Spin & Torsion Tensor on Gauge Gravity: A Re-examination of the Einstein–Cartan Spatio-Temporal Theory

This work aims to autonomously revisit some puncta salientia of the Einstein–Cartan (EC) theory, focusing wholly on the mathematical aspect, or, better still, emphasizing the differential geometry underlying the theory under examination, without the burden of sensible experiences (experiments) of Galilean heritage.

It is shown that it is possible to describe, or rather, derive an Einsteinian-like gravitational field starting from a Cartan h-subalgebra, and thus produce a couple of formulæ for a torsioning in a (1 + 3)-dimensional manifold. Some Cartan k-forms and J-bundles, along with other Clifford bundles, and a Clifford k-form field, will help to circumscribe a 4D torsional spin-space. Follows an overview of quantum Yang–Mills gravity according to a geometro-topological schema. This opens up the exciting issue, not addressed here, of the emergence of space-time, indicating a manifolded-structure including its spin plus torsional foundations.