Hydrodynamic Modes of Homogeneous and Isotropic Fluids

TL;DR

This work develops a comprehensive linearized framework for hydrodynamics without boost symmetry, applicable to fluids with general dynamical exponent $z$ and moving media. It derives the most general first-order constitutive relations, identifies five dissipative and one non-dissipative transport coefficients, and obtains the dispersion relations for sound, shear, and diffusion modes up to second order in momentum. A central finding is that Lifshitz scaling introduces a coupling between sound attenuation and thermal conduction, leading to a Lifshitz-specific attenuation formula $\Gamma_{\text{Lif}}$, while boost-invariant limits recover familiar relativistic and Galilean results. The study also clarifies how entropy production and Onsager relations constrain the transport coefficients and discusses potential holographic realizations and extensions to nonlinear regimes, providing a unified description of non-boost-invariant fluids relevant to critical and condensed-matter systems.

Abstract

Relativistic fluids are Lorentz invariant, and a non-relativistic limit of such fluids leads to the well-known Navier-Stokes equation. However, for fluids moving with respect to a reference system, or in critical systems with generic dynamical exponent z, the assumption of Lorentz invariance (or its non-relativistic version) does not hold. We are thus led to consider the most general fluid assuming only homogeneity and isotropy and study its hydrodynamics and transport behaviour. Remarkably, such systems have not been treated in full generality in the literature so far. Here we study these equations at the linearized level. We find new expressions for the speed of sound, corrections to the Navier-Stokes equation and determine all dissipative and non-dissipative first order transport coefficients. Dispersion relations for the sound, shear and diffusion modes are determined to second order in momenta. In the presence of a scaling symmetry with dynamical exponent z, we show that the sound attenuation constant depends on both shear viscosity and thermal conductivity.