Projections and covariant divergency of energy-momentun tensors

Abstract

The invariant projections of the energy-momentum tensors of Lagrangian densities for tensor fields over differentiable manifolds with contravariant and covariant affine connections and metrics [$(\bar{L}_n,g)$-spaces] are found by the use of an non-null (non-isotropic) contravariant vector field and its corresponding projective metrics. The notions of rest mass density, momentum density, energy current density and stress tensor are introduced as generalizations of these notions from the relativistic continuum media mechanics. The energy-momentum tensors are represented by means of the introduced notions and the corresponding identities are found. The notion of covariant differential operator along a contravariant tensor field is introduced. On its basis, as a special case, the notion of contravariant metric differential operator is proposed. The properties of the operators are considered. By the use of these operators the notion of covariant divergency of a mixed tensor field is determined. The covariant divergency of tensor fields of second rank of the types 1 and 2 is found. Invariant representations of the covariant divergency of the energy-momentum tensor are obtained by means of the projective metrics of a contravariant non-isotropic (non-null) vector field and the corresponding rest mass density, momentum density, and energy flux density. An invariant representation of the first Noether identity is found as well as relations between the covariant divergencies of the different energy-momentum tensors and their structures determining covariant local conserved quantities.