  We study the causality and stability of relativistic hydrodynamics with the
inclusion of the spin degree of freedom as a hydrodynamic field. We consider
two specific models of spin-hydrodynamics for this purpose. A linear mode
analysis for static background shows that a first-order dissipative
spin-hydrodynamics remains acausal and admits instabilities. Besides, it is
found that the inclusion of the spin field in hydrodynamics leads to new kinds
of linear modes in the system. These new modes also exhibit instability and
acausal behavior. The second model of the spin-hydrodynamics that we have
considered here is equivalent to a particular second-order conventional
hydrodynamics with no dissipative effects. 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, then the model admits instability and
acausality in certain situations. 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.
