Causality and Stability of First-Order Relativistic Spin Hydrodynamics with Conserved Charges

We study the causality and stability of first-order relativistic spin hydrodynamics with particle-number conservation. By deriving the complete dispersion relations of linear perturbations around a global equilibrium state, we find that conserved-charge dynamics modifies the sound sector and introduces additional non-hydrodynamic modes absent in the charge-neutral theory, while the structure of spin relaxation modes remains unchanged. Moreover, the stability conditions acquire new contributions from charge diffusion and thermodynamic susceptibilities. More importantly, a particle-number-induced mode is shown to violate the causality condition in the short-wavelength limit. We further demonstrate that particle-number conservation does not remove the instability inherent in first-order spin hydrodynamics. These results reveal a nontrivial interplay between spin and conserved-charge dynamics and provide important constraints on relativistic spin hydrodynamic theories at finite density.