  Using the second law of local thermodynamics and the first-order Palatini
formalism, we formulate relativistic spin hydrodynamics for quantum field
theories with Dirac fermions, such as QED and QCD, in a torsionful curved
background. We work in a regime where spin density, which is assumed to relax
much slower than other non-hydrodynamic modes, is treated as an independent
degree of freedom in an extended hydrodynamic description. Spin hydrodynamics
in our approach contains only three non-hydrodynamic modes corresponding to a
spin vector, whose relaxation time is controlled by a new transport
coefficient: the rotational viscosity. We study linear response theory and
observe an interesting mode mixing phenomenon between the transverse shear and
the spin density modes. We propose several field-theoretical ways to compute
the spin relaxation time and the rotational viscosity, via the Green-Kubo
formula based on retarded correlation functions.
