Canonical and gravitational stress-energy tensors

TL;DR

This paper examines the conditions under which the canonical and gravitational stress-energy tensors are equivalent in various gravity theories, including general relativity, Poincare gauge theory, and for matter fields like spinors and the Dirac-Maxwell system.

Contribution

It provides a comprehensive analysis of the equivalence between canonical and gravitational stress-energy tensors across different gravity frameworks and matter fields, including spinors and gauge theories.

Findings

Full equivalence for matter fields not coupling to metric derivatives

Inclusion of spinor fields via tetrad formulation

Detailed analysis of Dirac-Maxwell system energy separation

Abstract

It is dealt with the question, under which circumstances the canonical Noether stress-energy tensor is equivalent to the gravitational (Hilbert) tensor for general matter fields under the influence of gravity. In the framework of general relativity, the full equivalence is established for matter fields that do not couple to the metric derivatives. Spinor fields are included into our analysis by reformulating general relativity in terms of tetrad fields, and the case of Poincare gauge theory, with an additional, independent Lorentz connection, is also investigated. Special attention is given to the flat limit, focusing on the expressions for the matter field energy (Hamiltonian). The Dirac-Maxwell system is investigated in detail, with special care given to the separation of free (kinetic) and interaction (or potential) energy. Moreover, the stress-energy tensor of the gravitational field itself is briefly discussed.