# Non-Boost Invariant Fluid Dynamics

### Submission summary

 As Contributors: Niels Obers · Watse Sybesma Preprint link: scipost_202005_00012v4 Date accepted: 2020-08-04 Date submitted: 2020-07-07 02:00 Submitted by: Sybesma, Watse Submitted to: SciPost Physics Discipline: Physics Subject area: Fluid Dynamics 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 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 perturba- tions 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.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

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 non-boost 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 energy-momentum tensor (they appear in $\eta_{\text{rot}}$). The general form of the constitutive relations concern the decomposition of what we call the eta tensor. Non-negativity 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 non-boost 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 off-shell or on-shell. Off-shell we get the Ward identity given in (4.24) and when one considers the theory on-shell (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 momentum-stress 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.

### Submission & Refereeing History

Resubmission scipost_202005_00012v4 on 7 July 2020
Submission scipost_202005_00012v1 on 25 May 2020

## Reports on this Submission

### 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.

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### Report

The authors have addressed satisfactorily the concerns raised in the report. I recommend the paper for publication.

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### 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.

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