The covariant formulation of f(T) gravity

TL;DR

This paper reformulates $f(T)$ gravity in a covariant manner by incorporating the spin connection, resolving frame dependence and Lorentz invariance issues present in the traditional pure-tetrad approach.

Contribution

It introduces a fully covariant $f(T)$ gravity framework using both tetrad and spin connection as dynamical variables, ensuring frame independence and consistency.

Findings

Resolved frame dependence and Lorentz invariance issues in $f(T)$ gravity.

Provided methods to find solutions in Minkowski, FRW, and spherical cases.

Demonstrated arbitrary tetrad and coordinate system use with consistent field equations.

Abstract

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, instead of the "pure-tetrad" teleparallel gravity, from the covariant teleparallel gravity, using both the tetrad and the spin connection as dynamical variables, resulting in the 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 FRW and the spherically-symmetric ones. We show that in the 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 to the same physically relevant field equations.