Local well-posedness of the Einstein-Yang-Mills system in CMCSHGC gauge

TL;DR

This paper proves the local well-posedness of the Einstein-Yang-Mills equations in a specific gauge, demonstrating the existence and uniqueness of solutions for a coupled elliptic-hyperbolic system, with implications for future continuation criteria.

Contribution

It establishes the local existence and uniqueness of solutions to the Einstein-Yang-Mills system in CMCSHGC gauge, using a coupled elliptic-hyperbolic formulation and existing methods.

Findings

Existence of unique local solutions in the specified gauge

Development of an in-time continuation criterion

Foundation for future improved continuation criteria

Abstract

We study the local well-posedness of the Einstein-Yang-Mills equations in constant mean extrinsic curvature spatial harmonic generalized Coulomb gauge (CMCSHGC). In this choice of gauge, the complete Einstein-Yang-Mills equations reduce to a coupled elliptic-hyperbolic system. Utilizing the method developed by Andersson and Moncrief \cite{andersson2003elliptic}, we establish the existence of a unique, local, continuous-in-time solution of this coupled system. This yields an `in time' continuation criteria of the solutions which is to be used in the potential future proof of an improved continuation criteria for this coupled system utilizing Moncrief's light cone estimate technique.