On the uniquenessof the Einstein-Straus model

Abstract

We show that the Einstein-Straus model does not give a robust answer tothe problem of the influence of the cosmic expansion on the local physics.This is done by finding the most general static region embeddable in aFriedmann-Lemaître-Robertson-Walker expanding cosmology andshowing that the model must be `almost spherically symmetric'. Moreprecisely, we show that the boundary of the static region must be a2-sphere at each instant of cosmic time. The motion of this 2-sphere inspacetime is as follows:its `would-be' centre (if there were no static region) moves along a pathwhose projection on any 3-space of constant cosmic time is a geodesicwith respect to the induced 3-metric. The velocity of this geodesic isdetermined from the matching. In particular, this centre must be at restwith respect to the cosmological flow (thus giving a spherically symmetricmodel) when any of the standard energy-momentum tensors inside thestatic region is imposed.