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This work aims to analyze the action of spinning particles in general relativity.

In general relativity the affine connection is required to be symmetric, so torsion is zero, we want to verify if a spinor field can be considered a torsion source. The broader reference context is Einstein Cartan’s gravitational theory, which is a generalization of general relativity; in this theory in addition to curvature there is torsion associated with intrinsic angular momentum density. The affine connection is not restricted to be symmetric as required in general relativity, torsion is included due to the antisymmetric part of the affine connection. In Einstein Cartan’s theory torsion is connected to the spin tensor as ex-pressed by the Cartan equations. These equations are obtained through the variation of the total action with respect to the torsion; the total action is intended as the sum of the action for the gravitational field and the action for the fermionic field,  according to the minimal action principle 

We consider these important hints about torsion and spin tensor to revisit general relativity with spinor fields, we focus on the requirement of symmetric affine connection and develop the calculation of the spin coefficients.

In order to include fermions in general relativity we introduce a local reference frame and define a tetrad of basis vectors. We refer to the Hamiltonian formulation and calculate the canonical momenta associated with the temporal variation of the tetrads, we find a fermionic rotational term. This term is connected to torsion as suggested by Cartan’s equations. Starting from a torsion-less theory we get a rotational current that would generate a torsion contribution.

In the conclusions we turn back to the Lagrangian and consider the interaction terms in comparison with linearized theory. We argue that gravitomagnetism is well described in the linearized theory while the term of spin connection giving rise to fermionic current is canceled out, so we mean these terms are describing different interactions, it will be interesting to investigate in further analysis. 

spinor, tetrad gravity, fermionic rotational current, torsion, Cartan.