Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-12T13:28:28.107Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Beem, J. K.Some examples of incomplete space–times. General Relativity and Gravitation 7 (1976), 501–509.CrossRefGoogle Scholar
(2)Beem, J. K.Conformal changes and geodesic completeness. Comm. Math. Phys. 49 (1976), 179–186.CrossRefGoogle Scholar
(3)Beem, J. K. and Ehrlich, P. E.Distance lorentzienne finie et géodésiques f-causales incomplètes. C. R. Acad. Sci. Paris Sér. A 285 (1977), 1129–1131.Google Scholar
(4)Beem, J. K. and Erhlich, P. E.Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic spacetimes. Math. Proc. Cambridge Philos. Soc. 84 (1978), 159–175.CrossRefGoogle Scholar
(5)Geroch, R. P.What is a singularity in general relativity? Ann. Physics (1968), 526–540.CrossRefGoogle Scholar
(6)Geroch, R. P., Kronheimer, E. H. and Penrose, R.Ideal points in space–time. Proc. Roy. Soc. London Ser. A 327 (1972), 545–567.Google Scholar
(7)Gromoll, D. and Meyer, W.On complete open manifolds of positive curvature. Ann. of Math. 90 (1969), 75–90.CrossRefGoogle Scholar
(8)Hawking, S. W.The occurrence of singularities in cosmology III. Causality and singularities. Proc. Roy. Soc. London Ser. A 300 (1967), 187–201.Google Scholar
(9)Hawking, S. W. and Penrose, R.The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. London Ser. A 314 (1970), 529–548.Google Scholar
(10)Hawking, S. W. and Ellis, G. F. R.The large-scale structure of space–time (Cambridge University Press, 1973).CrossRefGoogle Scholar
(11)Kundt, W.Note on the completeness of space–times. Z. Physik 172 (1963), 488–489.CrossRefGoogle Scholar
(12)Lerner, D.Techniques of topology and differential geometry in general relativity. Springer Lect. Notes Phys. 14 (1972), 1–44.CrossRefGoogle Scholar
(13)Nomizu, K. and Ozeki, H.The existence of complete riemannian metrics. Proc. Amer. Math. Soc. 12 (1961), 889–891.CrossRefGoogle Scholar
(14)Penrose, R. Structure of space–time. In Battelle Rencontres; 1967 Lectures in Mathematics and Physics, ed. Witt, de and Wheeler, (New York, W. A. Benjamin), 121–235.Google Scholar
(15)Penrose, R. Techniques of differential topology in relativity. Regional Conference Series in Applied Math. 7 (SIAM, Philadelphia, 1972).Google Scholar
(16)Tipler, F.Causally symmetric space–times. J. Math. Phys. 18 (1977), 1568–1573.CrossRefGoogle Scholar
(17)Tipler, F.Singularities in universes with negative cosmological constant. Astrophys. J. 209 (1976), 12–15.CrossRefGoogle Scholar
(18)Tipler, F.Singularities and causality violation. Ann. Physics 108 (1977), 1–36.CrossRefGoogle Scholar
(19)Wolf, J. A.Spaces of constant curvature (New York, McGraw-Hill, 1967).Google Scholar