General relativistic orthonormal frame approach

The dynamical equations of an extended 1 + 3 orthonormal frame approach to the relativistic description of spacetime geometries are explicitly presented and discussed in detail. In particular, the Bianchi identities for the Weyl curvature tensor occur in a fully expanded form, as they are given a central role in the extended formalism. It is shown how one can naturally introduce local coordinates, both in the 1 + 3 threading and the ADM 3 + 1 slicing context. By specializing the general 1 + 3 dynamical equations it is demonstrated how a number of problems of interest can be obtained. In particular, the simplest choices of spatial frames for spatially homogeneous cosmological models, locally rotationally symmetric spacetime geometries, cosmological models with an Abelian isometry group and `silent' dust cosmological models are discussed.