Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism symmetry

TL;DR

This paper develops a general, covariant framework for defining invariant conserved currents in gravity theories with local Lorentz and diffeomorphism invariance, applicable in any dimension and for various matter couplings.

Contribution

It introduces a novel approach to construct invariant conserved currents using an ambiguity in the covariant Lie derivative, applicable to diverse gravity theories and matter couplings.

Findings

Formulated covariant conserved currents for gravity with local symmetries.

Applied regularization to obtain finite conserved quantities for nonflat solutions.

Provided explicit examples demonstrating the formalism's effectiveness.

Abstract

We discuss conservation laws for gravity theories invariant under general coordinate and local Lorentz transformations. We demonstrate the possibility to formulate these conservation laws in many covariant and noncovariant(ly looking) ways. An interesting mathematical fact underlies such a diversity: there is a certain ambiguity in a definition of the (Lorentz-) covariant generalization of the usual Lie derivative. Using this freedom, we develop a general approach to construction of invariant conserved currents generated by an arbitrary vector field on the spacetime. This is done in any dimension, for any Lagrangian of the gravitational field and of a (minimally or nonminimally) coupled matter field. A development of the "regularization via relocalization" scheme is used to obtain finite conserved quantities for asymptotically nonflat solutions. We illustrate how our formalism works by some explicit examples.