Stability of low-momentum rest frame modes in first- and second-order spin hydrodynamics

The low-momentum rest frame stability of the second-order spin hydrodynamics is analyzed in detail. A truncation scheme of the Israel-Stewart formalism derived in our earlier work is proposed that extends the minimal causal formulation. It consists of altogether 40 interconnected relaxation-type dynamical equations -- 16 (24) of them correspond to the independent components of the energy-momentum (spin) tensor. Similarly to previous studies, we find that the stability of the perturbations depends on the form of the spin equation of state. The stability is achieved if the equation of state satisfies the generalized Frenkel condition demanding that the "electric" and "magnetic" components of the spin density tensor have opposite signs. We then re-examine the study of the first-order (Navier-Stokes) dissipative spin hydrodynamics and find a similar type of behavior, which is consistent with the overall stability of low-momentum rest frame modes in conventional relativistic hydrodynamics.