Mode analysis of Nambu-Goldstone modes in U(1) charged first-order relativistic viscous hydrodynamics

We conduct a mode analysis of a general $U(1)$-charged first-order relativistic hydrodynamics within the framework of effective field theory for dissipative fluids in flat Minkowski spacetime. We derive the most general quadratic action for hydrodynamic modes, including stochastic noise, and analyze the corresponding dispersion relations in a consistent gradient expansion. We argue that spontaneous breaking of spacetime symmetry arises in the presence of a local thermal state specified by a local timelike four-vector. We demonstrate that hydrodynamical perturbations can be identified as Nambu-Goldstone (NG) modes, analogous to their embedding in global $U(1)$-invariant theories. We find that frame-invariant combinations of hydrodynamic transport coefficients determine the first-order dispersion relations in the low-energy limit, making the mode analysis manifestly independent of the choice of hydrodynamic frame. Assuming local Kubo-Martin-Schwinger (KMS) symmetry and unitarity of the underlying UV theory, we show that first-order hydrodynamics is stable if the enthalpy density is positive.