A Note on the Gannon-Lee Theorem

TL;DR

This paper extends the Gannon-Lee theorem to $C^1$ Lorentzian metrics, the broadest class for classical singularity theorems, and shows maximizing causal curves are geodesics with $C^2$ regularity.

Contribution

It proves a Gannon-Lee theorem for $C^1$ Lorentzian metrics and establishes that maximizing causal curves in such spacetimes are geodesics with $C^2$ regularity.

Findings

Gannon-Lee theorem holds for $C^1$ Lorentzian metrics.

Maximizing causal curves in $C^1$ spacetimes are geodesics.

Such geodesics are of $C^2$ regularity.

Abstract

We prove a Gannon-Lee theorem for non-globally hyperbolic Lo\-rentzian metrics of regularity $C^1$, the most general regularity class currently available in the context of the classical singularity theorems. Along the way we also prove that any maximizing causal curve in a $C^1$-spacetime is a geodesic and hence of $C^2$-regularity.