General Relativity and Gravitation

, Volume 17, Issue 5, pp 475–491 | Cite as

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

  • G. S. Hall
  • M. S. Hickman
  • C. B. G. McIntosh
Research Articles

Abstract

This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by McIntosh and Hickman [1]. Tetrads of real metrics can be formally complexified by complex coordinate transformations and tetrad rotations and their properties investigated from the viewpoint of complex relativity. First, complex bivectors are examined and classified, partly by using the fundamental quadric surface of a metric in projective complex 3-space Pℂ3-an elegant but not well-known method of investigating the null structure of a metric. A generalization of the Mariot-Robinson theorem from real relativity is then given and related to various canonical forms of complex bivectors. The second part of the paper discusses four classes of complex metrics. Real metrics of the first class are ones with a null congruence whose wave surfaces have equal curvature. The second class, a subcase of the first one, is the main one; it contains integrable double Kerr-Schild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typs-D vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) Kerr-Schild form.

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References

  1. 1.
    McIntosh, C. B. G., and Hickman, M. S. (1985). Complex relativity solutions, I: Introduction,Gen. Rel. Grav., to appear.Google Scholar
  2. 2.
    Plebanski, J. F., and Schild, A. (1976).Nuovo Cimenta,35B, 35.Google Scholar
  3. 3.
    Ehlers, J., and Kundt, W. (1962). InGravitation: an Introduction to Current Research, L. Witten, ed. (Wiley, New York), p. 49.Google Scholar
  4. 4.
    Robinson, I. (1982).Spacetime and Geometŕy (University of Texas Press, Austin), p. 26.Google Scholar
  5. 5.
    Hall, G. S. (1985). Preprint.Google Scholar
  6. 6.
    Semple, J. G., and Kneebone, G. T. (1952).Algebraic Protective Geometry (Oxford University Press, Oxford).Google Scholar
  7. 7.
    Kramer, D., Stephani, H., MacCallum, M., and Herlt, E. (1980).Exact Solutions of Einstein's Field Equations (VEB Deutscher Verlag der Wissenschaften, Berlin/Cambridge University Press, Cambridge).Google Scholar
  8. 8.
    Torres del Castillo, G. F. (1983).J. Math. Phys.,24, 590.Google Scholar
  9. 9.
    Plebanski, J. F., and Robinson, I. (1976).Phys. Rev. Lett.,37, 493.Google Scholar
  10. 10.
    Hall, G. S. (1978).Z. Naturforsch,33a, 559.Google Scholar
  11. 11.
    Hall, G. S. (1983). Classification of Second Order Symmetric Tensors in General Relativity Theory, inDifferential Geometry, Banach Centre Publications, Vol. 12, p. 53. (Polish Scientific Publishers, Warsaw)Google Scholar
  12. 12.
    McIntosh, C. B. G., and Hall, G. S. (1985). Symmetries and Kerr-Schild type metrics, preprint.Google Scholar
  13. 13.
    Plebański, J. F., and Hacyan, S. (1975).J. Math. Phys.,16, 2403.Google Scholar
  14. 14.
    Hickman, M. S. (1983). Vacuum Spacetimes from a Complex Viewpoint II: Quarter Flat Spaces, inContributed papers of the Tenth International Conference on General Relativity and Gravitation, Padova, Vol. 1, B. Bertotti, F. de Felice, and A. Pascolini, eds. (Consiglio Nazionale delle Ricerche, Roma), p. 259.Google Scholar
  15. 15.
    Kundt, W. (1961).Z. Phys.,163, 77.Google Scholar
  16. 16.
    Boyer J. P., Finley, J. D., and Plebański, J. F. (1980).General Relativity and Gravitation, Vol. 2, A. Held, ed. (Plenum Press, New York), p. 241.Google Scholar
  17. 17.
    Finley, J. D., and Plebański, J. F. (1981).J. Math. Phys.,22, 667.Google Scholar
  18. 18.
    Debney, G. C., Kerr, R. P., and Schild, A. (1969).J. Math. Phys.,10, 1842.Google Scholar
  19. 19.
    McIntosh, C. B. G. (1984). InClassical General Relativity, W. B. Bonnor, J. N. Islam, and M. A. H. MacCullum, eds. (Cambridge University Press, Cambridge), p. 183.Google Scholar
  20. 20.
    Kinnersley, W. (1969).J. Math. Phys.,10, 1195.Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • G. S. Hall
    • 1
  • M. S. Hickman
    • 2
  • C. B. G. McIntosh
    • 2
  1. 1.Department of MathematicsUniversity of Aberdeen, Edward Wright BuildingAberdeenScotland
  2. 2.Mathematics DepartmentMonash UniversityClaytonAustralia

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