On the Global Well-Posedness of the Einstein-Yang-Mills System

TL;DR

This paper investigates the global well-posedness of the Einstein-Yang-Mills system, establishing small-data global existence in higher dimensions and explaining why similar results are not achievable in four-dimensional spacetime due to conformal invariance.

Contribution

It provides a small-data global existence theorem for higher-dimensional Einstein-Yang-Mills systems and demonstrates the limitations of energy methods in four dimensions.

Findings

Global existence proven for n+1 dimensions with n≥4

Energy methods fail for 3+1 dimensions due to conformal invariance

A gauge-covariant formulation shows energy arguments are insufficient in 4D

Abstract

In this paper, we present a partial result on the global well-posedness of the Cauchy problem for the Einstein-Yang-Mills system in the constant mean extrinsic curvature spatial harmonic and generalized Coulomb gauges as introduced in [Mondal, arXiv:2112.14273]. We give a small-data global existence theorem for a family of $n+1$ dimensional spacetimes with $n\geq4$, utilizing energy arguments presented in [Andersson and Moncrief, arXiv:0908.0784]. We observe that these energy arguments will fail for $n=3$ due to the conformal invariance of the $3+1$ Yang-Mills equations and present a gauge-covaraiant formulation of the Einstein-Yang-Mills system in $3+1$ dimensions to show that an energy argument cannot be used to prove the global well-posedness result, regardless of the choice of gauge.