联系我们
Isaac Scientific Publishing
Theoretical Physics
TP > Volume 5, Number 4, December 2020

Eikonal Equations for Null Radial Geodesics in the Schwarzschild Metric

Download PDF  (2677.4 KB)PP. 41-49,  Pub. Date:January 7, 2021
DOI: 10.22606/tp.2020.54001

Author(s)
Miguel Socolovsky
Affiliation(s)
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Cd. Universitaria, 04510, Ciudad de México, México
Abstract
We study the eikonal function φ corresponding to outgoing and ingoing radial null geodesics (light rays in the short wave length limit) in the Schwarzschild spacetime. Contrary to the behavior of the expansion scalar θ at the singularities (past and future), φ turns out to be finite at r = 0 (except for light travelling along the horizons) and inversely proportional to M, the mass of the black hole, and so proportional to the Hawking temperature.
Keywords
Schwarzschild spacetime, eikonal function, radial light rays
References
  • [1]  1. Blandford, R.D. and Thorne, K.S. Applications of Classical Physics, Chapter 7, Caltech, (2013).
  • [2]  2. Landau, L.D. and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics Vol. 2, Elsevier, (1975).
  • [3]  3. Kruskal, M.D., Maximal extension of Schwarzschild metric, Phys. Rev. 119, 1743-1745, (1960).
  • [4]  4. Szekeres, G. On the singularities of a Riemannian manifold, Publ. Math. Debrecen 7, 285-301, (1960).
  • [5]  5. Kar. S. and Sengupta, S. The Raychaudhuri equations: A brief review, Pramana 69, 49-76, (2007).
  • [6]  6. Baez, J.C. and Bunn, E.F. The meaning of Einstein’s equation, Am. J. Phys. 73, 644-652, (2005).
  • [7]  7. Poisson, E. A Relativist’s Toolkit, The Mathematics of Black Hole Mechanics, Cambridge University Press, (2004).
  • [8]  8. Finkelstein, D. Past-future asymmetry of the gravitational field of a point particle, Phys. Rev. 110, 965-967, (1958).
  • [9]  9. Penrose, R. The Light Cone at Infinity. In Proceedings of the 1962 Conference on Relativistic Theories of Gravitation, Warsaw; Polish Academy of Sciences, (1965).
  • [10]  10. Carter, B. Complete analytic extension of the symmetry axis of Kerr’s solution of Einstein’s equations, Phys. Rev. 141, 1242-1247, (1966).
  • [11]  11. Ni, W.-T. “Equivalence principles, spacetime structure and the cosmic connection” in One Hundred Years of General Relativity, Vol 1, Chapter 5, World Scientific, (2017).
  • [12]  12. Frolov, V.P. and Zelnikov, A.Z. “Introduction to Black Hole Physics”, Oxford University Press, (2015).
Copyright © 2020 Isaac Scientific Publishing Co. All rights reserved.