Metric completeness of C(p, q) in a globally hyperbolic spacetime

Abstract

We consider a Lorentzian manifold M which is globally hyperbolic. We define a metric on C(p, q), the set of all equivalence classes of causal curves connecting two causally related points p and q. We show that C(p, q) is a complete metric space with the metric thus defined. Here, by completeness we mean that every Cauchy sequence (a sequence with a tendency to converge) in C(p, q) finds a point in it to converge. We also give an example to show that the result does not hold in general when the spacetime is not globally hyperbolic. The work is in line with research on causality in relativistic spacetimes.