Null injectivity estimate under an upper bound on the curvature

TL;DR

This paper derives a uniform estimate for the injectivity radius of the past null cone in Lorentzian manifolds with bounded null curvature, applicable even when curvature is large or unbounded, aiding in understanding spacetime geometry in general relativity.

Contribution

It introduces a novel null injectivity estimate under an upper curvature bound without restrictions on curvature size, extending geometric control in Lorentzian manifolds.

Findings

Establishes a uniform injectivity radius estimate under upper null curvature bounds.

Demonstrates applicability to spacetimes with unbounded curvature, including plane-symmetric models.

Provides tools for geometric analysis in general relativity with less restrictive curvature conditions.

Abstract

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation, and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the curvature or deformation tensors, and allow for metrics that are "far" from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the L2 norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity.