Relativistic spin hydrodynamics revisited with general rotation by entropy-current analysis

We revisit the canonical formulation of spin hydrodynamics for Dirac fermions with a general thermal vorticity. The orders of the general thermal vorticity and the corresponding spin variables are considered independently from those of the conventional hydrodynamic variables and their perturbative gradients. Assuming a totally antisymmetric spin current of Dirac fermions, the entropy-current analysis with a general spin potential indicates that the constitutive relations of the stress-energy tensor have to involve spin variables, particularly those linked to boost symmetry, to adhere to the entropy principle. In the presence of the degree of freedom associated with boost symmetry, we choose the constitutive relations of the canonical formulation to be connected to those of the phenomenological formulation through pseudogauge transformation. Subsequently, a linear-mode analysis is conducted using the resulting spin hydrodynamic equations. It is observed that the spin and hydrodynamic modes in this canonical formulation display different characteristics compared to those in the phenomenological formulation up to the second order of gradient.