General Relativity and Gravitation

, Volume 25, Issue 10, pp 1009–1018 | Cite as

AG 2 II algebraically general twistfree vacuum spacetime

  • Duong Phan
Research Articles


Several authors, e.g., Kerr and Debney (1970), Lun (1978), have obtained severalG 2 II algebraically special vacuum solutions. NoG 2 II algebraically general vacuum solutions in explicit form have been found before. In this paper, we start from a system of first order partial differential equations, obtained by using a triad formalism, which determines twistfree vacuum metrics with a spacelike Killing vector. The method of group-invariant solutions is then used and aG 2 II algebraically general twistfree vacuum solution is obtained. The solution also admits a homothetic Killing vector and is non-geodesic. It is believed to be new. The following explicit solutions are also obtained: (1) A Petrov type II with aG1-group of motions solution which belongs to Kundt's class. (2) A Petrov type III,G3 Robinson-Trautman solution. All these solutions are known.


Differential Equation Partial Differential Equation Triad Explicit Form Differential Geometry 
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  1. 1.
    Hall, G. S., Morgan, T., and Perjés, Z. (1987).Gen. Rel. Grav. 19, 1137.Google Scholar
  2. 2.
    Hearn, A. (1987). REDUCEUser's Manual, version 3.3 (Rand Corporation CP78, Santa Monica, California).Google Scholar
  3. 3.
    Kersten, P. H. M. (1985). “Infinitesimal Symmetries: a Computational Approach”. Ph.D. thesis, University of Twente, Enschede, The Netherlands.Google Scholar
  4. 4.
    Kramer, D., and Neugebauer, G. (1968).Commun. Math. Phys. 7, 173.Google Scholar
  5. 5.
    Kramer, D., Stephani, H., MacCallum, M. A. H., and Herlt, E. (1980).Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge).Google Scholar
  6. 6.
    Olver, P. W. (1986).Applications of Lie Groups to Differential Equations (Springer-Verlag, Berlin).Google Scholar
  7. 7.
    Ovsiannikov, L. V. (1982).Group Analysis of Differential Equations (Academic Press, New York).Google Scholar
  8. 8.
    Phan, D. (1991).Gen. Rel. Grav. 23, 269.Google Scholar
  9. 9.
    Phan, D. (1992). “Vacuum spacetimes with a spacelike Killing vector”. Ph.D. thesis, University of Sydney, Australia.Google Scholar
  10. 10.
    McIntosh, C. B. G. Private communication.Google Scholar
  11. 11.
    Kerr, R. P. and Debney, G. (1970).J. Math. Phys. 11, 2807.Google Scholar
  12. 12.
    Lun, W. C. (1978).Phys. Lett. A69, 79.Google Scholar
  13. 13.
    Newman, E. T., and Tamburino, L. A. (1962).J. Math. Phys. 3, 902.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Duong Phan
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of SydneyAustralia

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