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We study the causality and stability of the relativistic hydrodynamics when the spin degree of freedom is included as a hydrodynamic field, by considering two specific models of spinhydrodynamics. A linear mode analysis for the simplest static background with no flow velocity shows that the model of dissipative first-order spin-hydrodynamics remains acausal and admits unphysical instabilities as reported for the relativistic Navier-Stokes (NS) equation.Besides, the inclusion of the spin field in hydrodynamics can lead to new kinds of linear modes in the system. These new modes also exhibit instability and an acausal behavior. The second model of the spin-hydrodynamics that we have considered has no dissipative fluxes and is equivalent to a particular second-order conventional hydrodynamics [1]. For a static background, it is found that the linear modes of this model support the sound waves only.However, when the background has constant vorticity, the model can have instability in certain situations and the mode also has acausal behavior. It is found that the spin-dynamics have an effect on the hydrodynamic response of the fluid. These findings point toward the need for a causal and stable theory with spin as a hydrodynamic field to describe the spin-polarized fluid.

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The role of slow out of equilibrium modes (OEM), introduced to extend the validity of hydrodynamics near the QCD critical point on the power spectrum of dynamical density fluctuations has been studied. We have used the equation of motion of slow modes for the situation when the extensive nature of thermodynamics is not altered due to the introduction of OEM. We find that the extensivity condition puts an extra constraint on the coupling of OEM with the four divergence of velocity. The dynamic structure factor (S nn (k, ω)) in presence of the OEM shows four Lorentzian peaks asymmetrically positioned about ω(frequency) = 0, whereas the dynamic structure factor without the presence of any OEM shows three well-known Lorentzian peaks. The width of the peaks are reduced in presence of OEM. We find that the asymmetric peaks originate due to the coupling of the out-of-equilibrium modes with the hydrodynamic modes. It is also shown that the OEM has negligible effects on S nn (k, ω) if first order hydrodynamics (relativistic Navier-Stokes) is used. The introduction of OEM reduces the width of the Rayleigh peak indicating the reduction in the decay rate of the fluctuation which leads to slowing down, a well-known characteristics of the critical end point (CEP).