Spatially Homogeneous Teleparallel Gravity: Bianchi I

Using a recently developed algorithm that chooses preferred coordinates and a preferred co-frame, we will determine the completely general Bianchi type I teleparallel geometry. In using this algorithm, any remaining gauge freedom is allocated to the choice of spin connection. We then solve the symmetry constraints placed on the spin connection to derive a general class of Bianchi type I teleparallel geometries. We find that this general class of Bianchi type I teleparallel geometries breaks naturally into two sub-classes. We then illustrate some simple power-law solutions in $F(T)$ teleparallel gravity for each case to illustrate the differences.