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. (paper)