Topology of the future chronological boundary: universality for spacelike boundaries

Abstract

A method is presented for imputing a topology for any chronological set, i.e. a set with a chronology relation, such as a spacetime or a spacetime with some sort of boundary. This topology is shown to have several good properties, such as replicating the manifold topology for a spacetime and replicating the expected topology for some simple examples of spacetime-with-boundary; it also allows for a complete categorical characterization, in topological categories, of the future causal boundary construction of Geroch, Kronheimer and Penrose (GKP), showing that construction to have a universal property for future-completing chronological sets with spacelike boundaries. Rigidity results are given for any reasonable future-completion of a spacetime, in terms of the GKP boundary: in the imputed topology, any such boundary must be homeomorphic to the GKP boundary (if all points have indecomposable pasts) or to a topological quotient of a closely related boundary (if boundaries are spacelike). A large class of warped-product-type spacetimes with spacelike boundaries is examined, calculating the GKP and other possible boundaries and showing that the imputed topology gives the expected results; included among these are the Schwarzschild singularity and those Robertson-Walker singularities which are spacelike.