The regularity of geodesics in impulsive pp-waves

  • Alexander Lecke
  • Roland Steinbauer
  • Robert Švarc
Research Article


We consider the geodesic equation in impulsive pp-wave space-times in Rosen form, where the metric is of Lipschitz regularity. We prove that the geodesics (in the sense of Carathéodory) are actually continuously differentiable, thereby rigorously justifying the \({\mathcal C}^1\)-matching procedure which has been used in the literature to explicitly derive the geodesics in space-times of this form.


Impulsive pp-waves Geodesics Carathéodory-solutions 

Mathematics Subject Classification (2010)

83C15 34A36 83C35 



We thank Jiří Podolský for kindly sharing his expertise and Clemens Sämann, and Milena Stojković for helpful discussions. This work was supported by FWF-grant P25326 and OeAD WTZ-project CZ15/2013 resp. 7AMB13AT003.


  1. 1.
    Aliev, A.N., Nutku, Y.: Impulsive spherical gravitational waves. Class. Quant. Grav. 18(5), 891–906 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Balasin, H.: Geodesics for impulsive gravitational waves and the multiplication of distributions. Class. Quant. Grav. 14(2), 455–462 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Colombeau, J.F.: Elementary Introduction to New Generalized Functions. North Holland, Amsterdam (1985)zbMATHGoogle Scholar
  4. 4.
    D’Eath, P.D.: High-speed black-hole encounters and gravitational radiation. Phys. Rev. D (3) 18(4), 990–1019 (1978)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Erlacher, E., Grosser, M.: Inversion of a ‘discontinuous coordinate transformation’ in general relativity. Appl. Anal. 90(11), 1707–1728 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht (1988)CrossRefGoogle Scholar
  7. 7.
    Ferrari, V., Pendenza, P., Veneziano, G.: Beam like gravitational waves and their geodesics. J. Gen. Rel. Grav. 20(11), 1185–1191 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Volume 537 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  9. 9.
    Griffiths, J., Podolský, J.: Exact Space-Times in Einstein’s General Relativity. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hogan, P.A.: A spherical gravitational wave in the de Sitter universe. Phys. Lett. A 171(1–2), 21–22 (1992)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hogan, P.A.: A spherical impulse gravity wave. Phys. Rev. Lett. 70(2), 117–118 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hortacsu, M.: Quantum fluctuations in the field of an impulsive spherical gravitational wave. Class. Quant. Grav. 7(8), L165–L169, (1990). Erratum: Class. Quant. Grav. 9(3), 799 (1992)Google Scholar
  13. 13.
    Kunzinger, M., Steinbauer, R.: A note on the Penrose junction conditions. Class. Quant. Grav. 16(4), 1255–1264 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kunzinger, M., Steinbauer, R.: A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves. J. Math. Phys. 40(3), 1479–1489 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nutku, Y., Penrose, R.: On impulsive gravitational waves. Twist. Newslett. 34, 9–12 (1992)Google Scholar
  16. 16.
    Penrose, R.: The geometry of impulsive gravitational waves. In: O’Raifeartaigh, L. (ed.) General Relativity, pp. 101–115. Clarendon Press, Oxford (1972)Google Scholar
  17. 17.
    Podolský, J.: Exact impulsive gravitational waves in spacetimes of constant curvature. In: Semerák, O., Podolský, J., Žofka, M. (eds.) Gravitation: Following the Prague Inspiration. A Volume in Celebration of the 60th Birthday of Jiří Bičák., pp. 205–246. World Scientific, Singapore (2002)Google Scholar
  18. 18.
    Podolský, J., Griffiths, J.B.: Expanding impulsive gravitational waves. Class. Quant. Grav. 16(9), 2937–2946 (1999)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Podolský, J., Griffiths, J.B.: Nonexpanding impulsive gravitational waves with an arbitrary cosmological constant. Phys. Lett. A 261(1–2), 1–4 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Podolský, J., Griffiths, J.B.: The collision and snapping of cosmic strings generating spherical impulsive gravitational waves. Class. Quant. Grav. 17(6), 1401–1413 (2000)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Podolský, J., Ortaggio, M.: Symmetries and geodesics in (anti-) de Sitter spacetimes with non-expanding impulsive waves. Class. Quant. Grav. 18(14), 2689–2706 (2001)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Podolský, J., Steinbauer, R.: Geodesics in spacetimes with expanding impulsive gravitational waves. Phys. Rev. D (3), 67(6), 064013, 13 (2003)Google Scholar
  23. 23.
    Podolský, J., Švarc, R.: Refraction of geodesics by impulsive spherical gravitational waves in constant-curvature spacetimes with a cosmological constant. Phys. Rev. D (3), 81(12), 124035, 19 (2010)Google Scholar
  24. 24.
    Podolský, J., Veselý, K.: Continuous coordinates for all impulsive pp-waves. Phys. Lett. A 241(3), 145–147 (1998)ADSCrossRefGoogle Scholar
  25. 25.
    Podolský, J., Veselý, K.: Smearing of chaos in sandwich pp-waves. Class. Quant. Grav. 16(11), 3599–3618 (1999)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Steinbauer, R.: Geodesics and geodesic deviation for impulsive gravitational waves. J. Math. Phys. 39(4), 2201–2212 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Steinbauer, R.: On the geometry of impulsive gravitational waves. arXiv:gr-qc/9809054v2Google Scholar
  28. 28.
    Walter, W.: Ordinary Differential Equations, volume 182 of Graduate Texts in Mathematics. Springer, New York, 1998. Translated from the sixth German (1996) edition by Russell Thompson, Readings in MathematicsGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander Lecke
    • 1
  • Roland Steinbauer
    • 1
  • Robert Švarc
    • 2
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Institute of Theoretical Physics, Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic

Personalised recommendations