Uniqueness of Maximal Spacetime Boundaries

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 [22], where he obtains local uniqueness results for anchored spacetime extensions of similar character to earlier work for conformal boundaries by Chruściel [2]. 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 Chruściel for the conformal boundary.