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

DOI: 10.1007/BF01210929

Cite this article as:
Eardley, D., Isenberg, J., Marsden, J. et al. Commun.Math. Phys. (1986) 106: 137. doi:10.1007/BF01210929


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.

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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