# Timelike completeness as an obstruction to $C^0$-extensions

## TL;DR

This paper proves that a timelike complete and globally hyperbolic Lorentzian manifold cannot be extended as a continuous metric, using recent results on length-maximizing causal curves in such spacetimes.

## Contribution

It establishes timelike completeness as an obstruction to $C^0$-extensions of globally hyperbolic Lorentzian manifolds, advancing understanding of low regularity spacetime extendibility.

## Key findings

- Timelike complete, globally hyperbolic Lorentzian manifolds are $C^0$-inextendible.
- Existence of length-maximizing causal curves in continuous, globally hyperbolic spacetimes.
- Timelike completeness prevents $C^0$-extensions of certain Lorentzian manifolds.

## Abstract

The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike complete and globally hyperbolic Lorentzian manifold is $C^0$-inextendible. For the proof we make use of the result, recently established by S\"amann [17], that even for \emph{continuous} Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.