The $C^0$-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian Geometry

TL;DR

This paper proves the Schwarzschild spacetime cannot be extended as a Lorentzian manifold with a continuous metric, introducing the concept of spacelike diameter to understand inextendibility related to curvature singularities.

Contribution

It establishes the inextendibility of Schwarzschild spacetime with a continuous metric and introduces the spacelike diameter concept to analyze low-regularity inextendibility.

Findings

Schwarzschild spacetime is inextendible with a continuous metric

Introduces the notion of spacelike diameter for Lorentzian manifolds

Provides a criterion for finiteness of spacelike diameter

Abstract

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.