Differential form description of the Noether-Lagrange machinery, vielbein/gauge-field analogies and energy-momentum complexes

TL;DR

This paper develops a coordinate-independent variational framework for generally covariant field theories using exterior calculus, clarifying the relation between diffeomorphism symmetries, gauge fields, and energy-momentum constructs.

Contribution

It introduces a novel exterior calculus approach to derive Noether's theorem in covariant theories and relates diffeomorphism currents to energy-momentum complexes in vielbein formalism.

Findings

Diffeomorphism Noether currents relate to the matter energy-momentum tensor.

Decomposition in the vielbein basis is essential for certain relations.

Connections established between Noether currents, Hamiltonian boundary terms, and energy-momentum complexes.

Abstract

We derive the variational principle and Noether's theorem in generally covariant field theory in an explicitly coordinate-independent way by means of the exterior calculus over the space-time manifold. We then focus on the symmetry of active diffeomorphisms, that is, the pushforwards along the integral lines of any vector field, and its analogies with internal gauge symmetries. For instance, it is well known that a class of Noether currents associated to a gauge symmetry can be obtained by taking the partial derivative of the Lagrangian with respect to the corresponding gauge field. Here we show that this relation also holds for the Noether currents associated to diffeomorphisms and the vielbein, but only if one decomposes all forms in the vielbein basis. We also relate the diffeomorphism Noether currents to the matter energy-momentum tensor of General Relativity, to Hamiltonian boundary terms and to two known energy-momentum complexes of the vielbein.