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Singularities, incompleteness and the Lorentzian distance function

Published online by Cambridge University Press:  24 October 2008

John K. Beem  and
Paul E. Ehrlich
John K. Beem
Affiliation:
University of Missouri, Columbia, Missouri 65201
Paul E. Ehrlich
Affiliation:
University of Missouri, Columbia, Missouri 65201
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 161 - 178
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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