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NonBoost Invariant Fluid Dynamics
by Jan de Boer, Jelle Hartong, Emil Have, Niels A. Obers, Watse Sybesma
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Submission summary
Authors (as registered SciPost users):  Emil Have · Niels Obers · Watse Sybesma 
Submission information  

Preprint Link:  scipost_202005_00012v4 (pdf) 
Date accepted:  20200804 
Date submitted:  20200707 02:00 
Submitted by:  Sybesma, Watse 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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 rota tional 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 closedform expression for the energymomentum tensor in Landau frame which splits into three parts: a dissipative (10), a hydrostatic nondissipative (2) and a nonhydrostatic non dissipative part (4), where in parenthesis we have indicated the number of allowed transport coefficients. The nonhydrostatic nondissipative transport coefficients can be thought of as the generalization of coefficients that would vanish if we were to restrict to linearized perturba tions and impose the Onsager relations. For the two hydrostatic and the four nonhydrostatic nondissipative 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 nonhydrostatic nondissipative transport coefficients.
Author comments upon resubmission
We thank the referees for their time. In this letter we respond to comments and suggestions of the referee reports point by point. We are confident that the improvements will satisfy the referee and will increase the quality of the paper.

First point of report 1: We agree that the formulation here is confusing and it was by no means intended as a general statement about superfluids. We therefore changed the formulation to make it clear that we are merely describing one illustrative example.

Second point of report 1: From the conservation equation, by contracting (as usual) with a Killing vector of the (absolute space) background, one can obtain conserved currents giving rise to conserved charges. While it is not essential for the rest of the paper, we have added a footnote around eq.(2.27) pointing this out, as an aid to the reader.

First point of report 2: As is also the case in relativistic fluids, in the nonboost invariant case we similarly do not get any new transport coefficients at first hydrodynamic order as a consequence of considering the background to be curved (viz also the statements around eq. (2.18)). This should also be clear from the results of Section 5, where we restrict to a flat background. We believe the utility of considering the theory on a curved background should be apparent from the covariance entering the computations and results of the paper. We have added a remark in the introduction stressing the first point above, namely that the number of transport coefficients is unaffected by the introduction of background curvature to first order in derivatives.

Second point of report 2: Our method of analysis diverges in fact from [12] already at an earlier stage. We agree that there are 16 derivative structures to begin with. Two of these 16 only appear in the analysis of the HS part of the energymomentum tensor (they appear in $\eta_{\text{rot}}$). The general form of the constitutive relations concern the decomposition of what we call the eta tensor. Nonnegativity of entropy production restricts the symmetry properties of the eta tensor. We only implement constitutive relations after taking these symmetry properties into account which reduces the number of D and NHS coefficients from 16 down to 14. This by itself makes it already challenging to compare the results. Furthermore, given the fact that the paper [12] does not provide explicit final results for the most general nonboost invariant case (these have to be extracted from Mathematica files) we believe it is beyond our responsibility to do such a comparison. Should paper [12] be updated to present the results in an explicit way, such that we can make a meaningful comparison, we are happy to do so.

First point of report 3: What we described is the distinction between the consequence of diffeomorphism invariance depending on whether one considers the theory offshell or onshell. Offshell we get the Ward identity given in (4.24) and when one considers the theory onshell (i.e. obey the fluid equations of motion) this reduces to the one given in (4.25). We believe that we already added this distinction to the accompanying text around these equations.

Second point of report 3: There is indeed an ambiguity in the HS terms since, as far as the antisymmmetric $\eta$tensor is concerned, such terms can both be in the HS and NHS part. We have added a remark after (3.11) making this more clear from the outset. Moreover, we also point out that we will make a specific (convenient) choice in fixing this ambiguity. This choice was already stated at the end of Section 4.4 (last part of paragraph below eq. (4.110)).
Third point of report 3: The answer to this question is negative. The reason being that as we are in Landau frame the energy current (vector) is fixed in terms of the momentumstress tensor. Thus there is no such separation of transport coefficients.
List of changes
 Added the following sentence to the introduction (top of page 3): "We also show that the number of transport coefficients is unaffected by the introduction of background curvature to first order in derivatives"
 Adapted text in the introduction regarding the connection to superfluids (bottom of page 4)
 Added footnote 7 on page 8, showing that contracting Eq. (2.27) with a vector $K^\rho$ gives rise to a conserved current if $K^\rho$ is Killing in the Aristotelian sense we define later.
 Added a remark below Eq. (3.11) noting that HS terms are only defined up to the addition of NHS terms.
Published as SciPost Phys. 9, 018 (2020)
Reports on this Submission
Anonymous Report 3 on 2020717 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202005_00012v4, delivered 20200717, doi: 10.21468/SciPost.Report.1832
Report
In order to avoid the proliferation of conventions and definitions, it would have been useful for the authors to phrase the constraints that they have found in terms of the 16 transport coefficients that were already defined in the literature. These are given (explicitly and in closed form) in equation 2.24 of reference [12] (after taking the neutral limit).
However I do not wish to further belabour this point and delay publication of the paper. I can recommend that the paper now be published.
Anonymous Report 1 on 2020710 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202005_00012v4, delivered 20200710, doi: 10.21468/SciPost.Report.1818
Report
The authors have answered satisfactorily the points raised in the report. The results of the paper can potentially have multiple aplications in systems with broken boost invariance, and satisfies the general criteria regarding its presentation. I recommend its publication without need for further changes.