X iv : g r - qc / 9 50 80 18 v 3 2 4 Ju n 19 99 A Causal Order for Spacetimes with C 0 Lorentzian Metrics :

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 topol-ogy for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call K + , and in terms of it we formulate extended definitions of concepts like causal curve and global hy-perbolicity. In particular we prove that, in a spacetime M which is free of causal cycles, one may define a causal curve simply 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 for certain metrics which fail to be invertible in places). Using this feature, we prove for a general C 0 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 [1], and may also prove useful in the study of topology change [2]. We have tried to make our treatment 1 self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry. 1. Introduction Global geometrical properties of manifolds often condition the behavior of curves within the manifold. For example, in a riemannian manifold, the metric property of completeness implies the existence of a minimum length (and therefore geodesic) curve joining any two points of the manifold (Hopf-Rinow theorem [3]). In a lorentzian manifold, the situation is somewhat different. Here properties of order and topology tend to take precedence over strictly metric properties, and one is primarily interested, not in arbitrary curves, but in causal ones. In the lorentzian case there are two analogues of a minimal geodesic: a longest timelike curve and (cf. [4], [1]) a " fastest causal curve " (necessarily a null geodesic). An analogue of the Hopf-Rinow result does exist, but the causal property of global hyperbolicity replaces that of Cauchy completeness as the basic assumption. Moreover the compactness of the space of causal curves between two points of the manifold now becomes an essential part …