MOND as the weak field limit of an extended metric theory of gravity with torsion

In this article we construct a relativistic extended metric theory of gravity, for which its weak field limit reduces to the non-relativistic MOdified Newtonian Dynamics regime of gravity. The theory is fully covariant and the way to achieve this is to introduce torsion in the description of gravity as well as with the addition of a particular function of the matter lagrangian into the gravitational action.