A regularisation approach to causality theory for \(C^{1,1}\)-Lorentzian metrics

  • Michael Kunzinger
  • Roland Steinbauer
  • Milena Stojković
  • James A.  VickersEmail author
Research Article


We show that many standard results of Lorentzian causality theory remain valid if the regularity of the metric is reduced to \(C^{1,1}\). Our approach is based on regularisations of the metric adapted to the causal structure.


Causality theory Low regularity Regularisation Exponential map 



We would like to thank James D. E. Grant for helpful discussions. The authors acknowledge the support of FWF projects P23714 and P25326, as well as OeAD project WTZ CZ 15/2013. We are indebted to the referees of this paper for several comments that have led to substantial improvements in the presentation.


  1. 1.
    Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Klainerman, S., Rodnianski, I.: Rough solution for the Einstein vacuum equations. Ann. Math. 161(2), 1143–1193 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Maxwell, D.: Rough solutions of the Einstein constraint equations. J. Reine Angew. Math. 590, 1–29 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Klainerman, S., Rodnianski, I., Szeftel, J.: Overview of the proof of the bounded \(L^2\) curvature conjecture, arXiv:1204.1772 [gr-qc], 2012
  5. 5.
    Lichnerowicz, A.: Théories relativistes de la gravitation et de l’électromgnétisme. Masson, Paris (1955)zbMATHGoogle Scholar
  6. 6.
    Griffiths, J., Podolský, J.: Exact Space-Times in Einstein’s General Relativity. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Penrose, R.: Techniques of Differential Topology in Relativity (Conf. Board of the Mathematical Sciences Regional Conf. Series in Applied Mathematics vol 7). SIAM, Philadelphia, PA (1972)Google Scholar
  8. 8.
    O’Neill, B.: Semi-Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103. Academic Press, New York (1983)Google Scholar
  9. 9.
    Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry (Monographs and Textbooks in Pure and Applied Mathematics vol 202) 2nd edn. New York, Dekker (1996)Google Scholar
  10. 10.
    Kriele, M.: Spacetime. Foundations of General Relativity and Differential Geometry. Lecture Notes in Physics 59. Springer, Berlin (1999)Google Scholar
  11. 11.
    Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes in recent developments in pseudo-Riemannian geometry. ESI Lect. Math. Phys., Eur. Math. Soc. Publ. House, Zürich (2008)Google Scholar
  12. 12.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)CrossRefzbMATHGoogle Scholar
  13. 13.
    Senovilla, J.M.M.: Singularity theorems and their consequences. Gen. Relativ. Gravit. 30(5), 701–848 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    García-Parrado, A., Senovilla, J.M.M.: Causal structures and causal boundaries. Class. Quantum Gravity 22, R1–R84 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chruściel, P.T.: Elements of causality theory, arXiv:1110.6706
  16. 16.
    Clarke, C.J.S.: The analysis of spacetime singularities. Cambridge Lect. Notes Phys. 1. Cambridge University Press (1993)Google Scholar
  17. 17.
    Sorkin, R.D., Woolgar, E.: A causal order for spacetimes with \(C^0\) Lorentzian metrics: Proof of compactness of the space of causal curves class. Quantum Gravity 13, 1971–1994 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fathi, A., Siconolfi, A.: On smooth time functions. Math. Proc. Camb. Philos. Soc. 155, 1–37 (2011)zbMATHGoogle Scholar
  19. 19.
    Chruściel, P.T., Grant, J.D.E.: On Lorentzian causality with continuous metrics. Class. Quantum Gravity 29(14), 145001 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hartman, P.: On geodesic coordinates. Am. J. Math. 73, 949–954 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hartman, P., Wintner, A.: On the problems of geodesics in the small. Am. J. Math. 73, 132–148 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Whitehead, J.H.C.: Convex regions in the geometry of paths. Q. J. Math., Oxf. Ser. 3, 33–42, (1932). Addendum, ibid. 4, 226 (1933)Google Scholar
  23. 23.
    Kunzinger, M., Steinbauer, R., Stojković, M.: The exponential map of a \(C^{1,1}\)-metric. Differ. Geom. Appl. 34, 14–24 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chen, B.-L., LeFloch, P.: Injectivity radius of Lorentzian manifolds. Commun. Math. Phys. 278, 679–713 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Minguzzi, E.: Convex neighborhoods for Lipschitz connections and sprays, arXiv:1308.6675
  26. 26.
    Hirsch, M.W.: Differential Topology. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  27. 27.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Kluwer, Dordrecht (2001)zbMATHGoogle Scholar
  28. 28.
    Hörmann, G., Kunzinger, M., Steinbauer, R.: Wave equations on non-smooth space-times. Evolution equations of hyperbolic and Schrödinger type, pp. 163–186, Progr. Math., 301, Birkhäuser/Springer, Berlin (2012)Google Scholar
  29. 29.
    Amann, H.: Ordinary Differential Equations. De Gruyter, Berlin (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Michael Kunzinger
    • 1
  • Roland Steinbauer
    • 1
  • Milena Stojković
    • 1
  • James A.  Vickers
    • 2
    Email author
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.School of Mathematical SciencesUniversity of SouthamptonSouthamptonUK

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