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 ($I$, $II$, $V{I}_0$, $VI{I}_0$, $VIII$ 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 ($IV$, $V$, $V{I}_h$, $VI{I}_h$), 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.

28 pages

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Cited by in corpus (3)

• Spherically symmetric teleparallel geometries
• Teleparallel Geometry with Spherical Symmetry: The diagonal and proper frames
• On black holes in teleparallel torsion theories of gravity