  In this paper we present a theory of the gravitational field where this field
(a kind of square root of g) is represented by a (1,1)-extensor field h
describing a plastic distortion of the Lorentz vacuum (a real substance that
lives in a Minkowski spacetime) due to the presence of matter. The field h
distorts the Minkowski metric extensor in an appropriate way (see below)
generating what may be interpreted as an effective Lorentzian metric extensor g
and also it permits the introduction of different kinds of parallelism rules on
the world manifold, which may be interpreted as distortions of the parallelism
structure of Minkowski spacetime and which may have non null curvature and/or
torsion and/or nonmetricity tensors. We thus have different possible effective
geometries which may be associated to the gravitational field and thus its
description by a Lorentzian geometry is only a possibility, not an imposition
from Nature. Moreover, we developed with enough details the theory of multiform
functions and multiform functionals that permitted us to successfully write a
Lagrangian for h and to obtain its equations of motion, that results equivalent
to Einstein field equations of General Relativity (for all those solutions
where the manifold M is diffeomorphic to R^4. However, in our theory,
differently from the case of General Relativity, trustful energy-momentum and
angular momentum conservation laws exist. We express also the results of our
theory in terms of the gravitational potential 1-form fields (living in
Minkowski spacetime) in order to have results which may be easily expressed
with the theory of differential forms. The Hamiltonian formalism for our theory
(formulated in terms of the potentials) is also discussed. The paper contains
also several important Appendices that complete the material in the main text.
