SINGULARITIES, INCOMPLETENESS AND THE LORENTZIAN DISTANCE FUNCTION

A space-time (M, g) is singular if it is inextendible and contains an inextendible nonspacelike geodesic which is incomplete. In this paper nonspacelike incompleteness is studied using the Lorentzian distance d(p, q). A compact subset K of M causally disconnects two divergent sequences {pn} and {qn} if 0 < (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 C0-neighbourhood 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. © 1979, Cambridge Philosophical Society. All rights reserved.