Advertisement

General Relativity and Gravitation

, Volume 38, Issue 2, pp 365–380 | Cite as

Fermat principle in Finsler spacetimes

  • Volker Perlick
Research Article

Abstract

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point q (emission event) to a timelike curve γ (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here “arrival time” refers to a parametrization of the timelike curve γ. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.

Keywords

Geodesics Variational principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Landau, L., Lifshitz, E.: The Classical Theory of Fields. Pergamon, Oxford (1962)Google Scholar
  2. 2.
    Kline, M., Kay, I.W.: Electromagnetic Theory and Geometrical Optics. Interscience, New York (1965)Google Scholar
  3. 3.
    Kovner, I.: Astrophys. J. 351, 114 (1990)CrossRefADSGoogle Scholar
  4. 4.
    Perlick, V.: Class. Quantum Grav. 7, 1319 (1990)CrossRefMATHADSMathSciNetGoogle Scholar
  5. 5.
    Schneider, P., Ehlers, J., Falco, E.: Gravitational Lenses. Springer, Heidelberg (1992)Google Scholar
  6. 6.
    Perlick, V.: Ray Optics, Fermat's Principle, and Applications to General Relativity. Springer, Heidelberg (2000)Google Scholar
  7. 7.
    Perlick, V.: Gravitational lensing from a spacetime perspective, Living Rev. Relativity 7, 9 (2004) [Online Article]; cited August 2005 http://www.livingreviews.org/lrr-2004-9
  8. 8.
    Gordon, W.: Annalen der Physik 72, 421 (1923)CrossRefADSGoogle Scholar
  9. 9.
    Asanov, G.S.: Finsler Geometry, Relativity and Gauge Theories. Reidel, Dordrecht (1985)Google Scholar
  10. 10.
    Newcomb, W.A.: Amer. J. Phys. 51, 338 (1983)CrossRefADSGoogle Scholar
  11. 11.
    Glinskii, G.F.: Radiophysics and Quantum Electronics 23, 70 (1980)CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Babich, V.M.: In: Petrashev, G.I. (ed.) Problems of the Dynamic Theory of Propagation of Seismic Waves. Vol. V. p. 36. Publishing House of Leningrad University, Leningrad, (in Russian) (1961); English Translation in Geophys. J. Int. 118 (1994), 379Google Scholar
  13. 13.
    Epstein, M., Śniatycki, J.: J. Elasticity 27, 45 (1992)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Červený, V.: Stud. Geophys. Geod. 46, 567 (2002)CrossRefGoogle Scholar
  15. 15.
    Uginčius, P.: J. Acoust. Soc. Amer. 51, 1759 (1972)CrossRefMATHADSGoogle Scholar
  16. 16.
    Godin, O.A., Voronovich, A.G.: Proc. Roy. Soc. London, Ser. A 460, 1631 (2004)MATHADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Voronovich, A.G., Godin, O.A.: Phys. Rev. Lett. 91, 044302 (2003)CrossRefPubMedADSGoogle Scholar
  18. 18.
    Rund, H.: The Differential Geometry of Finsler Spaces. Springer, Berlin (1959)Google Scholar
  19. 19.
    Beem, J.: Canad. J. Math. 22, 1035 (1970)MATHMathSciNetGoogle Scholar
  20. 20.
    Perlick, V.: J. Math. Phys. 36, 6915 (1995)CrossRefMATHADSMathSciNetGoogle Scholar
  21. 21.
    Giannoni, F., Masiello, A., Piccione, P.: Ann. Inst. H. Poincaré. Physique Theorique 69, 359 (1998)MATHMathSciNetGoogle Scholar
  22. 22.
    Crampin, M.: Houston J. Math. 26, 255 (2000)MATHMathSciNetGoogle Scholar
  23. 23.
    Crampin, M.: Houston J. Math. 27, 807 (2001)MATHMathSciNetGoogle Scholar
  24. 24.
    Morse, M.: Variational Analysis. Wiley, New York (1973)Google Scholar
  25. 25.
    Perlick, V.: Nonlinear Analysis 47, 3019 (2001)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany

Personalised recommendations