Communications in Mathematical Physics

, Volume 106, Issue 1, pp 137–158

Homothetic and conformal symmetries of solutions to Einstein's equations

  • D. Eardley
  • J. Isenberg
  • J. Marsden
  • V. Moncrief
Article

Abstract

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.

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References

  1. 1.
    Gerhardt, C.: H-surfaces in Lorentcian manifolds. Commun. Math. Phys.89, 523 (1983)Google Scholar
  2. 2.
    Marsden, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep.66, 109 (1980)Google Scholar
  3. 3.
    Löbell, F.: Ber. Verhandl. Sächs Akad. Wiss. Leipzig. Math. Phys. K1.83, 167 (1931)Google Scholar
  4. 4.
    Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Gravitation: An introduction to current research. Witten, L. (ed.). New York: Wiley 1962Google Scholar
  5. 5.
    Fisher, A., Marsden, J.: Bull. Am. Math. Soc.79, 995 (1973)Google Scholar
  6. 6.
    Fischer, A., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einstein's equations. I. One Killing field. Ann. Inst. H. Poincaré33, 147 (1980)Google Scholar
  7. 7.
    Arms, J., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einstein's equations. II. Several Killing fields and the Einstein-Yang-Mills equations. Ann. Phys. (N.Y.)144, 81 (1982)Google Scholar
  8. 8.
    O'Murchadha, N., York, J. W.: Initial-value problem of general relativity. II. Stability of solutions of the initial-value equations. Phys. Rev.D 10, 437 (1974)Google Scholar
  9. 9.
    Isenberg, J., Nester, J.: Canonical gravity. In: General relativity and gravitation. Vol. 1. Held, A. (ed.). New York, London: Plenum 1980Google Scholar
  10. 10.
    Fischer, A., Marsden, J.: The initial value problem and the dynamical formulation of general relativity. In: General relativity. Einstein centenary survey. Hawking, S., Israel, W. (eds.). Cambridge: Cambridge University Press 1979Google Scholar
  11. 11.
    Choquet-Bruhat, Y., York, J.: The Cauchy problem. In: General relativity and gravitation. Vol. 1. Held, A. (ed.). New York, London: Plenum 1980Google Scholar
  12. 12.
    Berger, B. K.: Homothetic and conformal motions in spacelike slices of solutions of Einstein's equations. J. Math. Phys.17, 1268 (1976)Google Scholar
  13. 13.
    Misner, C., Thorne, K. Wheeler, J.: Gravitation. San Francisco: W. H. Freemann 1973Google Scholar
  14. 14.
    Hawking, S. W., Ellis, G. F. R.: The large scale structure of spacetime. Cambridge: Cambridge University Press 1983Google Scholar
  15. 15.
    Eardley, D. M.: Self-similar spacetimes: Geometry and dynamics. Commun. Math. Phys.37, 287 (1974)Google Scholar
  16. 16.
    Yano, K., Bochner, S.: Curvature and Betti numbers. In: Ann. Math. Stud. No.32. Princeton: Princeton University Press 1953Google Scholar
  17. 17.
    DeWitt, B.: In: Relativity, group and topology. DeWitt, C. DeWitt, B. (eds.) Gordon and Breach 1964Google Scholar
  18. 18.
    Yano, K.: The theory of Lie derivatives and its applications. New York: Interscience 1957Google Scholar
  19. 19.
    Garfinkle, D., Tiem, Q. J.: private communication 1985Google Scholar
  20. 20.
    Schoen, R., Yau, S. -T.: Proof of the positive mass theorem. II. Commun. Math. Phys.79, 231 (1981), and their previous papers cited thereinGoogle Scholar
  21. 21.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381 (1981)Google Scholar
  22. 22.
    Parker, T., Taubes, C. H.: On Witten's proof of the positive energy theorem. Commun. Math. Phys.84, 223 (1982)Google Scholar
  23. 23.
    Yip, P.: private communication 1985Google Scholar
  24. 24.
    Maithreyan, T. Eardley, D.: Gravitational collapse of spherically symmetric scale invariant scalar fields. ITP preprint 1985Google Scholar
  25. 25.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differen. Geom. (To appear).Google Scholar
  26. 26.
    Isenberg, J., Marsden, J.: Geom. Phys.1, 85 (1984)Google Scholar
  27. 27.
    Fischer, A., Marsden, J.: Can. J. Math.29, 193 (1977)Google Scholar
  28. 28.
    Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations. I. J. Math. Phys.16, 493 (1975)Google Scholar
  29. 29.
    Jantzen, R., Rosquist, K.: Adapted Slicings of space-times possessing simply transitive similarity groups. J. Math. Phys.27, 1191 (1986).Google Scholar
  30. 30.
    Hsu, L., Wainwright, J.: Self-similar spatially homogeneous cosmologies I: Orthogonal perfect fluid and vacuum solutions. Class. Quantum Gravity. (To appear).Google Scholar
  31. 31.
    Garfinkle, D.: Asymptotically flat spacetimes have no conformal Killing fields. J. Math. Phys. (To appear).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • D. Eardley
    • 1
  • J. Isenberg
    • 2
  • J. Marsden
    • 3
  • V. Moncrief
    • 4
  1. 1.Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  4. 4.Department of Mathematics and Department of PhysicsYale UniversityNew HavenUSA

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