Abstract
We formulate the Schwinger-Keldysh effective field theory of hydrodynamics without boost symmetry. This includes a spacetime covariant formulation of classical hydrodynamics without boosts with an additional conserved particle/charge current coupled to Aristotelian background sources. We find that, up to first order in derivatives, the theory is characterised by the thermodynamic equation of state and a total of 29 independent transport coefficients, in particular, 3 hydrostatic, 9 non-hydrostatic non-dissipative, and 17 dissipative. Furthermore, we study the spectrum of linearised fluctuations around anisotropic equilibrium states with non-vanishing fluid velocity. This analysis reveals a pair of sound modes that propagate at different speeds along and opposite to the fluid flow, one charge diffusion mode, and two distinct shear modes along and perpendicular to the fluid velocity. We present these results in a new hydrodynamic frame that is linearly stable irrespective of the boost symmetry in place. This provides a unified covariant stable approach for simultaneously treating Lorentzian, Galilean, and Lifshitz fluids within an effective field theory framework and sets the stage for future studies of non-relativistic intertwined patterns of symmetry breaking.
Highlights
In contrast with all the previous literature, we present our results in a new hydrodynamic frame, which we call density frame, that is linearly stable irrespective of the boost symmetry in place (Galilean or Lorentzian), or absence thereof, and is better suited for potential numerical simulations
We start with the energy, momentum, and charge/particle-number conservation equations and use the second law of thermodynamics to derive the constitutive relations of an ideal fluid without boost symmetry
Recall that we had an immense amount of redefinition freedom on our hands in the choice of hydrodynamic variables uμ, T, and μ that we left unfixed at the end of section 2.2.3
Summary
Hydrodynamics describes the long-wavelength collective behaviour of low-energy excitations in a broad range of physical systems. Schwinger-Keldysh EFT provides a controlled framework for developing such hydrodynamic theories and studying stochastic corrections to classical hydrodynamics It was shown recently in the context of isotropic relativistic fluids that stochastic corrections break the hydrodynamic derivative expansion at third derivative order, leading to non-classical contributions to hydrodynamic correlation functions [23]. In contrast with all the previous literature, we present our results in a new hydrodynamic frame, which we call density frame, that is linearly stable (in the sense of [30,31,32]) irrespective of the boost symmetry in place (Galilean or Lorentzian), or absence thereof, and is better suited for potential numerical simulations This frame choice aligns the fluid velocity with the flow of momentum, rather than the flow of internal energy (as in the Landau frame) or charge/particle-number (as in the Eckart frame). Appendix B provides the interaction Lagrangian for the linearised effective field theory of hydrodynamics without boosts, which can be used for studying stochastic contributions to hydrodynamic correlation functions
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