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

TL;DR

This paper analyzes the Nambu-Goldstone modes in U(1)-charged relativistic hydrodynamics, deriving a general quadratic action and dispersion relations, and establishing stability conditions based on symmetry and thermodynamic positivity.

Contribution

It introduces a comprehensive mode analysis framework for first-order relativistic hydrodynamics, including stochastic noise and frame-invariant transport coefficients, with implications for stability and symmetry breaking.

Findings

Hydrodynamic perturbations are identified as Nambu-Goldstone modes.

Dispersion relations depend on frame-invariant transport coefficients.

Hydrodynamics is stable if enthalpy density is positive under certain symmetries.

Abstract

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.