On the uniqueness of the Einstein-Straus model

We show that the Einstein-Straus model does not give a robust answer to the problem of the influence of the cosmic expansion on the local physics. Thi s is done by finding the most general static region embeddable in a Friedma nn-Lemaitre-Robertson-Walker expanding cosmology and showing that the model must be `almost spherically symmetric'. More precisely, we show that the b oundary of the static region must be a 2-sphere at each instant of cosmic t ime. The motion of this 2-sphere in spacetime is as follows: its `would-be' centre (if there were no static region) moves along a path whose projectio n on any 3-space of constant cosmic time is a geodesic with respect to the induced 3-metric. The velocity of this geodesic is determined from the matc hing. In particular, this centre must be at rest with respect to the cosmol ogical flow (thus giving a spherically symmetric model) when any of the sta ndard energy-momentum tensors inside the static region is imposed.