Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)

TL;DR

This paper introduces a novel field-theoretical method for defining conserved quantities directly from equations of motion, applicable to gravitation theories like General Relativity and Einstein-Cartan, providing new insights and recovering known results.

Contribution

It develops an alternative approach based on the equations of motion and Lie derivatives, offering a systematic way to derive conserved quantities in gauge natural theories.

Findings

Recovered known energy variation formulas in General Relativity

Applied the formalism to Einstein-Cartan theory in tetrad formalism

Gained new insights on the Kosmann lift in gauge natural theories

Abstract

We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles). The contraction of the Euler-Lagrange equations with Lie derivatives of the dynamical fields allows one to derive a variational Lagrangian for any given set of Lagrangian equations. A two steps algorithmical procedure can be thence applied to the variational Lagrangian in order to produce a general expression for the variation of all quantities which are (covariantly) conserved along the given dynamics. As a concrete example we test this new formalism on Einstein's equations: well known and widely accepted formulae for the variation of the Hamiltonian and the variation of Energy for General Relativity are recovered. We also consider the Einstein-Cartan (Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new insight on the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.