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Teleparallel gravity represents a viable alternative to general relativity where gravitation comes from torsion rather that curvature. The theory is based on a new modified connection, and the curvature for this theory is zero and in the geodesic equations there is a term which represents a force "field". However, the question would be how the basic tensor algebra is developed, such as raising/lowering indices in 1-3 rank tensors on tangent and general frame and how the Einstein field equations are derived from the corresponding action.

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    $\begingroup$ "Recently, it has been shown" is that a widely accepted mainstream result, or is it more in the area of self-published whacky stuff? (I have no idea, since not familiar with the field.) Are there say 10 other papers confirming the idea? $\endgroup$
    – Fattie
    May 12, 2016 at 14:03
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    $\begingroup$ @ Joe Blow The paper linked is a review of teleparallel theory and contains many references. I would like to study it from dark energy point of view such as non-minimal coupling, perturbations, etc. $\endgroup$
    – Nikey Mike
    May 12, 2016 at 14:21

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The basic techniques related to teleparallel gravity can be found in Appendix C from "Teleparallel Gravity - An Introduction" by Aldrovandi Ruben, and Pereira Jose G. The variational calculus in detail is presented in the recent paper "Generalized Teleparallel Theory" by Ednaldo L. B. Junior, and Manuel E. Rodrigues. These references offer a start to the unexperienced reader in teleparallel theory, and the basic relations are derived. From there one can find its own way into these theories. The basic requirements would be the mastering of the derivation of the Einstein equations from the Einstein-Hilbert action, and the derivation of the Friedmann equations in different gravity theories, applying the variational calculus. The latter one can be seen in this recent article "Brans-Dicke Galileon and the Variational Principle", by Israel Quiros, Ricardo García-Salcedo, Tame Gonzalez, F. Antonio Horta-Rangel, Joel Saavedra, published also on arxiv.

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