A note on spacelike and timelike compactness

Abstract

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