Energy-momentum pseudotensor and superpotential for generally covariant theories of gravity of general form

TL;DR

This paper develops a general framework for energy-momentum pseudotensors and superpotentials in covariant gravity theories, providing explicit formulas and analyzing conservation conditions across various gravity models.

Contribution

It introduces a universal method to derive energy-momentum pseudotensors and superpotentials for any generally covariant theory, regardless of derivative order or variable transformations.

Findings

Derived explicit superpotential expressions for broad class of theories

Established conditions for conservation laws independent of equations of motion

Applied framework to General Relativity, mimetic gravity, and embedding gravity

Abstract

The current paper is devoted to the investigation of the general form of the energy-momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpotential is also given. We discuss the structure of the obtained expressions and the conditions for the derived pEMT conservation laws to be satisfied independently (fully or partially) of the equations of motion. Deformations of the superpotential form for theories with a change of the independent variables in the action are also considered. We apply these results to some interesting particular cases: General Relativity and its modifications, particularly mimetic gravity and Regge-Teitelboim embedding gravity.