Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, Eab and Hab, represent the locally free curvature – i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of ∇cEab and ∇cHab, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace–free spatial derivatives of Eab and Hab together with a spatial antisymmetric part which generalises the vector curl. 1 The problem In a cosmological context, where there is a preferred four–velocity field u, the electric and magnetic parts of the Weyl tensor, Eab and Hab, represent covariantly the locally free gravitational field, in the sense that they are not pointwise determined by the matter fields (see e.g. [1] – [4]). They do not arise explicitly in the Einstein equations. Although they are not really independent of the matter fields, being constrained by integrability conditions in the form of the Bianchi identities, it is nevertheless useful to think of them as being locally free. The Bianchi identities involve the time derivatives and the spatial divergences and curls of Eab and Hab (see the Appendix). The locally free parts of the curvature are not pointwise determined by the matter fields via Einstein’s equations. Similarly, the locally free parts of the derivative of the curvature are those parts not pointwise determined via the Bianchi identities by the matter fields and their derivatives and by Eab, Hab. The main aim of this paper is to determine these locally free parts. We find via a complete covariant decomposition that the parts of ∇cEab and ∇cHab which are missing from the Bianchi identities are exactly the projected totally symmetric and trace–free parts ∇ and ∇. 5 These terms represent the part of the derivative of the curvature which is uncoupled from the Bianchi identities and does not locally interact with the matter fields. The divergences are determined pointwise by the matter terms and the locally free field. The curls (together with the matter terms and the locally free field) determine the time derivatives of Eab and Hab pointwise. Therefore the symmetric trace–free parts and the curls provide a covariant characterisation of the locally free part of the space–time gradient of the gravitational field. This is consistent with the fact that the curls are [email protected] [email protected] [email protected] The terms divergence and curl as applied to rank–2 tensors are defined by equation (1). Here and in the sequel, angle brackets < · · · > on indices represent their totally symmetric trace-free and spatially projected part. Round and square brackets denote, respectively, symmetrisation and anti–symmetrisation.