Topology of the Causal Boundary for Standard Static Spacetimes

The causal boundary of a strongly causal spacetime, introduced by Geroch, Kronheimer, and Penrose in [GKP], is a tool for examining the causal nature of a spacetime “at infinity”. As it is conformally invariant, it is a property of the conformal structure of the spacetime. The addition of the causal boundary to a spacetime results in an object whose structure might be most naturally characterized as a “chronological set”, i.e., a set with a chronology relation, a relation extending that on the spacetime. (In a spacetime, p is said to chronologically precede q, or p ≪ q, if there is a future-directed timelike curve from p to q. There is also the causality relation: p causally precedes q, or p ≺ q, if there is a futuredirected causal curve from p to q.) What is desired is a topology on this object so that the causal boundary puts appropriate endpoints on timelike curves which are endless in the spacetime; but there is a deal of controversy on how this might best be done. The topology suggested in [GKP] has some severe problems, most notable of which is that in the simplest of all cases, Minkowski n-space L, it fails to give what most consider the obvious topology for the spacetime-cum-boundary, that obtained from the conformal embedding of L into the Einstein static space, L × S (see [HE]). Some attempts have been made to ameliorate the general problems with this topology (such as [BS] and [S]), but they have no effect on the issue with Minkowski space. A new approach can be found in [MR], suggesting a substantial modification of the GKP method in a somewhat complex but also elegant manner; a related approach that also combines methods used here is to be found in [F]. None the less, the basic ideas in the causal boundary have much to recommend them; in particular, the separate elements of the future and past causal boundaries—that portion of the GKP construction designed to place future endpoints on future-endless timelike curves or, separately, past endpoints on pastendless timelike curves—seem both simple and natural. A project to regularize this