Yang--Mills Monopoles in Extremal Reissner--Nordstr\"{o}m Black Hole Metric

TL;DR

This paper analyzes Yang--Mills monopoles in extremal Reissner--Nordström black hole spacetime, proving uniqueness of a key solution, deriving related equations, and exploring various solutions with different behaviors.

Contribution

It establishes the uniqueness of a specific monotone solution linking vacuum states and derives the interior wave equation within the same framework.

Findings

Proves the uniqueness of the monotone solution

Derives the interior Yang--Mills wave equation

Identifies multiple closed-form solutions with different behaviors

Abstract

We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang--Mills wave equation over the extremal Reissner--Nordstr\"{o}m spacetime found in a recent work of Bizo\'{n} and Kahl, is the unique solution to an associated domain-wall type energy minimization problem. We also derive the accompanying interior Yang--Mills wave equation within the same formalism. We obtain a few more closed-form solutions, regular and singular, among other solutions of oscillatory behavior, and discuss several interesting features of the solutions based on some energy consideration.