\omega_{\mu\nu} is the vorticity tensor (antisymmetric),
a=uua^\mu = u^\nu \nabla_\nu u^\mu is the 4-acceleration (vanishes for geodesic flow),
h=g+uuh_{\mu\nu} = g_{\mu\nu} + u_\mu u_\nu is the projection tensor.
Torsion modifies the connection and thus affects the computation of all these quantities. In particular, vorticity may be non-zero even in cases where it would vanish under standard GR, and evolves according to:
+23=torsion-induced terms.\dot{\omega}_{\mu\nu} + \frac{2}{3} \theta \omega_{\mu\nu} = \text{torsion-induced terms}.
3.3 Energy-Momentum Tensor with Spin Corrections
We now include the effects of spin in the energy-momentum tensor. For a spin-polarized perfect fluid, the canonical form becomes:
T=(+p)uu+pg+,T_{\mu\nu} = (\rho + p) u_\mu u_\nu + p g_{\mu\nu} + \tau_{\mu\nu},
where \tau_{\mu\nu} is the spin-torsion correction term, emerging from the intrinsic angular momentum of the fluid.
In Einstein--Cartan theory, this leads to effective quantities:
eff=s2,peff=ps2,\rho_{\text{eff}} = \rho - \alpha s^2, \quad p_{\text{eff}} = p - \alpha s^2,
