G+g=8GT(eff),G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T^{\text{(eff)}}_{\mu\nu},
where T(eff)T^{\text{(eff)}}_{\mu\nu} includes torsion corrections.
Variation with respect to ~\tilde{\Gamma}^\lambda_{\;\mu\nu} leads to an algebraic relation:
T=8G(S+SS),T^\lambda_{\;\mu\nu} = 8\pi G \left( S^\lambda_{\;\mu\nu} + \delta^\lambda_\mu S^\sigma_{\;\nu\sigma} - \delta^\lambda_\nu S^\sigma_{\;\mu\sigma} \right),
where SS^\lambda_{\;\mu\nu} is the spin angular momentum density tensor. Unlike curvature, torsion in EC does not propagate; it is localized and vanishes in the absence of spin.
3.2 Inclusion of Expansion, Shear, and Vorticity Tensors
In a cosmological setting with a general anisotropic and rotating metric (e.g. Bianchi IX), the kinematics of a fluid congruence uu^\mu is described by decomposing its covariant derivative:
u=13h++au,\nabla_\nu u_\mu = \frac{1}{3} \theta h_{\mu\nu} + \sigma_{\mu\nu} + \omega_{\mu\nu} - a_\mu u_\nu,
where:
=u\theta = \nabla_\mu u^\mu is the expansion scalar,
\sigma_{\mu\nu} is the shear tensor (symmetric, trace-free),
