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Homothetic and Conformal Symmetries of Solutions to Einstein's Equations

Eardley, D. and Isenberg, J. and Marsden, J. and Moncrief, V. (1986) Homothetic and Conformal Symmetries of Solutions to Einstein's Equations. Communications in Mathematical Physics, 106 (1). pp. 137-158. ISSN 0010-3616. doi:10.1007/BF01210929.

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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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Additional Information:© Springer-Verlag 1986. Received: 16 December 1985. Communicated by S.-T. Yau. This work was supported in part by National Science Foundation under Grant Nos. DMS 83·03998 at the University of Oregon. DMS 84·04506 at the University of California at Berkeley, PHY85·03072 at Yale. and PHY85-06686 and PHY82·17853 (supplemented by founds from the National Aeronautics and Space Administration) at the University of California at Santa Barbara. Note added in proof. For recent work on solutions admitting a proper homothetic Killing vector field see [29.30]. For a result on nonexistence of spacetimcs that arc asymptotically nat at null infinity and that admit a conformal Killing vector field see [31].
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NSFDMS 83·03998
NSFDMS 84·04506
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ID Code:19241
Deposited By: Ruth Sustaita
Deposited On:02 Aug 2010 17:05
Last Modified:08 Nov 2021 23:50

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