Rotation, Embedding and Topology for the Szekeres Geometry

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.