Maximal Geodesics and Causally Disconnected Space–Times

Many basic properties of complete, noncompact Riemannian manifolds stem from the principle that a limit curve of a sequence of minimal geodesics is itself a minimal geodesic. This chapter gives methods for constructing families of almost maximal curves whose limit curves in strongly causal space–times are maximal geodesics. The strong causality is needed to ensure the upper semicontinuity of arc length in the C⁰ topology on curves and also so that Proposition may be applied. It explains the class of causally disconnected space–times. The chapter shows how geodesics may be constructed as limits of “almost maximal” curves in strongly causal space–times. The chapter concludes by studying conditions on the global geodesic structure of a given space–time which imply that is causally disconnected.