Uniqueness and non-uniqueness results for spacetime extensions

TL;DR

This paper investigates the conditions under which extensions of globally hyperbolic Lorentzian manifolds are unique, establishing a sharp regularity threshold at Lipschitz continuity for local uniqueness near boundary points.

Contribution

It provides a new uniqueness criterion for Lorentzian manifold extensions at low regularities and demonstrates the optimality of Lipschitz continuity for such results.

Findings

Unique local extension near boundary points for Lipschitz continuous extensions.

Non-uniqueness can occur for extensions with only Hölder continuity.

Sharp regularity threshold identified at Lipschitz continuity.

Abstract

Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on $\mathbb{R}^m$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions which are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma$ has limit points in both extensions, must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions which are only H\"older continuous do in general not enjoy this local uniqueness result.