Eikonal Approximation to 5D Wave Equations and the 4D Space-Time Metric


We apply a method analogous to the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold, using a method which identifies the symplectic structure of the corresponding mechanics, to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelberg's covariant classical and quantum dynamics. In this way, we demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. We then discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg–Schrödinger equation. The locally symplectic structure which emerges is that of a generally covariant form of Stueckelberg's mechanics on this manifold.

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  1. 1.

    H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, MA, 1950).

    Google Scholar 

  2. 2.

    M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

    Google Scholar 

  3. 3.

    P. Piwnicki, “Geometrical approach to light in inhomogeneous media, ” gr-qc/0201007.

  4. 4.

    M. Visser, Class. Quant. Grav. 15, 1767(1998). L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. A 63, 026311(2001); Phys. Rev. Lett. 85, 4643 (2000).

    Google Scholar 

  5. 5.

    M. Visser, C. Barceló, and S. Liberati, “Analogue models of and for gravity, ” gr-qc/0111111 (2001).

  6. 6.

    C. Barceló and M. Visser, Int. J. Mod. Phys. D 10, 799(2001).

    Google Scholar 

  7. 7.

    C. Barceló, S. Liberati, and M. Visser, Class. and Quantum Grav. 18, 3595(2001).

    Google Scholar 

  8. 8.

    U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301(1999).

    Google Scholar 

  9. 9.

    V. A. De Lorenci and R. Klippert, Phys. Rev. D 65, 064027(2002).

    Google Scholar 

  10. 10.

    Y. N. Obukhov and F. W. Hehl, Phys. Lett. B 458, 466(1999). See also, M. Schönberg, Riv. Brasileira de Fisica 1, 91 (1971). A. Peres, Ann. Phys. (N.Y.) 19, 279 (1962).

    Google Scholar 

  11. 11.

    H. Urbantke, J. Math. Phys. 25, 2321(1984); H. Urbantke Acta Phys. Austriaca Suppl. XIX, 875 (1978).

    Google Scholar 

  12. 12.

    E. C. G. Stueckelberg, Helv. Phys. Acta 14, 322, 588 (1941). J. S. Schwinger, Phys. Rev. 82, 664 (1951). R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 80, 440 (1950). C. Piron and L. P. Horwitz, Helv. Phys. Acta 46, 316 (1973), extended this theory to the many-body case throught he postulate of a universal invariant evolution parameter.

    Google Scholar 

  13. 13.

    O. Oron and L. P. Horwitz, Phys. Lett. A 280, 265(2001).

    Google Scholar 

  14. 14.

    P. A. M. Dirac, Proc. Roy. Soc. London Ser. A 167, 148(1938).

    Google Scholar 

  15. 15.

    R. I. Arshansky and L. P. Horwitz, J. Math. Phys. 30, 66and 380 (1989).

    Google Scholar 

  16. 16.

    M. A. Trump and W. C. Schieve, Classical Relativistic Many-Body Dynamics (Kluwer Academic, Dordrecht, 1999).

    Google Scholar 

  17. 17.

    D. Saad, L. P. Horwitz, and R. I. Arshansky, Found. Phys. 19, 1126(1989).

    Google Scholar 

  18. 18.

    M. C. Land, N. Shnerb, and L. P. Horwitz, J. Math. Phys. 36, 3263(1995).

    Google Scholar 

  19. 19.

    N. Shnerb and L. P. Horwitz, Phys. Rev. A 48, 4058(1993).

    Google Scholar 

  20. 20.

    M. Visser, “Birefringence versus bi-metricity, ” in Festschrift in Honor of Mario Novello (2002).

  21. 21.

    M. C. Land and L. P. Horwitz, Found. Phys. Lett. 4, 61(1991).

    Google Scholar 

  22. 22.

    J. D. Jackson, Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975).

    Google Scholar 

  23. 23.

    A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1964).

    Google Scholar 

  24. 24.

    H. Brooks, Adv. Electronics 7, 85(1955). C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963), p. 131.

    Google Scholar 

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Oron, O., Horwitz, L.P. Eikonal Approximation to 5D Wave Equations and the 4D Space-Time Metric. Foundations of Physics 33, 1323–1338 (2003). https://doi.org/10.1023/A:1025693311737

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  • eikonal approximation
  • relativistic quantum theory
  • space-time metric