Spin & Torsion Tensors on Gauge Gravity: a Re-examination of the Einstein–Cartan Spatio-Temporal Theory
Résumé
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 $\mathfrak{h}$-subalgebra, and thus produce a couple of formulæ for a torsioning in a $(1 + 3)$-dimensional manifold. Some Cartan $k$-forms and $\mathcal{J}$-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.
Mots clés
spin-torsion tensor
Yang–Mills–Euler–Lagrange equation(s)
Yang–Mills–Higgs equations
Yang–Mills-like geometry
Yang–Mills–Higgs bundle
black (and white?) hole(s)
Cartan $\mathfrak{h}$-subalgebra
Cartanian affine connections
Clifford bundles
Einstein–Cartan space-time
gauge gravity
$\mathcal{J}$-bundles
$k$-forms
Lagrangian density
Lorentz space(-time)
Minkowski space(-time)
Poincaré group
Riemann–Cartan geometry
singularity
space-time manifold in a 4-dimensional spin-torsion balancing
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