Maximizers in Lipschitz spacetimes are either timelike or null

TL;DR

This paper proves that in $C^{0,1}$ spacetimes, causal maximizers are exclusively timelike or null, resolving a previously open question and demonstrating inextendibility of certain geodesically complete spacetimes.

Contribution

It establishes that causal maximizers in $C^{0,1}$ spacetimes cannot be mixed timelike and null segments, filling a gap in the understanding of spacetime regularity effects.

Findings

Causal maximizers in $C^{0,1}$ spacetimes are either purely timelike or null.

Bubbling regions do not occur in $C^{0,1}$ spacetimes, unlike in less regular spacetimes.

Timelike geodesically complete spacetimes are $C^{0,1}$-inextendible.

Abstract

We prove that causal maximizers in $C^{0,1}$ spacetimes are either timelike or null. This question was posed in [17] since bubbling regions in $C^{0,\alpha}$ spacetimes ($\alpha <1$) can produce causal maximizers that contain a segment which is timelike and a segment which is null, cf. [3]. While $C^{0,1}$ spacetimes do not produce bubbling regions, the causal character of maximizers for spacetimes with regularity at least $C^{0,1}$ but less than $C^{1,1}$ was unknown until now. As an application we show that timelike geodesically complete spacetimes are $C^{0,1}$-inextendible.