Initial-value constraints and generation of isometries and homothetic motions.

Abstract

We discuss the generation of spacetime symmetries from an initial-value point of view. After reviewing the effective set of initial-value constraints (a combination of the familiar gravitational constraints and an additional set arising from projections of ${\mathcal{L}}_{\ensuremath{\xi}}g=\ensuremath{\alpha}g$ on the initial surface), we focus our attention on the construction of initial data generating spacetimes admitting homothetic motion. From the analysis of the constrained equations we show that there are only two classes of homothetic, spherically symmetric spacetimes ($M, g$) where (a) $M$ admits a flat Cauchy hypersurface $\ensuremath{\Sigma}$, (b) $g$ is a perfect-fluid solution of Einstein's equation with $P=k\ensuremath{\rho}$, $0\ensuremath{\le}k\ensuremath{\le}1$ as the equation of state and with an initial fourvelocity parallel to the normal of $\ensuremath{\Sigma}$, and (c) the projection of the infinitesimal homothetic generator $\ensuremath{\xi}$ along the normal of $\ensuremath{\Sigma}$ is everywhere a nonvanishing constant. The first class is the spatially flat Friedmann-Robertson-Walker (FRW) class of metrics. The second is a particular class of the marginally bound Tolman class of metrics. In combination with recent work by Eardley et al. [which shows the absence of spacetimes ($M, g$) with $g$ a nonvacuum homothetic solution of Einstein's equations and $M$ admitting a compact Cauchy hypersurface of constant extrinsic curvature], these results appear to suggest that only a "few" metrics (a) are compatible with Einstein's (nonvacuum) equations, and (b) allow the existence of a homothetic symmetry.