General Relativity and Gravitation

, Volume 12, Issue 9, pp 693–707 | Cite as

A review of the geometrical equivalence of metrics in general relativity

  • A. Karlhede
Research Articles

Abstract

We review the solution to the equivalence problem in general relativity given by Cartan and Brans and present a practically useful method to obtain a coordinate-invariant description of a geometry. The method, which can be seen as a generalized Petrov classification, automatically gives the dimensions of the isometry group and its isotropy subgroup. Finally, we illustrate the method using the Schwarzschild solution as a very simple example.

Keywords

General Relativity Isotropy Differential Geometry Isometry Group Equivalence Problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cartan, E. (1946).LeÇons sur la Geometrie des Espaces de Riemann (Gauthier-Villars, Paris).Google Scholar
  2. 2.
    Thomas, T. Y. (1934).The Differential Invariants of Generalized Spaces (Cambridge U.P., Cambridge).Google Scholar
  3. 3.
    Brans, C. H. (1965).J. Math. Phys.,6, 94.Google Scholar
  4. 4.
    Ehlers, J., and Kundt, W. (1962). InGravitation: An Introduction to Current Research, Ed. L. Witten (Wiley, New York).Google Scholar
  5. 5.
    Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1972),Gravitation (Freeman, San Francisco).Google Scholar
  6. 6.
    Petrov, A. Z. (1955).Dokl. Akad. Nauk SSSR,105, 905.Google Scholar
  7. 7.
    Penrose, R. (1960).Ann. Phys. (N. Y.),10, 171.Google Scholar
  8. 8.
    Pirani, F. A. E. (1965). InLectures on General Relativity, Brandeis 1964, Vol. 1 (Prentice-Hall, Englewood Cliffs, New Jersey).Google Scholar
  9. 9.
    Clarke, C. J. S., and Schmidt, B. G. (1977).Gen. Rel. Grav.,8, 129.Google Scholar
  10. 10.
    Brans, C. H. (1970).J. Math. Phys.,11, 1210.Google Scholar
  11. 11.
    Brans, C. H. (1971).J. Math. Phys.,12, 1616.Google Scholar
  12. 12.
    Brans, C. H. (1977).J. Math. Phys.,18, 1378.Google Scholar
  13. 13.
    Karlhede, A. (1979).USIP Report, 79-05.Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • A. Karlhede
    • 1
  1. 1.Institute of Theoretical PhysicsUniversity of StockholmStockholmSweden

Personalised recommendations