Torsion and Chern-Simons gravity in 4D space-times from a Geometrodynamical four-form

In Hermann Minkowski's pioneering formulation of special relativity, the space-time geometry in any inertial frame is described by the line-element ds²= ημ ν dx^μ dx^ν. It is interesting to note that not only the Minkowski metric ημ ν is invariant under proper Lorentz transformations, the totally antisymmetric Levi-Civita tensor eμ ν α β too is. In general relativity (GR), ημ ν of the flat space-time gets generalized to a dynamical, space-time dependent metric tensor gμ ν that characterizes a curved space-time geometry. In the present study, it is put forward that the flat space-time Levi-Civita tensor gets elevated to a dynamical four-form field w̃ in curved space-time manifolds, i.e. eμ ν α β → wμ ν α β (x) = φ (x) \ eμ ν α β so that w̃ = 1 4! \ wμ ν ρ σ \ d̃ x^μ d̃ x^ν d̃ x^ρ d̃ x^σ. It is shown that this geometrodynamical four-form field, extends GR by leading naturally to a torsion in the theory as well as to a Chern-Simons gravity. In order to subject the hypothesized four-form field w̃ to observational evidence, we first argue that the associated scalar-density φ (x) corresponds to an axion-like pseudo-scalar field in the Minkowski space-time, and that it can also masquerade as dark matter. Thereafter, we provide a simple semi-classical analysis in which a self-gravitating Bose-Einstein condensate of such ultra-light pseudo-scalars leads to the formation of a supermassive black hole.