This question will be answered by a new paper. The result is that Einstein’s field equation can be derived as combination of curvature vectors negelcting torsion. Doug Lindstrom is currently developing a paper on this subject using totally antisymmetric torsion. He writes:
In this paper, my plan was to link Einstein to ECE at the Riemann tensor level, in so doing obtaining the Einstein field equations (first Bianchi identity).
In the next paper, I would introduce the tetrad and spin connection and have torsion and curvature respecified, with the limitations, determined in this paper, placed on the metric connection. The second Bianchi and Maurer equations, etc. would be added to the mix as I think you are suggesting.
To summarize my reasoning steps in this paper – all of this on a Riemann-Cartan manifold.
Part I. If torsion is assumed to be totally antisymmetric, then there are four and only four scalars, each one associated with a basis element the base manifold.
Part II. The hodge dual of a totally antisymmetric tensor is a vector which satisfies the requirements of Part I.
Part III. Torsion is equal to twice the value of the antisymmetric connection component, whether we look at it as a rank 3 tensor, or as a dual vector.
Part IV Assuming that the metric connection is antisymmetric as given by the commutator’s antisymmetric nature then there is an array ( don’t know if it is a tensor) with possible non-vanishing diagonal entries only. This is the most general symmetric component , given the antisymmetry of the commutator .
Part V. Sums along the diagonals of the symmetric connection generates three one forms, of which two are equal (and equal to the four dimensional gradient of a scalar that is a function of the metric). The third one is related to the other two with the difference being the 4-divergence of the metric.
Part VI A Ricci-like rank two tensor can be made by applying the results of Part V above to the curvature tensor. This entity (not proven to be a tensor) is composed of an antisymmetric part, a symmetric part with vanishing diagonals, and a purely diagonal (symmetric) component. Taking the Hodge Dual of this reduced curvature generates a relatively simple vector based expression – the symmetric part disappears. A scalar curvature which follows is also quite simple. Both of these expressions carry non-linear terms in the vector for the antisymmetric connection (or torsion).
Part VII Reduction to Einstein – in the limit of torsion vanishing, this becomes the Einstein equation including a wave-like disturbance in a function of the metric. (something I gather Einstein’s original relativity did not)
Part VIII Einstein with Torsion – A linearized version of the equation including torsion looks like the Einstein-Cartan-Sciami-Kibble equation. The non-linearity terms in torsion that were neglected were not discussed.
I will spend some time this week considering a totally covariant torsion, and see how that propagates through the paper.
Thanks
Doug