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 $d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. 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. $e_{\mu \nu \alpha \beta }\to w_{\mu \nu \alpha \beta }(x)=\varphi (x)e_{\mu \nu \alpha \beta }$, so that $\tilde{w}=\frac{1}{4!}{w}_{\mu \nu \rho \sigma }\tilde{d}{x}^{\mu }\wedge \tilde{d}{x}^{\nu }\wedge \tilde{d}{x}^{\rho }\wedge \tilde{d}{x}^{\sigma }$. 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 $\varphi (x)$ 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 $\varphi (x)$ 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.

Appeared in the Proceedings of the Third Minkowski Meeting that was held at Albena (Bulgaria) in September, 2023