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The Einstein-Hilbert action density 0-form ~ *(R^I^J/e^K/e^L)
R is the curvature 2-form = D(spin connection 1-form)
D = d + (spin connection)/
d^2 = 0
e is the tetrad 1-form
* is the Hodge star dual operation
The Yang-Mills torsion field action density 0-form ~ *(T^I/T^J)
T^I = De^I = torsion 2-form.
In the teleparallel theory
*(R^I^J/e^K/e^L) + *(T^I/T^J) = 0
The torsion field is Yang-Mills spin 1.
spin 1/spin 1 ~ spin 2 + spin 1 + spin 0

Gravitational Lorentz force and the description of the gravitational interaction
V. C. de Andrade and J. G. Pereira
Instituto de F?sica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 S Sao Paulo, Brazil
Received 21 March 1997
"In the context of a gauge theory for the translation group, we have obtained, for a spinless particle, a gravitational analogue of the Lorentz force. Then, we have shown that this force equation can be rewritten in terms of magnitudes related to either the teleparallel or the Riemannian structures induced in spacetime by the presence of the gravitational field. In the first case, it gives a force equation, with torsion playing the role of force. In the second, it gives the usual geodesic equation of general relativity. The main conclusion is that scalar matter is able to feel any one of the above spacetime geometries, the teleparallel and the metric ones. Furthermore, both descriptions are found to be completely equivalent in the sense that they give the same physical trajectory for a spinless particle in a gravitational field."

Feb 2009arXiv:0902.0560v1 [gr-qc] A formal framework for a nonlocal generalization of Einstein’s theory of gravitation
Friedrich W. Hehl∗
Institute for Theoretical Physics, University of Cologne, 50923 K¨oln, Germany and
Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA
Bahram Mashhoon†
Department of Physics and Astronomy, University o
"it is known that a gauge theory of the translation group, for spinless matter, yields a teleparallelism theory of gravity that, for a suitably chosen Lagrangian, is equivalent to Einstein’s theory"