A CAUSAL ORDER FOR SPACETIMES WITH C-0 LORENTZIAN METRICS - PROOF OF COMPACTNESS OF THE SPACE OF CAUSAL CURVES

We recast the tools of 'global causal analysis' in accord with an appr oach to the subject animated by two distinctive features: a thoroughgo ing reliance on order-theoretic concepts, and a utilization of the Vie toris topology for the space of closed subsets of a compact set. We ar e led to work with a new causal relation which we call K+, and in term s of it we formulate extended definitions of concepts like causal curv e and global hyperbolicity. In particular we prove that, in a spacetim e M which is free of causal cycles, one may define a causal curve simp ly as a compact connected subset of M which is linearly ordered by K+. Our definitions all make sense for arbitrary C-0 metrics (and even fo r certain metrics which fail to be invertible in places). Using this f eature, we prove for a general C-0 metric the familiar theorem that th e space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in additi on to yielding a more general theorem, simplifies and clarifies the re asoning involved. Our results have application in a recent positive-en ergy theorem, and may also prove useful in the study of topology chang e. We have tried to make our treatment self-contained by including pro ofs of all the facts we use which are not widely available in referenc e works on topology and differential geometry.