Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. I. General Consideration

TL;DR

This paper develops covariant methods to derive identities and conserved quantities in metric-torsion gravity theories, providing explicit formulas without auxiliary structures, and analyzing superpotential ambiguities.

Contribution

It introduces generalized covariant identities and conserved currents in metric-torsion gravity, including explicit superpotential constructions and analysis of their ambiguities.

Findings

Derived complete covariant Klein-Noether identities.

Constructed three types of conserved currents: canonical, Belinfante, Hilbert-Bergmann.

Proved the generalized boundary Klein theorem.

Abstract

Arbitrary diffeomorphically invariant metric-torsion theories of gravity are considered. It is assumed that Lagrangians of such theories contain derivatives of field variables (tensor densities of arbitrary ranks and weights) up to a second order only. The generalized Klein-Noether methods for constructing manifestly covariant identities and conserved quantities are developed. Manifestly covariant expressions are constructed without including auxiliary structures like a background metric. In the Riemann-Cartan space, the following \emph{manifestly generally covariant results} are presented: (a) The complete generalized system of differential identities (the Klein-Noether identities) is obtained. (b) The generalized currents of three types depending on an arbitrary vector field displacements are constructed: they are the canonical Noether current, symmetrized Belinfante current and identically conserved Hilbert-Bergmann current. In particular, it is stated that the symmetrized Belinfante current does not depend on divergences in the Lagrangian. (c) The generalized boundary Klein theorem (third Noether theorem) is proved. (d) The construction of the generalized superpotential is presented in details, and questions related to its ambiguities are analyzed.