Thus, while curvature couples to energy--momentum, torsion couples to intrinsic spin. Importantly, the torsion does not propagate in vacuum---it is algebraically determined by the spin content of matter and vanishes in the absence of spin.
2.2 Torsion Tensor and Spin Density Formalism
The torsion tensor TT^\lambda_{\;\mu\nu} is defined as the antisymmetric part of the affine connection:
T=~~.T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;\mu\nu} - \tilde{\Gamma}^\lambda_{\;\nu\mu}.
In a cosmological context, particularly for a universe filled with spinning matter (such as fermionic fields), the spin density SS^{\mu\nu\lambda} can be introduced as:
S=us,S^{\mu\nu\lambda} = \epsilon^{\mu\nu\lambda\sigma} u_\sigma s,
where uu^\mu is the four-velocity of the cosmic fluid and ss is the scalar spin density.
Substituting this into the Einstein--Cartan field equations results in effective corrections to the stress-energy tensor:
Teff=T+U(s),T^{\text{eff}}_{\mu\nu} = T_{\mu\nu} + U_{\mu\nu}(s),
where the torsion correction term U(s)U_{\mu\nu}(s) behaves like a stiff matter contribution or effective negative pressure at high spin densities.
In particular, for a spin-polarized perfect fluid, the effective pressure and energy density become:
