General Relativity and Gravitation

, Volume 39, Issue 12, pp 2053–2059

Noether versus Killing symmetry of conformally flat Friedmann metric

Research Article


In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.


Noether symmetry Killing vector Conformal Killing vector 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation Benjamin, New York (1973)Google Scholar
  2. 2.
    Petrov A.Z. (1969). Einstein Spaces. Pergamon, Oxford MATHGoogle Scholar
  3. 3.
    Stephani H., Kramer D., MacCallum M.A.H., Hoenselaers C. (2003). Exact solutions of Einstein Field Equations. Cambridge University Press, Cambridge MATHGoogle Scholar
  4. 4.
    Katzin, G.H., Levine, J.: Coloq. Math. 26, (21) (1972)Google Scholar
  5. 5.
    Hall, G.S.: Symmetries and curvature structure in general relativity. World Scientific, (2004)Google Scholar
  6. 6.
    Bokhari A.H., Kashif A.R. (1996). J. Math. Phys. 37(7): 3496 CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Giachetta, G., Sardanashvily, G.: Stree-energy-momentum tensors in Lagrangian field theory, arXiv:gr-qc/9510061Google Scholar
  8. 8.
    Camci U., Barnes A. (2002). Class. Quantum Grav. 19: 393 MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Nunez L.A., Percoco U., Villalba V.M. (1990). J. Math. Phys. 31: 137 MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Bokhari A.H. (1992). Int. J. Th. Phys. 31: 2091 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Amer M.J., Bokhari A.H., Qadir A. (1994). J. Math. Phys. 35(6): 3005 CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Marteen, R., Maharaj, S.D.: Class. Quantum Grav., 3, 1005 (1986)Google Scholar
  13. 13.
    Fatibene L., Ferraris M., Francaviglia M., McLenaghan R.G. (2002). J. Math. Phys. 43(6): 3147 MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Mangiarotti L., Sardanashvily G. (2000). Connections in classical and quantum field theory. World Scientific, Singapore MATHGoogle Scholar
  15. 15.
    Bokhari A.H., Kara A.H., Kashif A.R., Zaman F.D. (2006). Int. J. Th. Phys. 45(6): 1063 MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Wolf, T.: Crack, LiePDE, ApplySym and ConLaw, section 4.3.5 and computer program on CD-ROM. In: Grabmeier, J., Kaltofen, E., Weispfenning, V. (eds.) Computer Algebra Handbook, vol. 465. Springer, Heidelberg (2002)Google Scholar
  17. 17.
    Wolf T. (2004). Applications of CRACK in the classification of integrable systems. CRM Proc. Lect. Notes 37: 283 Google Scholar
  18. 18.
    Bokhari, A.H.: Conformal extension of Pseudo-Newtonian Formalis, PhD Thesis, Quaid-i-Azam University (1985)Google Scholar
  19. 19.
    Kara A.H., Mahomed F.M., Vawda F.E. (1994). Lie groups and their applications 2: 27 MathSciNetGoogle Scholar
  20. 20.
    Kara A.H., Khalique C.M. (2005). J. Phys. A 38: 4629 MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.School of Mathematics and Centre for Differential Equations, Continuum Mechanics and ApplicationsUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations