Communications in Mathematical Physics

, Volume 359, Issue 3, pp 937–949 | Cite as

Timelike Completeness as an Obstruction to C 0-Extensions

Open Access


The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike complete and globally hyperbolic Lorentzian manifold is C 0-inextendible. For the proof we make use of the result, recently established by Sämann (Ann Henri Poincaré 17(6):1429–1455, 2016), that even for continuous Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.


  1. 1.
    Anderson M.: Existence and stability of even-dimensional asymptotically de Sitter spaces. Ann. Henri Poincaré 6, 801–820 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry, second ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker Inc., New York (1996)Google Scholar
  3. 3.
    Burago D., Burago Y., Ivanov S.: A Course in Metric Geometry. American Mathematical Society, Rhode Island (2001)CrossRefMATHGoogle Scholar
  4. 4.
    Christodoulou D.: The Formation of Black Holes in General Relativity. European Mathematical Society, Mandralin (2009)CrossRefMATHGoogle Scholar
  5. 5.
    Christodoulou D., Klainerman S.: The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  6. 6.
    Chruściel, P.T.: Elements of causality theory, (2011), arXiv:1110.6706
  7. 7.
    Chruściel, P.T., Grant, J.D.E.: On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29(14), (2012) 145001, 32Google Scholar
  8. 8.
    Dafermos M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations. Ann. of Math. (2) 158(3), 875–928 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eschenburg J.-H., Galloway G. J.: Lines in space-times. Commun. Math. Phys. 148(1), 209–216 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Friedrich H.: Existence and structure of past asymptotically simple solutions of Einsteins field equations with positive cosmological constant. J. Geometry Phys. 3, 101–117 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Friedrich H.: On the existence of n-geodesically complete or future complete solutions of Einsteins field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Galloway G. J.: Curvature, causality and completeness in space-times with causally complete spacelike slices. Math. Proc. Cambridge Philos. Soc. 99(2), 367–375 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Galloway, G.J., Ling, E.: Some remarks on the C0-(in)extendibility of spacetimes, Annales Henri Poincaré (2017)Google Scholar
  14. 14.
    Minguzzi, E.: Limit curve theorems in Lorentzian geometry, J. Math. Phys. 49(9), (2008) 092501, 18Google Scholar
  15. 15.
    O’Neill, B.: Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, (1983)Google Scholar
  16. 16.
    Penrose, R.: Techniques of differential topology in relativity, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1972, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7Google Scholar
  17. 17.
    Sämann Clemens: Global hyperbolicity for spacetimes with continuous metrics. Ann. Henri Poincaré 17(6), 1429–1455 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sbierski, J.: The C 0-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian Geometry, (2015), arXiv:1507.00601v2 (to appear in J. Diff. Geom.)
  19. 19.
    Seifert H.-J.: Global connectivity by timelike geodesics. Z. Naturforsch 22a, 1356–1360 (1967)ADSMathSciNetMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MiamiCoral GablesUSA
  2. 2.Department for Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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