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

Abstract

The space-time geometry in any inertial frame is described by the line-element $ds^2= \eta_{\mu \nu} dx^\mu dx^\nu$. Now, not only the Minkowski metric $\eta_{\mu \nu} $ is invariant under proper Lorentz transformations, the totally antisymmetric Levi-Civita tensor $e_{\mu \nu \alpha \beta} $ too is. In general relativity (GR), $\eta_{\mu \nu} $ of the flat space-time gets generalized to a dynamical, space-time dependent metric tensor $ g_{\mu \nu} $ 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 $\tilde {w} $ in curved space-time manifolds, i.e. $e_{\mu \nu \alpha \beta} \rightarrow w_{\mu \nu \alpha \beta} (x) = \phi (x) \ e_{\mu \nu \alpha \beta} $, so that $\tilde {w} = {1\over {4!}} \ w_{\mu \nu \rho \sigma} \ \tilde{d} x^\mu \wedge \tilde{d} x^\nu \wedge \tilde{d} x^\rho \wedge \tilde{d} x^\sigma$. 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. It is demonstrated that the scalar-density $\phi (x)$ associated with $\tilde {w} $ may be used to construct a generalized exterior derivative that converts a p-form density to a (p+1)-form density of identical weight. It is argued that the scalar-density $\phi (x)$ associated with $\tilde {w}$ 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. A brief analysis of propagation of weak gravitational waves in the presence of $\tilde{w} $ is also considered in this article.