Torsion Field Equation and Spinor Source

We propose a geometrical theory of gravitation based on a non-orthogonal tetrad field which plays the role of a torsion potential. Starting from the gravitational Lagrangian defined by scalar curvature, we obtain two field equations, the Einstein equation and the torsion field equation. The latter is a second-order differential equation for the torsion potential, and therefore the torsion has acquired a dynamical property. When we consider the torsion coupling to spinor fields, the torsion field equation has a difficulty. We show that this difficulty can be overcome by adding appropriate quadratic terms of the torsion tensor to the Lagrangian. Using the generalized Lagrangian, we study the torsion field equation and its solution in the weak field approximation.