A note on spacelike and timelike compactness

TL;DR

This paper clarifies different formulations of spacelike and timelike compactness in globally hyperbolic spacetimes, analyzing their interrelations and providing precise characterizations relevant for causal propagation of fields.

Contribution

It compares various definitions of spacelike and timelike compactness, establishing their logical relations and providing criteria for their equivalence and distinctions.

Findings

A closed set intersects all Cauchy surfaces in a compact set iff it is contained in J(K) for some compact K.

A closed set is between two Cauchy surfaces iff its intersection with J(K) is compact for all K.

Characterization of future and past compact sets in terms of intersections with J(K).

Abstract

When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the complementary idea of timelike compactness, and we clarify their interrelations. E.g., a closed subset A of M has a compact intersection with all Cauchy surfaces if and only if A is contained in J(K) for some compact set K. (However, it does not suffice to consider only those Cauchy surfaces that partake in a given foliation of M.) Similarly, a closed subset A of M is contained in a region between two Cauchy surfaces if and only if the intersection of A with J(K) is compact for all compact K. We also treat future and past compact sets in a similar way.