First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory

TL;DR

This work derives a first-order $3+1$ reduction of the teleparallel equivalent of general relativity (TEGR) in its pure-tetrad (Weitzenböck) formulation and analyzes its hyperbolicity in vacuum. By decomposing torsion into electric and magnetic components relative to a chosen observer, introducing a Legendre-transformed potential and new dynamical variables, the authors obtain a symmetric-hyperbolic-like structure akin to the ERWBB tetrad formulation of GR under suitable gauge conditions. The resulting evolution system for the tetrads, torsion, and associated energy-momentum quantities exhibits a clear analogy with relativistic electrodynamics and fits into a framework suitable for structure-preserving numerical discretization and potential comparison with dGREM. While hyperbolicity is established in the vacuum case, the paper identifies remaining questions for matter coupling and full equivalence to GR in a fully constrained, numerically stable setting. The methodology provides a path toward a computational TEGR-based approach to numerical relativity and potential extensions to $f(\mathcal{T})$ theories and covariant teleparallel geometries.

Abstract

This paper presents a derivation of a first-order reduction and 3+1 decomposition of the teleparallel equivalent of general relativity (TEGR) in the pure-tetrad formulation (no spin connection). Our analysis demonstrates that in vacuum spacetimes, our 3+1 TEGR equations has the principal part of the differential operator equivalent to the one of tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen, and therefore the presented 3+1 decomposition of TEGR also admits a symmetric hyperbolic formulation, a desirable property for ensuring well-posedness of the initial value problem. Furthermore, the structure of the 3+1 equations possess a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity.