Non-Boost Invariant Fluid Dynamics

The paper presents a complete and detailed analysis of the hydrodynamics of a fluid without any
boost symmetry. They use the method of coupling the system to an external background geometry, analogously to the Newton-Cartan geometry that has been used to derive Lifshitz hydrodynamics.
They derive the complete structure of transport coefficients at first order in derivatives but non-linearly in the fluctuations. In addition, the constraints given by the positivity of entropy production are analyzed.
The exposition is clear and all the steps of the derivation are given. The results are interesting and potentially useful, even though at present it is not clear if the formalism has any specific application.

I only have a few minor points to raise:
1) I could not completely follow the logic of the derivation of the conservation equation, (4.25), from the action. Before (4.23), it is stated that the fluid equations of motion follow from a diffeo transformation of the action, but then later it is stated that they follow from varying only \beta^\mu. Perhaps the author can elaborate more on this point.

2) There is an ambiguity in separating the HS from the NHS coefficients, as the NHS terms can be seen as solutions of the homogeneous part of eq. (3.11). However in section (4.4) they are identified with parts of the tensors having specific symmetry properties, symmetric or antisymmetric, so seemingly a specific choice has been made but it’s not clear to me how the ambiguity was fixed.

3) The full energy-momentum tensor can be decomposed in a vector and a tensor part, as in (2.24). Can the transport coefficients be separated according to this decomposition?

This paper studies hydrodynamics to first-order in gradients in the case where there is no boost invariance, i.e. theories with a privileged reference frame. In such theories the transport coefficients are functions also of the velocity with respect to that frame.

The novelty of the construction in this paper is to take the uncharged fluid case and generalise it to a curved background. The geometric data is discussed in terms of Aristotelian geometry. Some of the transport coefficients are non-dissipative, and the authors pay close attention to this fact and go as far as to construct an action for these theories.

I recommend that the paper be published as it is an important contribution to the area, particularly for their detailed analysis of how to couple to background geometry in this context, as well as their treatment of the hydrostatic partition function. However I have some minor points that should be addressed first:

1. In the relativistic fluid context, coupling to background geometry does not introduce new transport coefficients at first order. One covariantizes the first order expressions, but new transport coefficients appear at second order in gradients with a curvature term. It would be useful for the authors to clarify whether this is true also in the non-boost-invariant setting. Does the curved background allow for new transport coefficients at first hydrodynamic order, as compared with flat space? Indeed, the authors do state that curvatures first appear at second order in gradients, however I think some further discussion of this point in the manuscript would help clarify the significance of the calculations they have presented.

2. If the answer to the question in point 1. above is negative, then all the transport coefficients studied by the authors should be enumerated previously in [12]. The authors do acknowledge agreement with [12] at the level of the constitutive relations, however I think it would be valuable for readers if they could provide the explicit relations between the transport coefficients appearing in their constitutive relations and the existing definitions.

The authors do an exhaustive classification of first order transport coefficients in fluids which may lack boost invariance and study how they are constrained in some particular cases, imposing Onsager relations, with Lifshitz scale invariance or with Lorentz boost invariance. The results could be of interest in systems with broken boost invariance that appear in condensed matter or in general for fluids in contact with an external medium that sets a fixed reference frame.

The paper completes the classification of possible transport coefficients of fluids of this type to first order (assuming there is no parity breaking), introducing a new set of coefficients that are non-linear in the velocities and would be absent in more symmetric fluids. The analysis is carefully done and the results could be interesting for a variety of researchers working on effective field theories and hydrodynamics.

I have just two small comments/questions that do not affect to the main content of the paper:

-In the introduction the authors state that a superfluid "that appears in nature" would have an expectation value for a gauge potential. I'm not sure what do they have in mind exactly. To mention some cases that I'm sure are known to the authors, one could have a superfluid at zero density if one has a complex scalar with a Higgs-like potential. Similarly, one could have a superfluid at finite density just by introducing a chemical potential for a complex scalar, without gauge fields ever entering the picture. Regarding real superfluids, Helium can be described as having (spontaneously broken) Galilean boost invariance. So I find this discussion a bit confusing.

-Equation (2.27) is declared as the conservation equation extrapolating from the diffeomorphism Ward identity in the action. Maybe assuming some conditions over $\tau_\mu$, would it be possible to define a conserved charge as one does for instance in other cases as the integral of a density, and derive the local conservation equation of the charge density from this definition?