The Causal Boundary of spacetimes revisited

TL;DR

This paper revisits the causal boundary of spacetimes, proposing a minimal boundary completion that overcomes previous issues and retains desirable topological and chronological properties.

Contribution

It introduces a new minimal boundary completion for the causal boundary of spacetimes, resolving issues of excess boundary and improving the construction's optimality.

Findings

Minimal boundary completions always exist.

Previous deficiencies are due to overly large boundaries.

The new construction has desirable properties and examples.

Abstract

We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an ``excessively big'' boundary. In fact, a notion of ``completion with minimal boundary'' is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples.