Rotation, Embedding and Topology for the Szekeres Geometry

TL;DR

This paper investigates the rotation and embedding properties of Szekeres inhomogeneous cosmological models, demonstrating their consistency and constructing models with closed topologies, supported by explicit examples.

Contribution

It provides a detailed analysis of the embedding and rotation effects in Szekeres models, including models with torus topology, and clarifies their geometric properties.

Findings

Embedding properties are consistent with rotation effects.

Constructed models with closed 'radial' topology.

Explicit examples illustrating rotation and tilt effects.

Abstract

Recent work on the Szekeres inhomogeneous cosmological models uncovered a surprising rotation effect. Hellaby showed that the angular $(\theta, \phi)$ coordinates do not have a constant orientation, while Buckley and Schlegel provided explicit expressions for the rate of rotation from shell to shell, as well as the rate of tilt when the 3-space is embedded in a flat 4-d Euclidean space. We here investigate some properties of this embedding, for the quasi-spherical recollapsing case, and use it to show that the two sets of results are in complete agreement. We also show how to construct Szekeres models that are closed in the 'radial' direction, and hence have a 'natural' embedded torus topology. Several explicit models illustrate the embedding as well as the shell rotation and tilt effects.