A causal order for spacetimes with Lorentzian metrics: proof of compactness of the space of causal curves

Abstract

We recast the tools of `global causal analysis' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call , and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime which is free of causal cycles, one may define a causal curve simply as a compact connected subset of which is linearly ordered by . Our definitions all make sense for arbitrary metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general metric the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive-energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.