Constructing maximal geodesics in strongly causal space-times

Let (M, g) be an arbitrary space-time of dimension ≥ 2 and let d = d(g): M × M → ℝ ∪ {∞} (where d(p, q) = 0 for q∉J+(p)) denote the Lorentzian distance function of (M, g). Also let C(M, g) denote the space of Lorentzian metrics for M globally con-formal to g. Here g1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g1 = Ωg.

Cut points, conjugate points and Lorentzian comparison theorems

1. Introduction. The purpose of this paper is to study the global geometry of a space–time (M, g), which is related to the Lorentzian distance function induced on the manifold M by the Lorentzian structure. We will use the signature convention (−, +, …, +) for g and assume that (M, g) is time orientated. The first part of this paper deals with cut points and maximal geodesies, both of which were defined in (1) using the Lorentzian distance function in analogy to the standard concepts in Rie-mannian geometry. In (1), sections 2 and 3, some elementary properties of maximal geodesies were established. In particular, the principle that, for strongly causal space-times, a limit curve of a sequence of future-directed nonspacelike ‘almost maximal’ curves is a maximal geodesic was used to prove nonspacelike incompleteness ((1), theorem 6·3). Also null cut points were used to obtain results on null incompleteness ((1), section 5). In (2) we studied deeper properties of maximal geodesies and cut points using the technical tools developed in (1), sections 2 and 3. The first part of the present paper continues these investigations.

Singularities, incompleteness and the Lorentzian distance function

A space–time (M, g) is singular if it is inextendible and contains an inex-tendible nonspacelike geodesic which is incomplete. In this paper nonspacelike incompleteness is studied using the Lorentzian distance d(p, q). A compact subset Kof M causally disconnects two divergent sequences {pn} and {qn} if 0 < d(pn,qn) < ∞ for all n and all nonspacelike curves from pn to qn meet K. It is shown that no space–time (M, g) can satisfy all of the following three conditions: (1) (M, g) is chronological, (2) each inextendible nonspacelike geodesic contains a pair of conjugate points and (3) there exist two divergent sequences {pn} and {qn} which are causally disconnected by a compact set K. This particular result extends a theorem of Hawking and Penrose. It also implies that if (M, g) satisfies conditions (1) and (3), then there is a Co-neigh-bourhood of g in the space of metrics conformal to g such that any metric in this neighbourhood which satisfies the generic condition and the strong energy condition is nonspacelike incomplete.