Effective Field Theory of ideal-fluid Hydrodynamics

Starting from a standard description of an ideal, isentropic fluid, we derive the effective theory governing a gapless non-relativistic mode—the sound mode. The theory, which is dictated by the requirement of Galilei invariance, entails the entire set of hydrodynamic equations. The gaplessness of the sound mode is explained by identifying it as the Goldstone mode associated with the spontaneous breakdown of Galilei invariance. Differences with a superfluid are pointed out. Typeset using REVTEX 1 The hydrodynamics of an ideal, classical fluid was already well understood in the 19th century. The case of isentropic flow, for which the entropy per unit mass is constant, is particularly simple. The pressure P is here a function of the mass density ρ only, and the flow is automatically a potential flow. A property of such a fluid is that it supports unattenuated sound waves, i.e., propagating density oscillations. The waves are unattenuated because viscosity and thermal conductivity, which usually serve to dissipate the energy of a propagating mode, are absent. Sound waves belong to the class of modes having a gapless energy spectrum. The purpose of this essay is to argue that their presence in an isentropic fluid is an emergent property [1]. That is, we do not take the existence of this gapless mode for granted, but wish to explain it from an underlying principle, namely that of broken symmetry. To describe the hydrodynamics of an isentropic fluid, we use Eckart’s variational principle [2] and start with the Lagrange density L = 1 2 ρv − ρe+ θ[∂0ρ+∇ · (ρv)], (1) where v is the velocity field, ρ the mass density, and e the internal energy per unit mass. For isentropic flow e is a function of ρ alone. The first and second term in (1) represent the kinetic and potential energy density, respectively. The variable θ is a Lagrange multiplier introduced to impose the conservation of mass: ∂0ρ+∇ · (ρv) = 0; (2) its dimension is [θ] = ms. The variation of (1) with respect to v yields the equation