Frame rotation in the Szekeres spacetimes

Abstract

The Szekeres metric is an inhomogeneous cosmological model without any symmetries. The standard Riemann-type coordinates can be transformed into spherical-type coordinates, but the metric is no longer diagonal, and the constant ‘radius’ 2-spheres, 2-hyperboloids or 2-planes are known to be ‘non-concentric’. Since the transformation into spherical-type coordinates is ‘radius’ dependent, we question whether these coordinates have the same orientation on each 2-surface. To answer this question, we set up an orthonormal tetrad (ONT), and investigate its variation. We find that a relative rotation of the tetrad is generic, and it can increase systematically under conditions that are not very restrictive. We search for paths along which the tetrad is constant, and find they only exist under very restrictive conditions. In the process, we create a systematic method for defining an ONT with chosen properties from a given metric.