General Relativity and Gravitation

, Volume 24, Issue 5, pp 477–500

Finding isometry groups in theory and practice

  • Marcelo E. Araujo
  • Tevian Dray
  • James E. F. Skea
Research Articles


An algorithm is given for determining the isometry group of an arbitrary spacetime (in four dimensions). Numerous examples are given and the partial implementation of this algorithm using the symbolic manipulation package CLASSI is discussed.


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  1. 1.
    Karlhede, A., and MacCallum, M. A. H. (1982).Gen. Rel. Grav. 14, 673.Google Scholar
  2. 2.
    Cartan, E. (1946).Leçons sur la Géométrie des Espaces de Riemann (Gauthier-Villars, Paris).Google Scholar
  3. 3.
    Brans, C. H. (1965).J. Math. Phys. 6, 94.Google Scholar
  4. 4.
    Karlhede, A. (1980).Gen. Rel. Grav. 12, 693.Google Scholar
  5. 5.
    Kobayashi, S., and Nomizu, K. (1963).Foundations of Differential Geometry (John Wiley and Sons, New York), vol. 1.Google Scholar
  6. 6.
    Spivak, M. (1979).A Comprehensive Introduction to Differential Geometry (2nd. ed., Publish or Perish, Houston) vol 2.Google Scholar
  7. 7.
    Karlhede, A. (1979). “A Review of the Equivalence Problem”, University of Stockholm preprint, Appendices 2 and 3. This paper was later published without the appendices as Ref. 4.Google Scholar
  8. 8.
    MacCallum, M. A. H., and Skea, J. E. F. (1991). InAlgebraic Computing in General Relativity. Lecture Notes from the First Brazilian School on Computer Algebra, M. J. Rebouças and W. L. Roque, eds. (Oxford University Press, Oxford), vol. 2, to appear.Google Scholar
  9. 9.
    Schmidt, B. (1968). Dissertation, Universität Hamburg.Google Scholar
  10. 10.
    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
  11. 11.
    MacCallum, M. A. H. (1980). “On Enumerating the Real Four Dimensional Lie Algebras”, Queen Mary College preprint.Google Scholar
  12. 12.
    Kruchkovich, G. I. (1954).Usp. Matem. Nauk SSSR, 9, part 1(59), 3; Petrov, A. Z. (1969).Einstein Spaces (Pergamon, Oxford); Bratzlavsky, F. (1959).Sur les algèbres et les groupes de Lie résolubles de dimension trois et quatre. Memoire de Licence, Université Libre de Bruxelles; Mubarakzyanov, G. M. (1963).Izv. Vyss. Uch. Zav. Mat. 1(32), 114;3(34), 99;4(35), 104; Patera, J., Sharp, R. T., Winternitz, P., and Zassenhaus, H. (1976).J. Math. Phys. 17, 986.Google Scholar
  13. 13.
    Gott III, J. R. (1985).Astrophys. J. 288, 422.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Marcelo E. Araujo
    • 1
  • Tevian Dray
    • 3
  • James E. F. Skea
    • 4
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasília DFBrazil
  2. 2.Department of PhysicsUniversity of Wales at CardiffCardiffUK
  3. 3.Department of MathematicsOregon State UniversityCorvallisUSA
  4. 4.School of Mathematical SciencesQueen Mary and Westfield CollegeLondonUK

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