Generalization of the Einstein-Straus model to anisotropic settings

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.

Matching of spatially homogeneous nonstationary space-times to vacuum in cylindrical symmetry

“…Thus we require the interior to be spatially homogeneous, with an axial symmetry and at least one Killing vector commuting with the axial symmetry. This is sufficient to force the interior to be an LRS space-time (see Mena, Tavakol & Vera [8]). …”

“…This model cannot be generalised (in a physically reasonable way) to cylindrical symmetry with spatially homogeneous interiors and static exteriors. In fact, Mena, Tavakol and Vera [8] have shown that the matching of such space-times leads to very restrictive interior solutions with no evolution along one spacelike direction. As a next step one therefore considers a non-static exterior, but for simplicity retains a spatially homogeneous interior.…”

Conformally flat sources for the Linet–Tian spacetime

“…In order to prove this impossibility they show that a Robertson-Walker spacetime, which is conformally flat, cannot be matched to any cylindrically symmetric static metric across a nonspacelike hypersurface preserving the symmetry. This result was subsequently generalised in [26,27,28]. Another result that might be linked to this trend was obtained by Di Prisco et al [15] and is the following: A cylindrically symmetric shear free collapsing anisotropic fluid can be matched to Einstein-Rosen spacetime as obtained in [15], however, by considering that the exterior spacetime reduces to the static Levi-Civita spacetime, it imposes through its matching conditions, that the cylindrical source must be static 1 .…”

Cylindrically symmetric inhomogeneous dust collapse with a zero expansion component

“…• As far as we know, the metrics (8) and (9) are new exact solutions to the EFEs. These metrics might be of interest to model physical situations, such as jets, where the spacetime dynamics along the z direction dominates with respect to the dynamics along the other directions.…”