Singularities, incompleteness and the Lorentzian distance function

Abstract

AbstractA 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.