SPACE-TIME CUT LOCUS

Let (M, g) be a space-time with Lorentzian distance function d. If (M, g) is distinguishing and d is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metrics g′ conformal to g. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut point q of p along the geodesic c must be either the first conjugate point of p or else there must be at least two maximal timelike (respectively, null) geodesics from p to q. If q is a closest cut point of p in a globally hyperbolic space-time, then either q is conjugate to p or else q is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point. © 1979 Plenum Publishing Corporation.