Internal translation gauging and local gravitational energy in Weyl gauge theory

Abstract

We investigate the local gravitational source and charges obtained by introducing translations as an internal Stückelberg redundancy in Weyl gauge theory. The internal-$T(4)$-invariant dressed coframe identifies the source of the translational potential with the existing coframe variational derivative, without introducing new dynamics. We derive an excitation/source equation and balance law whose structure is independent of the constitutive parameters of the parity-even quadratic torsion family. The gravitational self-current, defined relative to a specified gravitational/matter split and boundary representative, gives a density invariant under the internal gauge factors once its frame index is contracted with the coframe; this invariance neither establishes ordinary conservation nor identifies the density with the external translation Noether current. A surface charge follows by contraction with a covariantly constant internal direction, and we separately derive the three-sector Noether superpotential of generalized reducibility, with the asymptotic conditions for conservation of its flux. At the teleparallel equivalent of general relativity (TEGR) the excitation reduces to the teleparallel superpotential and the covariant coframe--connection pair gives the Schwarzschild ADM energy; flat cosmology and transverse-traceless waves give further checks. Away from TEGR, an exact solution of New General Relativity gives a charge whose fractional difference from the Kepler mass is half the deviation of the post-Newtonian spatial-curvature parameter from unity, at the adopted normalisation; independent Hamiltonian charge matching remains open. Finally, the integrable Hamiltonian charge is invariant under covariant Lagrangian boundary terms at fixed reference normalisation and presymplectic representative.