The covariant formulation of f(T) gravity

We show that the well-known problem of frame dependence and violation of local Lorentz invariance in the usual formulation of f(T) gravity is a consequence of neglecting the role of spin connection. We re-formulate f(T) gravity starting from, instead of the 'pure tetrad' teleparallel gravity, the covariant teleparallel gravity, using both the tetrad and the spin connection as dynamical variables, resulting in a. fully covariant, consistent, and frame-independent. version of f(T) gravity, which does not suffer from the notorious problems of the usual, pure tetrad, f(T) theory. We present the method to extract solutions for the most physically important cases, such as the Minkowski, the Friedmann-Robertson-Walker (FRW) and the spherically symmetric ones. We show that in. covariant f(T) gravity we are allowed to use an arbitrary tetrad in an arbitrary coordinate system along with the corresponding spin connection, resulting always in the same physically relevant field equations.