Summary
This theoretical framework sets the stage for a dynamical exploration of rotating hyperspherical universes within Einstein--Cartan gravity. By combining spin-torsion coupling, anisotropic Bianchi IX metrics, and observationally motivated initial conditions, we provide a robust alternative to CDM for addressing key cosmological tensions.
3. Field Equations with Rotation and Torsion
3.1 Derivation of Modified Einstein--Cartan Field Equations
Einstein--Cartan (EC) theory modifies the Einstein field equations by accounting for torsion, sourced by the spin density of matter. In a Riemann--Cartan spacetime (M,g,~)(M, g, \tilde{\Gamma}), the affine connection is no longer symmetric:
~~.\tilde{\Gamma}^\lambda_{\;\mu\nu} \neq \tilde{\Gamma}^\lambda_{\;\nu\mu}.
The antisymmetric part gives the torsion tensor:
T=~[].T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;[\mu\nu]}.
The field equations in EC theory are obtained by varying the total action with respect to the metric and connection:
S=12R(~)gd4x+Lmatter(g,,)gd4x.S = \frac{1}{2\kappa} \int R(\tilde{\Gamma}) \sqrt{-g}\, d^4x + \int \mathcal{L}_{\text{matter}}(g_{\mu\nu}, \psi, \nabla \psi) \sqrt{-g}\, d^4x.
Variation with respect to gg_{\mu\nu} yields:
