Uniqueness of maximal spacetime boundaries

TL;DR

This paper investigates the conditions under which maximal spacetime boundaries are unique, demonstrating that excluding certain pathological geodesic behaviors ensures a unique maximal future boundary extension.

Contribution

It establishes global uniqueness of maximal future boundaries for extendible spacetimes under regularity assumptions and the absence of intertwined timelike geodesics.

Findings

Maximal future boundaries are unique when certain geodesic pathologies are excluded.

The result generalizes previous local uniqueness results to a global setting.

Provides conditions under which spacetime extensions have a unique maximal boundary.

Abstract

Given an extendible spacetime one may ask how much, if any, uniqueness can in general be expected of the extension. Locally, this question was considered and comprehensively answered in a recent paper of Sbierski, where he obtains local uniqueness results for anchored spacetime extensions of similar character to earlier work for conformal boundaries by Chru\'sciel. Globally, it is known that non-uniqueness can arise from timelike geodesics behaving pathologically in the sense that there exist points along two distinct timelike geodesics which become arbitrarily close to each other interspersed with points which do not approach each other. We show that this is in some sense the only obstruction to uniqueness of maximal future boundaries: Working with extensions that are manifolds with boundary we prove that, under suitable assumptions on the regularity of the considered extensions and excluding the existence of such ''intertwined timelike geodesics'', extendible spacetimes admit a unique maximal future boundary extension. This is analogous to results of Chru\'sciel for the conformal boundary.