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Communications in Mathematical Physics

, Volume 285, Issue 2, pp 399–420 | Cite as

The Wave Equation on Singular Space-Times

  • James D.E. Grant
  • Eberhard Mayerhofer
  • Roland Steinbauer
Article

Abstract

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

Keywords

Wave Equation Energy Momentum Tensor Sobolev Norm Energy Integral Dominant Energy Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Clarke C.J.S.: The Analysis of Space-Time Singularities. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  2. 2.
    Clarke, C.J.S.: Singularities: boundaries or internal points?. In: Singularities, Black Holes and Cosmic Censorship, Joshi, P.S., Raychaudhuri, A.K., eds., Bombay: IUCCA, 1996, pp. 24–32Google Scholar
  3. 3.
    Clarke C.J.S.: Generalized hyperbolicity in singular spacetimes. Class. Quantum Grav. 15, 975–984 (1998)MATHCrossRefADSGoogle Scholar
  4. 4.
    Colombeau, J.-F.: New generalized functions and multiplication of distributions. Vol. 84 of North-Holland Mathematics Studies, Amsterdam: North-Holland Publishing Co., 1984Google Scholar
  5. 5.
    Colombeau, J.-F.: Multiplication of Distributions. A tool in mathematics, numerical engineering and theoretical physics, vol. 1532 of Lecture Notes in Mathematics, New York: Springer, 1992Google Scholar
  6. 6.
    Friedlander F.G.: The wave equation on a curved space-time. Cambridge University Press, Cambridge (1975)MATHGoogle Scholar
  7. 7.
    Geroch R., Traschen J.: Strings and other distributional sources in general relativity. Phys. Rev. D 36, 1017–1031 (1987)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric theory of generalized functions with applications to general relativity, Vol. 537 of Mathematics and its Applications, Dordrecht: Kluwer Academic Publishers, 2001Google Scholar
  9. 9.
    Grosser M., Kunzinger M., Steinbauer R., Vickers J.A.: A global theory of algebras of generalized functions. Adv. Math. 166, 50–72 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grosser, M., Kunzinger, M., Steinbauer, R., Vickers, J.A.: A global theory of algebras of generalized functions II: tensor distributions. Preprint 2007Google Scholar
  11. 11.
    Hanel, C.: Linear hyperbolic second order partial differential equations on space time. Master’s thesis, University of Vienna, 2006Google Scholar
  12. 12.
    Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge University Press, London (1973)MATHGoogle Scholar
  13. 13.
    Hörmann G.: Hölder-Zygmund regularity in algebras of generalized functions. Z. Anal. Anwendungen 23, 139–165 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Kunzinger M., Steinbauer R.: Foundations of a nonlinear distributional geometry. Acta Appl. Math. 71, 179–206 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kunzinger M., Steinbauer R.: Generalized pseudo-Riemannian geometry. Trans. Amer. Math. Soc. 354, 4179–4199 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Marsden, J.E.: Generalized Hamiltonian mechanics: A mathematical exposition of non-smooth dynamical systems and classical Hamiltonian mechanics. Arch. Rat. Mech. Anal. 28, 323–361 (1967/1968)Google Scholar
  17. 17.
    Mayerhofer, E.: On Lorentz geometry in algebras of generalized functions. Proc. Edinb. Math. Soc., to appear 2008. http://arxiv.org/list/math-ph/0604052, 2006
  18. 18.
    Mayerhofer, E.: The wave equation on singular space-times, Ph.D. thesis, University of Vienna, Faculty of Mathematics 2006. Available from http://arxiv.org/list/abs/0802.1616, 2008
  19. 19.
    Parker P.E.: Distributional geometry. J. Math. Phys. 20, 1423–1426 (1979)MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Penrose R., Rindler W.: Spinors and space-time. Vol. 1. Cambridge University Press, Cambridge (1987)Google Scholar
  21. 21.
    Podolský J., Griffiths J.B.: Expanding impulsive gravitational waves. Class. Quantum Grav. 16, 2937–2946 (1999)MATHCrossRefADSGoogle Scholar
  22. 22.
    Schwartz L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)MATHMathSciNetGoogle Scholar
  23. 23.
    Senovilla J.M.M.: Super-energy tensors. Class. Quantum Grav. 17, 2799–2841 (2000)MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Steinbauer R., Vickers J.: The use of generalized functions and distributions in general relativity. Class. Quantum Grav. 23, R91–R114 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Vickers J.A., Wilson J.P.: Generalized hyperbolicity in conical spacetimes. Class. Quantum Grav. 17, 1333–1260 (2000)MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • James D.E. Grant
    • 1
  • Eberhard Mayerhofer
    • 1
  • Roland Steinbauer
    • 1
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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