Universality of the Future Chronological Boundary

TL;DR

This paper proves that the Geroch-Kronheimer-Penrose (GKP) future causal boundary is categorically universal for spacetimes with certain chronological structures, establishing its uniqueness among similar boundary constructions.

Contribution

It demonstrates the categorical universality and uniqueness of the GKP future causal boundary for spacetimes with chronological boundaries, excluding topology considerations.

Findings

The GKP future causal boundary is categorically universal.

There exists a unique causal map from the boundary to any reasonable future completion.

The result applies to boundaries supporting a chronological structure, not the full causality relation.

Abstract

The purpose of this note is to establish, in a categorical manner, the universality of the Geroch-Kronheimer-Penrose causal boundary when considering the types of causal structures that may profitably be put on any sort of boundary for a spacetime. Actually, this can only be done for the future causal boundary (or the past causal boundary) separately; furthermore, only the chronology relation, not the causality relation, is considered, and the GKP topology is eschewed. The final result is that there is a unique map, with the proper causal properties, from the future causal boundary of a spacetime onto any ``reasonable" boundary which supports some sort of chronological structure and which purports to consist of a future completion of the spacetime. Furthermore, the future causal boundary construction is categorically unique in this regard.