR

This paper studies a magnetic SU(2) Yang-Mills field around an extremal Reissner-Nordström black hole, revealing infinitely many static solutions, their stability spectrum, and the dynamics of relaxation to these states.

Contribution

It demonstrates the existence of infinitely many static solutions and analyzes their stability and evolution, using conformal symmetry and hyperboloidal methods.

Findings

Existence of infinitely many static solutions

Spectrum of linear perturbations and quasinormal modes identified

Described relaxation process to static endstates

Abstract

We consider a spherically symmetric (magnetic) $SU(2)$ Yang-Mills field propagating on the exterior of the extremal Reissner-Nordstr\"om black hole. Taking advantage of the conformal symmetry, we reduce the problem to the study of the Yang-Mills equation in a geodesically complete spacetime with two asymptotically flat ends. We prove the existence of infinitely many static solutions (two of which are found in closed form) and determine the spectrum of their linear perturbations and quasinormal modes. Finally, using the hyperboloidal approach to the initial value problem, we describe the process of relaxation to the static endstates of evolution for various initial data.