Symmetries and exact solutions of Einstein's equations

  • C. B. G. McIntosh
Theoretical aspects of general relativity
Part of the Lecture Notes in Physics book series (LNP, volume 124)


The theorems in the last section which show that curvature collineations are almost always conformal motions are somewhat disappointing since they mean that the program of using curvature collineations (and indeed therefore many symmetries other than conformal motions) in the search for exact solution of Einstein's equations is not a very fruitful one. The relatively few spacetimes which have non-trivial curvature collineations are ones in general with lots of other symmetries or properties which mean that they are well known or probably easily found solutions. It is still interesting to see why such spacetimes have non-trivial curvature collineations and a discussion on this point will be published elsewhere.

In the cosmological case it has been shown that Eardley's suggested examination of models with homothetic motions is not as useful as may be hoped because of the restrictions on such motions as outlined in II. Thus, again, the program of looking at solutions with symmetries other than Killing ones is not quite as fruitful as may be hoped.

On the other hand, many authors when writing about symmetries only seem to think in terms of isometries, and so it must be stressed that it is important to think in wider terms and examine the role of other symmetries. There are still many useful and interesting results using such symmetries waiting to be found!


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    See for example Cosgrove, C., this volume.Google Scholar
  2. 2.
    Kinnersley, W., J.Math.Phys., 10, 1195 (1969).Google Scholar
  3. 3.
    Katzin, G.H., Levine, J. and Davis, W.R., J.Math.Phys., 10, 617 (1969).Google Scholar
  4. 4.
    Hughston, L.P. and Sommers, P., Commun.Math.Phys., 32, 147 (1973).Google Scholar
  5. 5.
    Collinson, C.D. and French, D.C., J.Math.Phys., 8, 701 (1967).Google Scholar
  6. 6.
    Halford, D. and Kerr, R.P., J.Math.Phys., to appear (1979).Google Scholar
  7. 7.
    Kerr, R.P. and Halford, W.D., J.Math.Phys., to appear (1979).Google Scholar
  8. 8.
    King, A.R. and Ellis, G.F.R., Corrsnun.Math. Phys., 31, 209 (1973).Google Scholar
  9. 9.
    Eardley, D.M., Concrrun.Math.Phys., 37, 287 (1974).Google Scholar
  10. 10.
    McIntosh, C.B.G., Phys.Letts.A, 50, 429 (1975) and Gen.Rel.Grav., 7, 199 (1976).Google Scholar
  11. 11.
    Wainwright, J., Ince, W.C.W. and Marshman, B.J., Gen.ReZ.Grav., to appear (1979).Google Scholar
  12. 12.
    McIntosh, C.B.G., Phys.Letts.A, 69, 1 (1978).Google Scholar
  13. 13.
    Buchdahl, H.A., Int.J.Theoret.Phys., 6, 407 (1972) and 7, 287 (1973).Google Scholar
  14. 14.
    Sneddon, G.E. and McIntosh, C.B.G., Aust.J.Phys., 27, 411 (1974).Google Scholar
  15. 15.
    McIntosh, C.B.G., Conmn.Math.Phys., 37, 335 (1974).Google Scholar
  16. 16.
    Collinson, C.D., J.Math.Phys., 11, 818 (1970).Google Scholar
  17. 17.
    Aichelburg, P.C., Gen.Rel.Grav., 3, 397 (1972).Google Scholar
  18. 18.
    Halford, D., McIntosh, C.B.G. and van Leeuwen, E.H., in preparation (1979).Google Scholar
  19. 19.
    Kundt, W., Zeitschrift für Physik, 163, 77 (1961).Google Scholar
  20. 20.
    Pirani, F.A.E., Lectures on General Relativity, Brandeis Summer Institute in Theoretical Physics, Prentice-Hall, New Jersey, p. 354 (1964).Google Scholar
  21. 21.
    Tariq, N. and Tupper, B.O.J., Tensor, 31, 42 (1977).Google Scholar
  22. 22.
    Hlavatý, V., J.Math.Mech., 8, 285 and 597 (1959).Google Scholar
  23. 23.
    Ihrig, E., J.Math.Phys., 16, 54 (1975).Google Scholar
  24. 24.
    Goldberg, J.N. and Kerr, R.P., J.Math.Phys., 2, 327 (1961).Google Scholar
  25. 25.
    Kerr, R.P. and Goldberg, J.N., T.Math.Phys., 2, 332 (1961).Google Scholar
  26. 26.
    McIntosh, C.B.G. and Halford, W.D., in preparation (1979).Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • C. B. G. McIntosh
    • 1
  1. 1.Department of MathematicsMonash UniversityClaytonAustralia

Personalised recommendations