Generalisation of the Einstein-Straus model to anisotropic settings

TL;DR

This paper investigates extending the Einstein--Straus model to anisotropic spacetimes, revealing restrictions on matching static cylindrical regions to LRS models and implications for cosmological objects and collapse models.

Contribution

It demonstrates the limitations of matching static cylindrically symmetric regions to anisotropic LRS spacetimes and explores consequences for cosmological models and gravitational collapse.

Findings

Matching is only possible for non-evolving LRS models.

Embedding static strings in Bianchi models is impossible.

Only specific LRS stiff fluid solutions are compatible with perfect fluid sources.

Abstract

We study the possibility of generalising the Einstein--Straus model to anisotropic settings, by considering the matching of locally cylindrically symmetric static regions to the set of $G_4$ on $S_3$ locally rotationally symmetric (LRS) spacetimes. We show that such matchings preserving the symmetry are only possible for a restricted subset of the LRS models in which there is no evolution in one spacelike direction. These results are applied to spatially homogeneous (Bianchi) exteriors where the static part represents a finite bounded interior region without holes. We find that it is impossible to embed finite static strings or other locally cylindrically symmetric static objects (such as bottle or coin-shaped objects) in reasonable Bianchi cosmological models, irrespective of the matter content. Furthermore, we find that if the exterior spacetime is assumed to have a perfect fluid source satisfying the dominant energy condition, then only a very particular family of LRS stiff fluid solutions are compatible with this model. Finally, given the interior/exterior duality in the matching procedure, our results have the interesting consequence that the Oppenheimer-Snyder model of collapse cannot be generalised to such anisotropic cases.