Bianchi type cosmological models in f(T) tele-parallel gravity

Abstract

Symmetry assumptions on the geometrical framework have provided successful mechanisms to develop physically meaningful solutions to many problems. In tele-parallel gravity, invariance of the frame and spin-connection under a group of motions defines an affine symmetry group. Here, we assume there exists a three-dimensional group of affine symmetries acting simply transitively on a spatial hypersurface and that this group of symmetry actions defines our affine frame symmetry group. We determine the general form of the co-frame and spin connection for each spatially homogeneous Bianchi type. We then construct the corresponding field equations for f(T) tele-parallel gravity. We show that if the symmetry group is of Bianchi type A (I, II, VI0, VII0, VIII or IX) then there exists a co-frame/spin connection pair that is consistent with the antisymmetric part of the field equations of f(T) tele-parallel gravity. For those geometries having a Bianchi type B symmetry group (IV, V, VIh, VIIh), we find that in general these geometries are inconsistent with the antisymmetric part of the f(T) tele-parallel gravity field equations unless the theory reduces to an analog of General Relativity with a cosmological constant.