Global behavior of solutions to the static spherically symmetric EYM equations

TL;DR

This paper analyzes the asymptotic behavior of globally bounded solutions to static spherically symmetric Einstein-Yang-Mills equations for arbitrary gauge groups, revealing finite or complex asymptotic value sets depending on the model.

Contribution

It derives asymptotic properties of global solutions for the Einstein-Yang-Mills equations with arbitrary gauge groups, extending understanding beyond known embedded solutions.

Findings

Asymptotic values of the Yang-Mills potential are finite in regular cases.

In irregular cases, asymptotic values form complex real varieties.

Global solutions with positive mass and no horizons exhibit specific asymptotic behaviors.

Abstract

The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a black hole horizon as well as those that are analytic and bounded near infinity were shown to exist. Some globally bounded solutions are also known to exist because they can be obtained by embedding solutions for the $G=SU(2)$ case which is well understood. Here we derive some asymptotic properties of an arbitrary global solution, namely one that exists locally near a radial value $r_{0}$, has positive mass $m(r)$ at $r_{0}$ and develops no horizon for all $r>r_{0}$. The set of asymptotic values of the Yang-Mills potential (in a suitable well defined gauge) is shown to be finite in the so-called regular case, but may form a more complicated real variety for models obtained from irregular rotation group actions.