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

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