General Relativistic 1+3 Orthonormal Frame Approach Revisited

TL;DR

This paper revisits the 1+3 orthonormal frame approach in general relativity, providing explicit equations, coordinate choices, and applications to various cosmological models, including silent dust and magnetic fields.

Contribution

It offers a detailed presentation of the 1+3 orthonormal frame equations with new applications to specific cosmological models and discusses properties like the 3-Cotton--York tensor in these contexts.

Findings

The 3-Cotton--York tensor vanishes for Szekeres dust models.

The tensor is nonzero for a generic silent model with magnetic fields.

Explicit equations for various cosmological models are derived.

Abstract

The equations of the 1+3 orthonormal frame approach are explicitly presented and discussed. Natural choices of local coordinates are mentioned. A dimensionless formulation is subsequently given. It is demonstrated how one can obtain a number of interesting problems by specializing the general equations. In particular, equation systems for ``silent'' dust cosmological models also containing magnetic Maxwell fields, locally rotationally symmetric spacetime geometries and spatially homogeneous cosmological models are presented. We show that while the 3-Cotton--York tensor is zero for Szekeres dust models, it is nonzero for a generic representative within the ``silent'' class.