Abstract
We consider uncharged fluids without any boost symmetry on an arbitrary curved background and classify all allowed transport coefficients up to first order in derivatives. We assume rotational symmetry and we use the entropy current formalism. The curved background geometry in the absence of boost symmetry is called absolute or Aristotelian spacetime. We present a closed-form expression for the energy-momentum tensor in Landau frame which splits into three parts: a dissipative (10), a hydrostatic non-dissipative (2) and a non-hydrostatic non-dissipative part (4), where in parenthesis we have indicated the number of allowed transport coefficients. The non-hydrostatic non-dissipative transport coefficients can be thought of as the generalization of coefficients that would vanish if we were to restrict to linearized perturbations and impose the Onsager relations. For the two hydrostatic and the four non-hydrostatic non-dissipative transport coefficients we present a Lagrangian description. Finally when we impose scale invariance, thus restricting to Lifshitz fluids, we find 7 dissipative, 1 hydrostatic and 2 non-hydrostatic non-dissipative transport coefficients.
Highlights
Hydrodynamics arises as the universal description of interacting systems near local thermal equilibrium in the long wavelength limit, which makes hydrodynamics indispensable as an effective theory for a broad class of physical phenomena
In Appendix B we show that the converse is true, i.e. starting from the most general non-canonical entropy current and demanding that its divergence obeys (3.11), where the energy-momentum tensor is the most general one allowed by symmetries, we find that the non-canonical entropy current is precisely of the form as given in (4.70) and (4.72)
Implementing the constraint of non-negativity of the divergence of the entropy current, we find 10 dissipative, 2 hydrostatic non-dissipative and 4 non-hydrostatic non-dissipative transport coefficients
Summary
Hydrodynamics arises as the universal description of interacting systems near local thermal equilibrium in the long wavelength limit, which makes hydrodynamics indispensable as an effective theory for a broad class of physical phenomena. Our hydrodynamic description starts from the perfect fluid energy-momentum tensor for non-boost invariant systems obtained in Ref. [9, 11] how the framework of non-boost invariant hydrodynamics can be adapted to (non-relativistic) scale invariant fluids with critical exponent z This includes particular expressions for the speed of sound in generic z Lifshitz fluids as well as specific results for the first-order transport coefficients in the linearized case. This is similar to the effective energy-momentum tensor one can associate to a cosmological constant, and the form of the energy-momentum tensor in the presence of additional ‘Carroll’ boost invariance [9] It is unclear whether any systems in nature properly realize Carroll symmetry and this observation may be equivalent to the non-existence of type I framids.
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