The field-theoretical formalism for the teleparallel equivalent of general relativity

Abstract

The teleparallel equivalent of general relativity (TEGR) is formulated in a field-theoretical form, in which tetrad and matter perturbations propagate on an arbitrary background solution of TEGR. The perturbations may be finite rather than infinitesimal and are treated as dynamical fields described by a Lagrangian. Gauge transformations are defined, and the gauge invariance of the dynamical Lagrangian and field equations is established. Applying the Noether procedure, we construct conserved currents, superpotentials, and charges. An identically conserved current and the associated superpotential are also derived from the Noether identity for the auxiliary linear gravitational Lagrangian density. The field-theoretical conserved quantities turn out to be insensitive to total-divergence terms in the original Lagrangian, including the term containing the inertial spin connection. Their gauge behavior at the asymptotic boundary is analyzed, and falloff and parity conditions ensuring well-defined charges are formulated. As an application, we consider linear gravitational perturbations on a Ricci-flat background and derive the gravitational-wave equations and tetrad perturbations in the transverse-traceless gauge. The antisymmetric part of the tetrad perturbation is explicitly shown to be pure local-Lorentz gauge and therefore does not represent additional gravitational-wave polarizations, leaving only the two standard tensor modes. Using the resulting charge formulae, we calculate the Schwarzschild and Kerr masses and the Kerr angular momentum. The required asymptotic conditions are verified for both black hole solutions in Cartesian coordinates of a flat background. The physically acceptable charge values demonstrate the consistency of the formalism and indicate its potential for extension to generalized teleparallel theories.