Regularity for Lorentz Metrics under Curvature Bounds

TL;DR

This paper proves that under certain curvature bounds, Lorentzian metrics in space-time manifolds can be locally expressed in coordinate systems with controlled Sobolev regularity, enhancing understanding of geometric regularity under curvature constraints.

Contribution

It establishes local regularity results for Lorentz metrics under curvature bounds, applicable to vacuum and certain matter models, with Sobolev space control.

Findings

Lorentz metrics are locally in L^{2,p} Sobolev spaces under curvature bounds.

Regularity results hold for vacuum and mild stress-energy conditions.

Provides a framework for analyzing geometric regularity in space-time manifolds.

Abstract

Let (M, g) be an (n+1) dimensional space-time, with bounded curvature with respect to a bounded framing. If (M, g) is vacuum or satisfies a mild condition on the stress-energy tensor, then we show that (M, g) locally admits coordinate systems in which the Lorentz metric is well-controlled in the (space-time) Sobolev space L^{2,p}, for any finite p.