Static Spherically Symmetric Solutions of the SO(5) Einstein Yang-Mills Equations

TL;DR

This paper explores new static, spherically symmetric solutions to the Einstein-Yang-Mills equations with gauge group SO(5), revealing complex solution structures and simplifying the analysis through gauge-invariant methods.

Contribution

It introduces an algebraic gauge condition for SO(5) EYM equations, enabling classification and numerical exploration of solutions with residual non-Abelian symmetry.

Findings

Discovery of three families of embedded SU(2) solutions within SO(5) EYM system

Numerical solutions classified by pairs of positive integers

Simplification of the system using gauge-invariant polynomials

Abstract

Globally regular (ie. asymptotically flat and regular interior), spherically symmetric and localised ("particle-like") solutions of the coupled Einstein Yang-Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang--Mills fields are classified by a choice of isotropy generator and SO(5) is distinguished as the simplest model with a \emph{non-Abelian} residual (little) group, $SU(2)\times U(1)$, and which admits globally regular particle-like solutions. We exhibit an algebraic gauge condition which normalises the residual gauge freedom to a finite number of discrete symmetries. This generalises the well-known reduction to the real magnetic potential $w(r,t)$ in the original SU(2) YM model. Reformulating using gauge invariant polynomials dramatically simplifies the system and makes numerical search techniques feasible. We find three families of embedded SU(2) EYM equations within the SO(5) system, one of which was first detected only within the gauge-invariant polynomial reduced system. Numerical solutions representing mixtures of the three SU(2) sub-systems are found, classified by a pair of positive integers.