Report on scipost_202005_00012v4 Article Swipe

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 nondissipative 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.An interesting feature of non-boost invariant fluids is the appearance of non-dissipative transport coefficients at first order, alongside dissipative transport coefficients [11,12].By applying the entropy current constraint to the full non-linear constitutive relations we show in this paper that there are 10 dissipative transport coefficients and 6 non-dissipative ones.We also show that the number of transport coefficients is unaffected by the introduction of background curvature to first order in derivatives.Following [25,26], one can further separate the non-dissipative ones into two types, hydrostatic and non-hydrostatic, which in the present case turns out to be 2 and 4 transport coefficients, respectively.For the case of Lifshitz fluids, these numbers become 7, 1 and 2, respectively.We will show that both the hydrostatic and non-hydrostatic transport coefficients can be obtained using Lagrangian methods.The hydrostatic transport coefficients feature in the non-canonical part of the entropy current and coincide with contributions that can be computed using an action principle obtained by allowing for time dependence in the hydrostatic partition function, see e.g.[27,28] and the earlier works [29,30].Furthermore, we find that when restricting to linearized perturbations all the non-hydrostatic transport coefficients vanish due to the Onsager relations.We now return to a brief discussion of the physical relevance of non-boost invariant hydrodynamics, before presenting an outline of the paper. Relevance of non-boost invariant hydrodynamicsAs remarked above, for many systems in nature one does not have the luxury of assuming boost symmetry.In condensed matter, for example, one can study critical points where boost symmetries are absent [31].The Lifshitz critical point [32] is an example of this and related recent papers include quantum critical transport in strange metals, see e.g.[33], electrons in graphene [34] and viscous electron fluids [35].With this application in mind, it is shown in the original Refs.[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.In particular, it was shown that the sound attenuation constant depends on both shear viscosity and thermal conductivity.The framework was also recently used in [36] to study out-of-equilibrium energy transport in a quantum critical fluid with Lifshitz scaling symmetry following a local quench between two semi-infinite fluid reservoirs.It is also interesting to note that Lifshitz hydrodynamics is relevant in connection with non-AdS holographic realizations of systems with Lifshitz thermodynamics [23,[37][38][39][40], see also [41][42][43][44][45].More generally non-boost invariant hydrodynamics is of relevance to any system with a reference frame, such as aether theories or in various active matter systems exhibiting e.g.flocking behavior, see e.g.[46] and [47] for a recent example.General active matter systems typically do not have conserved energy or momentum as the divergence of the energy and momentum currents is equal to 'driving' terms.Non-boost invariant hydrodynamics is only an approximate description for configurations that are close to equilibrium configurations at 'cruising speed' where the driving terms vanish.Assuming that the fundamental laws of physics are Lorentz invariant, the necessity of some type of reference frame to obtain a system with broken boosts is obvious.Dispersion relations