We reexamine the mode structure of first-order relativistic hydrodynamics in flat spacetime with a global $U(1)$ symmetry from the perspective of low energy effective field theory. Using the Schwinger-Keldysh formalism, we show that boost invariance is realized nonlinearly and that the associated Nambu-Goldstone modes resolve the apparent instabilities encountered in conventional treatments. In a consistent derivative expansion, our formulation yields a frame invariant, gapless infrared spectrum of hydrodynamic modes that is stable if the enthalpy density is positive. The issue of well-posedness of partial differential equations is ultraviolet and must be kept distinct from the question of first order hydrodynamics as an effective theory. Our results indicate that, first order hydrodynamics has no internal inconsistency on its domain of applicability, and with an appropriate cutoff one can carry out analytic applications such as loop and fluctuation analyses without the frame dependence.

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• Diffusion
• Effective field theory
• Fluctuations & noise
• Quantum field theory (low energy)
• Relativistic hydrodynamics