General Relativity and Gravitation

, Volume 42, Issue 12, pp 2957–2980 | Cite as

Lie and Noether symmetries of geodesic equations and collineations

Research Article

Abstract

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.

Keywords

Geodesics General relativity Classical mechanics Collineations Lie symmetries Projective collineations Noether symmetries First integrals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Katzin G.H., Levine J.: Colloquium Mathematicum, pp. 21. Wrocław, Poland (1972)Google Scholar
  2. 2.
    Katzin G.H., Levine J.: J. Math. Phys. 15, 1460 (1974)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Katzin G.H., Levine J.: J. Math. Phys. 17, 1345 (1976)MATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Katzin G.H., Levine J.: J. Math. Phys. 22, 1878 (1981)MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Aminova A.V.: Gravit. i. Teoriya Otmotisel 14, 4 (1978)MathSciNetADSGoogle Scholar
  6. 6.
    Aminova A.V.: Izv. - Vyssh. - Uchebn. - Zaved. - Mat. 2, 3 (1994)Google Scholar
  7. 7.
    Aminova A.V.: Sbornik Math. 186(12), 1711 (1995)MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Aminova, A.V.: Tensor, N.S. 65 (2000)Google Scholar
  9. 9.
    Prince G.E., Crampin M.: Gen. Relativ. Gravit. 16, 921 (1984)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Feroze T., Mahomed F.M., Qadir A.: Nonlinear Dyn. 45, 65 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bokhari A.H., Kara A.H., Kashif A.R., Zaman F.D.: Int. J. Theor. Phys. 45, 1063 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bokhari A.H., Kara A.H.: Gen. Relativ. Gravit. 39, 2053–2059 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Yano K.: The theory of Lie derivatives and its Applications. North Holland, Amsterdam (1956)Google Scholar
  14. 14.
    Scouten K.J.A.: Ricci Calculus. Springer, Berlin (1954)Google Scholar
  15. 15.
    Stephani H.: Differential Equations: Their Solutions Using Symmetry. Cambridge University Press, New York (1989)Google Scholar
  16. 16.
    Tsamparlis M., Nikolopoulos D., Apostolopoulos P.S.: Class. Quantum Grav. 15, 2909 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Barnes A.: Class. Quantum Grav. 10, 1139 (1993)MATHCrossRefADSGoogle Scholar
  18. 18.
    Knebelman M.S., Yano K.: Proc. Am. Math. Soc. 12, 300 (1961)MATHMathSciNetGoogle Scholar
  19. 19.
    Maartens R., Maharaj S.D.: Class Quant Grav. 3, 1005 (1986)MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Maartens R.: J. Math. Phys. 28, 2051 (1987)MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Physics, Department of Astronomy-Astrophysics-MechanicsUniversity of AthensAthensGreece

Personalised recommendations