Topology of the Future Chronological Boundary: Universality for Spacelike Boundaries

TL;DR

This paper introduces a topology for chronological sets that captures boundary structures in spacetimes, providing a universal characterization of the Future Causal Boundary with applications to various spacetime models.

Contribution

It presents a new topology for chronological sets that replicates manifold boundaries and characterizes the Geroch-Kronheimer-Penrose boundary categorically, establishing its universality.

Findings

The topology replicates manifold boundary topologies.

The GKP boundary has a universal property in this topology.

Applications to Schwarzschild and Robertson-Walker spacetimes show expected boundary structures.

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, 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 expected results; included among these are the Schwarzschild singularity and those Robertson-Walker singularities which are spacelike.