Invariant conserved currents for gravity

TL;DR

This paper develops a general method using Lagrange-Noether theory to define invariant conserved currents in gravity, applicable to various models and symmetries, enabling computation of physical quantities like energy and angular momentum.

Contribution

It introduces a unified approach to derive invariant conserved currents in gravity theories with coordinate and Lorentz symmetries, applicable to diverse models and interactions.

Findings

Constructed conserved currents from the Lagrangian and field momenta.

Proved invariance of currents, superpotentials, and charges under symmetries.

Applied method to compute energy and angular momentum in specific gravitational solutions.

Abstract

We develop a general approach, based on the Lagrange-Noether machinery, to the definition of invariant conserved currents for gravity theories with general coordinate and local Lorentz symmetries. In this framework, every vector field \xi on spacetime generates, in any dimension n, for any Lagrangian of gravitational plus matter fields and for any (minimal or nonminimal) type of interaction, a current J[\xi] with the following properties: (1) the current (n-1)-form J[\xi] is constructed from the Lagrangian and the generalized field momenta, (2) it is conserved, d J[\xi] = 0, when the field equations are satisfied, (3) J[\xi]= d\Pi[\xi] "on shell", (4) the current J[\xi], the superpotential \Pi[\xi], and the charge Q[\xi] = \int J[\xi] are invariant under diffeomorphisms and the local Lorentz group. We present a compact derivation of the Noether currents associated with diffeomorphisms and apply the general method to compute the total energy and angular momentum of exact solutions in several physically interesting gravitational models.