Covariant differential identities and conservation laws in metric-torsion theories of gravitation. II. Manifestly generally covariant theories

Abstract

The present paper continues the work of Lompay and Petrov [J. Math. Phys. 54, 062504 (2013)] where manifestly covariant differential identities and conserved quantities in generally covariant metric-torsion theories of gravity of the most general type have been constructed. Here, we study these theories presented more concretely, setting that their Lagrangians \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}$\end{document}L are manifestly generally covariant scalars: algebraic functions of contractions of tensor functions and their covariant derivatives. It is assumed that Lagrangians depend on metric tensor g, curvature tensor R, torsion tensor T and its first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\mathbf {T}$\end{document}∇T and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\mathbf {T}$\end{document}∇∇T covariant derivatives, besides, on an arbitrary set of other tensor (matter) fields \documentclass[12pt]{minimal}\begin{document}$\bm {\varphi }$\end{document}φ and their first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\bm {\varphi }$\end{document}∇φ and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\bm {\varphi }$\end{document}∇∇φ covariant derivatives: \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}= \mathscr {L}(\mathbf {g},\mathbf {R}; \;\mathbf {T},{\bm {\nabla }}\mathbf {T},{\bm {\nabla }}{\bm {\nabla }}\mathbf {T}; \;\bm {\varphi },{\bm {\nabla }}\bm {\varphi },{\bm {\nabla }}{\bm {\nabla }}\bm {\varphi })$\end{document}L=L(g,R;T,∇T,∇∇T;φ,∇φ,∇∇φ). Thus, both the standard minimal coupling with the Riemann-Cartan geometry and non-minimal coupling with the curvature and torsion tensors are considered. The studies and results are as follow: (a) A physical interpretation of the Noether and Klein identities is examined. It was found that they are the basis for constructing equations of balance of energy-momentum tensors of various types (canonical, metrical, and Belinfante symmetrized). The equations of balance are presented. (b) Using the generalized equations of balance, new (generalized) manifestly generally covariant expressions for canonical energy-momentum and spin tensors of the matter fields are constructed. In the cases, when the matter Lagrangian contains both the higher derivatives and non-minimal coupling with curvature and torsion, such generalizations are non-trivial. (c) The Belinfante procedure is generalized for an arbitrary Riemann-Cartan space. (d) A more convenient in applications generalized expression for the canonical superpotential is obtained. (e) A total system of equations for the gravitational fields and matter sources are presented in the form more naturally generalizing the Einstein-Cartan equations with matter. This result, being a one of the more important results itself, is to be also a basis for constructing physically sensible conservation laws and their applications.