Effective equivalence of the Einstein-Cartan and Einstein theories of gravity

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Abstract
I prove that, for any choice of minimally coupled source field Lagrangian for the Einstein-Cartan-Sciama-Kibble theory of gravity, there exists a related minimally coupled source field Lagrangian for the Einstein theory which produces the same field equations for the metric and source field. By using a standard first-order form for source Lagrangians, the converse is also demonstrated. This establishes a one-to-one correspondence between source Lagrangians for the two theories which clearly reveals their similarities and their differences. Because of this "equivalence," one can view either theory, in terms of the other, as minimal coupling for a related Minkowski source Lagrangian or as nonminimal coupling for the same Minkowski source Lagrangian. Consequently the two theories are, in this sense, indistinguishable. Some other implications of this "equivalence" are discussed.

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