Space tensors in general relativity III: The structural equations

Abstract

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), dT (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis $$(\tilde \nabla ,\tilde \nabla T)$$ (M,Γ). The structural equations of $$(\tilde \nabla ,\tilde \nabla T)$$ and the corresponding spatial Bianchi identities are discussed. The special case $$(\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)$$ is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν4,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.