First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory (2211.13659v3)

Abstract: Driven by the need for numerical solutions to the Einstein field equations, we derive a first-order reduction of the second-order $ f(T) $-teleparallel gravity equations in the pure-tetrad formulation (no spin connection). We then restrict our attention to the teleparallel equivalent of general relativity (TEGR) and propose a 3+1 decomposition of these equations suitable for computational implementation. We demonstrate that in vacuum (matter-free spacetime) the obtained system of first-order equations is equivalent to the tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen, and therefore also admits a symmetric hyperbolic formulation. However, the question of hyperbolicity of the 3+1 TEGR equations for arbitrary spacetimes remains unaddressed so far. Furthermore, the structure of the 3+1 equations resembles a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity.