Advertisement

Annali di Matematica Pura ed Applicata

, Volume 126, Issue 1, pp 379–396 | Cite as

Null electromagnetic fields, total gravitational radiation and collineations in general relativity

  • Zafar Ahsan
  • S. Izhar Husain
Article

Summary

The null electromagnetic field, or (F, g, r, S)-structure, and the corresponding Nijenhuis tensor have been studied in an invariant index-free manner. It is seen that the null electromagnetic fields are characterized by the relation F3= 0, and the Nijenhuis tensor plays a very natural role in the study of null electromagnetic fields. The Lichnerowicz contions for the total gravitational radiation have been given in the present setting, and the condition F3=0 has been translated into a corresponding condition on the Ricci tensor. Further, different types of collineations, for (F, g, r, S)-structure, along the propagation and polarization vectors S and T, respectively, have been studied. It is also shown that\(\mathop \pounds\limits_S F_{ij} = 0\) implies\(F_j^k \mathop \pounds\limits_S g_{ik} = 0\). Finally, a covariant conservation law generator has been given for the present structure.

Keywords

General Relativity Electromagnetic Field Polarization Vector Ricci Tensor Present Setting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Z. Ahsan,Studies of Radiation in General Relativity, Ph. D. Thesis, Aligarh Muslim University, Aligarh (1978).Google Scholar
  2. [2]
    Z.Ahsan,Gravitational Radiation and Nijenhuis Tensor, in General Relativity Physics and Contemporary Needs, Vol. 5 (1980), Ed. Riazuddin, Plenum Press.Google Scholar
  3. [3]
    B. Carter,Black Hole Equilibrium State, in Black Holes, Les Houches Lectures, Gordon Breach, N. Y. (1972), p. 57.Google Scholar
  4. [4]
    C. D. Collinson. GRG,1 (1970), p. 137.Google Scholar
  5. [5]
    K. L. Duggal, Tensor (N. S.),22 (1971), p. 328.Google Scholar
  6. [6]
    K. L.Duggal,Existence of two Killing vector fields on the space-time of general relativity, preprint (1978).Google Scholar
  7. [7]
    L. P.Eisenhert,Riemanian geometry, Princeton University Press (1949).Google Scholar
  8. [8]
    V. Hlavaty,Geometry of Einstein Unified Field Theory, P. Noordhoff, Groningen (1958).Google Scholar
  9. [9]
    W. Israel, Comm. Math. Phys.,8 (1968), p. 245.Google Scholar
  10. [10]
    G. H. Katzin -J. Levine -W. R. Davis, J. Math. Phys.,10 (1969), p. 617.Google Scholar
  11. [11]
    S. Kobayashi -K. Nomizu,Foundation of Differential Geometry, Vol. 1, Interscience publishers, N. Y. (1963).Google Scholar
  12. [12]
    A. Lichnerowicz, Ann. di Mat. Pura ed Appl.,50 (1960), p. 1.Google Scholar
  13. [13]
    M. Matsumuto, Tensor (N. S.),21 (1970), p. 15.Google Scholar
  14. [14]
    Y. Matsushima,Differentiable Manifolds, Marcel Dekker, Inc., N. Y. (1972), p. 142.Google Scholar
  15. [15]
    H. Michalaski -J. Wainwright,GRG,6 (1975), p. 289.Google Scholar
  16. [16]
    R. S. Mishra, Tensor (N.S.),30 (1976), p. 145.Google Scholar
  17. [17]
    H. Müller zum Hagen, D. C. Robinson andH. J. Seifert, GRG,5 (1974), p. 61.Google Scholar
  18. [18]
    L. Radhakrishna -V. D. Khade, J. Shiv. Univ.,6 (1973), p. 57.Google Scholar
  19. [19]
    R. K. Sachs, Z. Physik,157 (1960), p. 462.Google Scholar
  20. [20]
    A.Trautmann,Conservation laws in general relativity, in Gravitation: an introduction to current research, ed. L. Witten, J. Wiley and Sons (1962).Google Scholar
  21. [21]
    M. L. Woolley, Comm. Math. Phys.,31 (1973), p. 75.Google Scholar
  22. [22]
    K.Yano,Theory of Lie derivatives and its applications, North Holland Publications Co. (1957).Google Scholar
  23. [23]
    K. Yano,Differential geometry of complex and almost complex spaces, Pergamon Press, London (1965).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1980

Authors and Affiliations

  • Zafar Ahsan
    • 1
  • S. Izhar Husain
    • 1
  1. 1.India

Personalised recommendations