Flat spin connections in the Teleparallel equivalent of General Relativity

This work generalizes the treatment of flat spin connections in the teleparallel equivalent of general relativity. It is shown that a general flat spin connection form a subspace in the affine space of spin connections which is dynamically decoupled from the tetrad and the matter fields. A translation in the affine subspace introduces a torsion term without changing the tetrad. Instead, the change in the torsion is related to the introduction of a global acceleration field term that introduces Lorentz inertial effects in the reference frame. The dynamics of the gravitationally coupled matter fields remains however equivalent regardless of the flat spin connection chosen. The implications of the break of this invariance by a general $f(T)$ and $f(R)$ is discussed.