Annales Henri Poincaré

, Volume 16, Issue 9, pp 2163–2214 | Cite as

Smoothness of Compact Horizons

  • Eric LarssonEmail author


We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypothesis necessary in the original proof.


Null Vector Spacelike Hypersurface Null Energy Condition Timelike Curve Null Hypersurface 
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  1. 1.
    Beem J.K., Harris S.G.: The generic condition is generic. Gen. Relat. Gravit. 25(9), 939–962 (1993)zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Beem J.K., Królak A.: Cauchy horizon end points and differentiability. J. Math. Phys. 39(11), 6001–6010 (1998)zbMATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Borde, A.: Topology change in classical general relativity (1994). arXiv:gr-qc/9406053v1
  4. 4.
    Budzyński, R.J., Kondracki, W., Królak, A.: New properties of Cauchy and event horizons. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000), vol. 47, pp. 2983–2993 (2001)Google Scholar
  5. 5.
    Budzyński R.J., Kondracki W., Królak A.: On the differentiability of compact Cauchy horizons. Lett. Math. Phys. 63(1), 1–4 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chruściel, P.T.: Elements of causality theory (2011). arXiv:1110.6706v1 [gr-qc]
  7. 7.
    Chruściel P.T., Delay E., Galloway G.J., Howard R.: Regularity of horizons and the area theorem. Ann. Henri Poincaré 2(1), 109–178 (2001)CrossRefADSGoogle Scholar
  8. 8.
    Chruściel P.T., Fu J.H.G., Galloway G.J., Howard R.: On fine differentiability properties of horizons and applications to Riemannian geometry. J. Geom. Phys. 41(1-2), 1–12 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Chruściel P.T., Galloway G.J.: Horizons non-differentiable on a dense set. Commun. Math. Phys. 193(2), 449–470 (1998)zbMATHCrossRefADSGoogle Scholar
  10. 10.
    Clarke, F.H.: Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, vol. 5, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)Google Scholar
  11. 11.
    Federer H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Federer H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York Inc. (1969)Google Scholar
  13. 13.
    Galloway G.J.: Maximum principles for null hypersurfaces and null splitting theorems. Ann. Henri Poincaré 1(3), 543–567 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Galloway, G.J.: Null geometry and the Einstein equations. In: The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 379–400. Birkhäuser, Basel (2004)Google Scholar
  15. 15.
    Geroch R.P.: Topology in general relativity. J. Math. Phys. 8, 782–786 (1967)zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space–time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London (1973)Google Scholar
  17. 17.
    Isenberg J., Moncrief V.: Symmetries of cosmological Cauchy horizons with exceptional orbits. J. Math. Phys. 26(5), 1024–1027 (1985)zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Kriele, M.: Spacetime. In: Lecture Notes in Physics. New Series m: Monographs, vol. 59. Springer, Berlin (1999). Foundations of general relativity and differential geometryGoogle Scholar
  19. 19.
    Kupeli D.N.: On null submanifolds in spacetimes. Geom. Dedicata 23(1), 33–51 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Larsson, E.: Lorentzian cobordisms, compact horizons and the generic condition. Master’s thesis, KTH Royal Institute of Technology, Sweden (2014).
  21. 21.
    Lee, J.M.: Introduction to smooth manifolds. Volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, (2013)Google Scholar
  22. 22.
    Milnor, J.W., Stasheff, J.D.: Characteristic classes. In: Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1974)Google Scholar
  23. 23.
    Minguzzi, E.: Area theorem and smoothness of compact Cauchy horizons (2014). arXiv:1406.5919v1 [gr-qc]
  24. 24.
    Minguzzi, E.: Completeness of Cauchy horizon generators (2014). arXiv:1406.5909v1 [gr-qc]
  25. 25.
    Moncrief V., Isenberg J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89(3), 387–413 (1983)zbMATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Moncrief, V., Isenberg, J.: Symmetries of higher dimensional black holes. Class. Quant. Grav. 25(19), 195015, 37 (2008)Google Scholar
  27. 27.
    Morgan, F.: Geometric Measure Theory, A Beginner’s Guide, 3rd edn. Academic Press, Inc., San Diego (2000)Google Scholar
  28. 28.
    O’Neill, B.: Semi-Riemannian geometry. In: Pure and Applied Mathematics, vol. 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1983). With applications to relativityGoogle Scholar
  29. 29.
    Reinhart B.L.: Cobordism and the Euler number. Topology 2, 173–177 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Tipler F.J.: Singularities and causality violation. Ann. Phys. 108(1), 1–36 (1977)zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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