Space tensors in general relativity I: Spatial tensor algebra and analysis

Abstract

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation$\tilde \nabla$ and a time-derivative$\tilde \nabla _T$, both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair$\left( {\tilde \nabla ,\tilde \nabla _T } \right)$ are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail.