Homothetic and Conformal Symmetries of Solutions to Einstein
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Commun. Math. Phys. 106, t37-158 (1986) Communications in Mathematical Physics © Springer-Verlag 1986 Homothetic and Conformal Symmetries of Solutions to Einstein's Equations D. Eardley: J. Isenberg,I J. Marsden J and V. Moncrief t Institute for Theoretical Physics. University of California, Santa Barbara, California, 93t06. USA 2 Department of Mathematics. University of Oregon, Eugene. Oregon 97403. USA 3 Department of Mathematics. University of California. Berkeley. California 94720, USA 4 Department of Mathematics and Department of Physics. Yale University, P.O. Box 6666. New Haven, Connecticut 0651t. USA 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 admilting a hypersurface of constant mean extrinsic curvature, and also admilting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the malter 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 ifthe 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. I. Introduction Virtually all explicitly known spacetime solutions of Einstein's equations admit some nontrivial isometry group. This is not surprising since the Einstein equations are very difficult to solve, and isometries simplify them considerably. While much physical insight on astrophysical and cosmological questions has been obtained from the study of spacetimes with lots of symmetry, it clearly would be useful to examine solutions with a smaller isometry group, or even a trivial one. One possible way for reducing spacetime symmetry without giving up all the simplifications it provides is to replace spacetime isometries with spacetime c:oltformal symmelries or Itomothelic symmetries. A conformal symmetry preserves the metric up to a general point-dependent scale factor. while for a homothetic symmetry the scale factor must be constant.
Transcript of Homothetic and Conformal Symmetries of Solutions to Einstein
Commun. Math. Phys. 106, t37-158 (1986)
Communications inMathematical
Physics© Springer-Verlag 1986
Homothetic and Conformal Symmetries of Solutions toEinstein's Equations
D. Eardley: J. Isenberg,I J. MarsdenJ and V. Moncrieft Institute for Theoretical Physics. University of California, Santa Barbara, California, 93t06. USA2 Department of Mathematics. University of Oregon, Eugene. Oregon 97403. USA3 Department of Mathematics. University of California. Berkeley. California 94720, USA4 Department of Mathematics and Department of Physics. Yale University, P.O. Box 6666. NewHaven, Connecticut 0651t. USA
Abstract. We present several results about the nonexistence of solutions ofEinstein's equations with homothetic or conformal symmetry. We show that theonly spatially compact, globally hyperbolic spacetimes admilting a hypersurfaceofconstant mean extrinsic curvature, and also admilting an infinitesimal properhomothetic symmetry, are everywhere locally flat; this assumes that the malterfields either obey certain energy conditions, or are the Yang-Mills or masslessKlein-Gordon fields. We find that the only vacuum solutions admitting aninfinitesimal proper conformal symmetry are everywhere locally flat spacetimesand certain plane wave solutions. We show that ifthe dominant energy conditionis assumed, then Minkowski spacetime is the only asymptotically flat solutionwhich has an infinitesimal conformal symmetry that is asymptotic to a dilation.In other words, with the exceptions cited, homothetic or conformal Killing fieldsare in fact Killing in spatially compact or asymptotically flat spactimes. In theconformal procedure for solving the initial value problem, we show that datawith infinitesimal conformal symmetry evolves to a spacetime with full isometry.
I. Introduction
Virtually all explicitly known spacetime solutions of Einstein's equations admitsome nontrivial isometry group. This is not surprising since the Einstein equationsare very difficult to solve, and isometries simplify them considerably. While muchphysical insight on astrophysical and cosmological questions has been obtainedfrom the study of spacetimes with lots of symmetry, it clearly would be useful toexamine solutions with a smaller isometry group, or even a trivial one. One possibleway for reducing spacetime symmetry without giving up all the simplifications itprovides is to replace spacetime isometries with spacetime c:oltformal symmelries orItomothelic symmetries. A conformal symmetry preserves the metric up to a generalpoint-dependent scale factor. while for a homothetic symmetry the scale factor mustbe constant.
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