Enstrophy without boost symmetry

N. Pinzani-Fokeeva1,2*, A. Yarom3

1 Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

2 Department of Physics and Astronomy, University of Florence, Via G. Sansone
1, I-50019, Sesto Fiorentino (Firenze), Italy

3 Department of Physics, Technion, Haifa 32000, Israel

*<EMAIL_ADDRESS>
## Abstract

We construct an approximately conserved current for $2+1$ dimensional,
Aristotelian (non boost invariant), fluid flow. When Aristotelian symmetry is
enhanced to Galilean symmetry, this current matches the enstrophy current
responsible for the inverse cascade in incompressible fluids. Other
enhancements of Aristotelian symmetry discussed in this work include
Lorentzian, Carrollian and Lifshitz scale symmetry.

###### Contents

  1. 1 Introduction
  2. 2 A sufficient condition for the existence of an enstrophy current
  3. 3 Solving $i_{u}\Omega=0$ by constraining the equation of state
    1. 3.1 Recovering the Galilean invariant solution
    2. 3.2 Recovering the Lorentz invariant solution
    3. 3.3 Carrollian symmetry
    4. 3.4 Lifshitz symmetry
  4. 4 Solving $i_{u}\Omega=0$ for incompressible flow
  5. 5 Summary
  6. A Boost invariance
    1. A.1 Galilean boosts
    2. A.2 Carrollian boosts

##  1 Introduction

Enstrophy is an approximately conserved quantity in two dimensional,
incompressible, fluid flow. Its existence is crucial for the appearance of the
inverse energy cascade in two dimensional turbulence [1] and its dynamics is
used in modeling turbulent dependent phenomena. While enstrophy is well
understood in the context of incompressible fluid flow, barotropic flow and
relativistic fluids (see [2, 3, 4, 5]), little is known regarding its
existence for more general flows with varying degrees of symmetry.

The goal of this work is to study enstrophy conservation for generic fluids in
two spatial dimensions with little symmetry. More precisely, we will consider
fluids which possess translation invariance in space and time, rotational
invariance in space, and have a well defined thermodynamic limit. We will
refer to these fluids as non frame invariant or Aristotelian fluids. The
dynamics of such fluids is of interest for several reasons. In [6, 7, 8, 9] it
has been shown that Lorentz, Galilean and Corrolian fluids, as well as
Lifshitz invariant fluids, can all be obtained from Aristotelian fluids with
ehnanced symmetry. Thus, demonstrating enstrophy conservation in Aristotelian
fluids suggests that this property is a robust phenomenon which is not tied to
any particular symmetry of the fluid. From a more physical perspective, the
dynamics of Aristotelian fluids can be tied to the dynamics of flocking [10],
though, we have not yet found connections between enstrophy conservation and
flocking phenomenon.

In incompressible flow, the enstrophy current can be written in the form

$J^{\mu}_{\rm inc}=\frac{\omega^{2}}{n}\left(1,\,\vec{v}\right)$ (1)

with $\vec{v}$ the velocity field, $n$ the particle number density, assumed to
be constant, and
$\omega^{2}=\omega_{ij}\delta^{ii^{\prime}}\delta^{jj^{\prime}}\omega_{i^{\prime}j^{\prime}}$
where

$\omega_{ij}=\partial_{i}v_{j}-\partial_{j}v_{i}\,.$ (2)

Here and in what follows Greek indices $\mu,\nu,\dots$ denote spacetime
coordinates while Latin indices $i,j,\dots$ denote spatial coordinates.
(Often, $n$, being constant, is omitted from (1).) It is straightforward to
check that (1) is conserved under the Euler equations and is negative
semidefinite once viscous corrections are introduced. In fact, there is not
one, but an infinite set of conserved enstrophy currents,

$J^{\mu}_{(\alpha)}=q\left(\frac{s}{n}\right)\frac{(\omega^{2})^{\alpha}}{n^{2\alpha-1}}\left(1,\,\vec{v}\right)\,,$
(3)

with $\alpha$ a real number, $s$ the entropy density and $q$ an arbitrary
function. (Equivalently, one can combine powers of $J^{\mu}_{(\alpha)}$ into
$J^{\mu}_{Q}=n\,Q(s/n,\omega^{2}/n^{2})(1,\vec{v})$ with $Q$ a generic
function of its arguments, and parameterize the enstrophy currents with the
function $Q$.) These currents can be further generalized in the presence of
additional conserved $U(1)$ charges, or if the flow is barotropic. See, e.g.,
[5]. For $\alpha=1$ and $q=1$ we obtain (1).

In this work we generalize (1) and show that a fluid flow whose dynamics is
determined by space-time translation invariance, rotation invariance, and the
requirement of a thermodynamic limit possesses a family of conserved enstrophy
currents of the form

$J^{\mu}=q\left(\frac{s}{n}\right)\frac{\left(\Omega^{2}\right)^{\alpha}}{s^{2\alpha-1}}u^{\mu}\,.$
(4)

Here $u^{\mu}$ is an appropriate velocity field such that $su^{\mu}$ is the
leading contribution to the entropy current and $nu^{\mu}$ is a conserved
$U(1)$ current (which presumably exists). Also, $q$ is an arbitrary function
of $s/n$ and $\Omega^{2}$ a generalized squared vorticity.111 We are
intentionally vague here regarding the construction of $\Omega^{2}$ and the
normalization of the velocity $u^{\mu}$ since their explicit form depends on
the type of metric available and the symmetries we intend to keep. See table 2
and section 3 for details. Curiously, $J^{\mu}$ with $\alpha=0$ and $q=1$ is
the entropy current.

We claim that the current (4) is conserved either for a particular class of
equations of state (summarized in table 2) which in the Galilean invariant
limit reduce to the barotropic condition, or for a generalized incompressible
flow in which case (4) reduces to (1) when Galilean invariance becomes a
symmetry of the equations of motion. To be somewhat more explicit, in the
limit where Aristotelian symmetry is enhanced to conformal invariance, we find
that, in the absence of a chemical potential,

$\Omega_{\mu\nu}=\partial_{\mu}\left(Tu_{\nu}\right)-\partial_{\nu}\left(Tu_{\mu}\right)$
(5)

where $T$ is the temperature field, $u^{\mu}u_{\mu}=-1$, and

$\Omega^{2}=g^{\mu\alpha}g^{\nu\beta}\Omega_{\mu\nu}\Omega_{\alpha\beta}\,.$
(6)

This reproduces the result quoted in [2] which has been recently generalized
in [5].

Our work is organized as follows. In section 2 we identify sufficient
conditions for the existence of an enstrophy current in a flat background:
that there exists a closed two-form $\Omega$ satisfying $i_{u}\Omega=0$ under
the equations of motion (here $i_{u}$ is the interior product with the
velocity vector $u^{\mu}$). In section 3 we solve the condition
$i_{u}\Omega=0$ for Aristotelian fluids by constraining the equation of state.
In the same section we study various limits of the enstrophy current once
boost or scale symmetries are present. Later, in section 4 we solve
$i_{u}\Omega=0$ in the incompressible limit. We end with a summary and brief
discussion in section 5. A brief review of Galilean and Carrollian invariant
hydrodynamics is relegated to the appendix.

##  2 A sufficient condition for the existence of an enstrophy current

In this work we will be interested in constructing an enstrophy current which
is conserved in a flat background geometry. Since we are interested in
dynamical systems which do not necessarily possess any type of boost
invariance (Lorentz, Galilean or other), we start with a brief discussion of
Aristotelian geometry which will help us ensure Aristotelian covariance. It
will also help us to ensure that our theory is coordinate invariant and later,
if the theory does possess boost invariance, to ensure compatibility of the
isometries of space-time with the symmetries of the equations of motion. After
a brief discussion of a non boost invariant geometry we will turn our
attention to the construction of the enstrophy current. As stated earlier, we
will then argue that the existence of a closed two-form $\Omega$ with
$i_{u}\Omega=0$ is sufficient to obtain an enstrophy current in a hydrodynamic
setting.

Consider a manifold equipped with an inverse metric $h^{\mu\nu}$ which is
degenerate in the sense that there exists an $n_{\mu}$ such that
$h^{\mu\nu}n_{\nu}=0$. The manifold is also equipped with a vector
$\bar{n}^{\mu}$ which we refer to as the time direction and which we take,
without loss of generality, to satisfy $n_{\mu}\bar{n}^{\mu}=-1$. We refer to
$h^{\mu\nu}$ as the (inverse) spatial metric. In Cartesian coordinates, the
flat metric and time direction are given by

$h^{\mu\nu}=\delta^{\mu}_{i}\delta^{\nu}_{j}\delta^{ij}\,,\qquad\bar{n}^{\mu}=\delta^{\mu}{}_{0}\,.$
(7)

The barred notation used in the previous paragraph is to emphasize that
$\bar{n}^{\mu}$ is not obtained by raising the indices of $n_{\mu}$ with the
spatial metric. However, from $h^{\mu\nu}$ and $\bar{n}^{\mu}$ we can
construct $\gamma^{\mu\nu}=h^{\mu\nu}-\bar{n}^{\mu}\bar{n}^{\nu}$ and its
inverse $\gamma_{\mu\nu}$ from which $\bar{n}^{\mu}=\gamma^{\mu\nu}n_{\nu}$.
The tensor $\gamma^{\mu\nu}$ can also be used to define
$\bar{h}_{\mu\nu}=\gamma_{\mu\nu}+n_{\mu}n_{\nu}$. Note that

$\bar{n}^{\mu}\bar{h}_{\mu\nu}=0\,,\qquad P^{\mu}{}_{\nu}\equiv
h^{\mu\alpha}\bar{h}_{\alpha\nu}=\delta^{\mu}{}_{\nu}+\bar{n}^{\mu}n_{\nu}\,.$
(8)

Since there is no preferred choice of metric or connection on the manifold, we
will use the inverse metric

$g_{A}^{\mu\nu}={h}^{\mu\nu}-N\bar{n}^{\mu}\bar{n}^{\nu}$ (9a) to raise and
lower indices. In an Aristotelian geometry we may take $N$ to be $1$ or $0$
without loss of generality. In the limit where the spacetime symmetry is
enhanced to Lorentz invariance the metric must take the form
$g_{L}^{\mu\nu}=g_{A}^{\mu\nu}\Big{|}_{N=1}={h}^{\mu\nu}-\bar{n}^{\mu}\bar{n}^{\nu}\,.$
(9b) Likewise, the $N=0$ metric corresponds to theories where the spacetime
symmetry is enhanced to (massive) Galilean invariance,
$g_{G}^{\mu\nu}=g_{A}^{\mu\nu}\Big{|}_{N=0}={h}^{\mu\nu}\,.$ (9c) The somewhat
peculiar Carrollian symmetry is associated with a metric
$g_{C}^{\mu\nu}=h^{\mu\nu}-P^{\mu}_{\alpha}M_{C}^{\alpha}\bar{n}^{\nu}-P^{\nu}_{\alpha}M_{C}^{\nu}\bar{n}^{\mu}+\bar{n}^{\mu}\bar{n}^{\nu}M_{C}^{2}\,,$
(9d)

where $M_{C}^{\mu}$ is an auxiliary background field necessary to ensure
Carrollian invariance [11] and
$M_{C}^{2}=M^{\alpha}_{C}M^{\beta}_{C}\bar{h}_{\alpha\beta}$. We will discuss
these enhanced symmetries in section 3 when they will be more relevant. We
will often use $g^{\mu\nu}$ for the inverse metric to denote any one of (9).

In a Riemannian geometry there is a unique torsion free, metric compatible
connection. This is not the case for Aristotelian, Galilean or Carrollian
geometries. There, a well defined connection requires the introduction of
additional fields of which $M_{C}^{\mu}$ introduced in the context of
Carrollian geometries above is an example. Luckily, for the purpose of this
work, we only need to ensure that an appropriate metric compatible connection
exists. If $g^{\mu\nu}$ is invertible then we can use the Christoffel
connection as our connection. For metrics of the form
$g_{A}^{\mu\nu}\Big{|}_{N=0}$ we may use the Newton-Cartan connection

$\Gamma^{\mu}{}_{\nu\rho}=-\bar{n}^{\mu}\partial_{\nu}n_{\rho}+\frac{1}{2}h^{\mu\sigma}\left(\partial_{\nu}\bar{h}_{\rho\sigma}+\partial_{\rho}\bar{h}_{\nu\sigma}-\partial_{\sigma}\bar{h}_{\nu\rho}\right)\,,$
(10)

which is compatible with $h^{\mu\nu}$ and $n_{\mu}$ and is torsion free as
long as $d(n_{\alpha}dx^{\alpha})=0$. For the metric $g^{\mu\nu}_{C}$ we may
use the same connection (10) appropriately modified to include $M_{C}^{\mu}$
terms, see [11], such that it is compatible with $g^{\mu\nu}_{C}$ and
$n_{\mu}-\bar{h}_{\mu\alpha}M^{\alpha}_{C}$. We note that it is also possible
to construct a connection which is compatible with $h^{\mu\nu}$ and $n_{\mu}$
and with $\bar{h}_{\mu\nu}$ and $\bar{n}^{\mu}$. See Appendix A for details.
In what follows we will assume the background geometry to be torsion free.

A sufficient condition for

$J_{e}^{\mu}=\frac{\Omega^{2}}{s}u^{\mu}$ (11)

to be conserved is that

${S}^{\mu}=su^{\mu}$ (12)

is conserved ($\nabla_{\mu}S^{\mu}=0$) and that

$\Omega^{2}=-W^{\mu}{}_{\mu}$ (13)

with

$W^{\nu}{}_{\mu}=\Omega^{\nu\alpha}\Omega_{\alpha\mu}\,,\qquad\Omega^{\alpha\beta}=g^{\alpha\mu}g^{\beta\nu}\Omega_{\mu\nu}\,,$
(14)

such that

$\Omega^{\mu\nu}\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=0$ (15)

at least under the equations of motion.

Recall that the Galilean enstrophy current described in the introductory
section is conserved only in the absence of viscous terms. Therefore, the
current $J_{e}^{\mu}$ in (11) should be conserved in the absence of
dissipative terms as well. In a hydrodynamic setting the entropy current is
always conserved in the absence of dissipative terms and is therefore always
available in order to construct (12) with $s$ the entropy density and
$u^{\mu}$ the appropriate velocity field. In the presence of an additional
conserved $U(1)$ charge, $N^{\mu}=nu^{\mu}+\ldots$, one can construct a broad
class of enstrophy currents of the form (4) which are conserved due to the
conservation laws for $S^{\mu}$, $N^{\mu}$ and equation (15). In the remainder
of this section we will describe an operative algorithm for constructing a
two-form $\Omega_{\mu\nu}$ which satisfies (15). Indeed, as we will now show,
in order to maintain (15) in a flat background geometry, it is sufficient to
require that $\Omega$ is closed and orthogonal to $u^{\mu}$

$\displaystyle d\Omega$ $\displaystyle=0\,,$ (16a) $\displaystyle
u^{\alpha}\Omega_{\alpha\beta}$ $\displaystyle=0\,,$ (16b)

at least under the equations of motion.

Given (16) we have $\pounds_{u}\Omega_{\mu\nu}=0$ with $\pounds_{u}$ the Lie
derivative in the $u$ direction. Thus,

$\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=\Omega_{\mu\nu}\nabla_{\alpha}u^{\alpha}-\Omega_{\alpha\nu}\nabla_{\mu}u^{\alpha}-\Omega_{\mu\alpha}\nabla_{\nu}u^{\alpha}\,.$
(17)

It is convenient to decompose $\nabla_{\mu}u^{\alpha}$ into components which
are parallel and perpendicular to vectors $\bar{\tau}^{\alpha}$ and
$\tau_{\alpha}$ which satisfy $\tau_{\mu}\bar{\tau}^{\mu}=-1$, viz.,

$\nabla_{\mu}u^{\alpha}=\tau_{\mu}\bar{\tau}^{\alpha}S+P_{(\tau)}^{\alpha}{}_{\beta}\tau_{\mu}j^{\beta}+\bar{\tau}^{\alpha}P_{(\tau)}^{\gamma}{}_{\mu}\bar{j}_{\gamma}+\frac{1}{d}P_{(\tau)}^{\alpha}{}_{\mu}\Theta+\Sigma^{\alpha}{}_{\mu}\,,$
(18)

where

$P_{(\tau)}^{\alpha}{}_{\beta}=\delta^{\alpha}{}_{\beta}+\bar{\tau}^{\alpha}\tau_{\beta}\,,$
(19)

$d$ is the number of spatial dimensions,
$\Sigma^{\alpha}{}_{\mu}\bar{\tau}^{\mu}=\Sigma^{\alpha}{}_{\mu}{\tau}_{\alpha}=0$,
and $\Sigma^{\alpha}{}_{\alpha}=0$. A straightforward computation yields

$\Omega^{\mu\nu}\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=S\left(2\tau_{\alpha}W^{\alpha}{}_{\beta}\bar{\tau}^{\beta}-\Omega^{2}\right)+\frac{1}{d}\Theta\left({2\tau_{\alpha}W^{\alpha}{}_{\beta}\bar{\tau}^{\beta}}+(d-2)\Omega^{2}\right)\\\
+2\left(\Sigma^{\alpha}{}_{\beta}W^{\beta}{}_{\alpha}+\bar{j}_{\alpha}P_{(\tau)}^{\alpha}{}_{\beta}W^{\beta}{}_{\gamma}\bar{\tau}^{\gamma}+\tau_{\alpha}W^{\alpha}{}_{\beta}P_{(\tau)}^{\beta}{}_{\gamma}j^{\gamma}\right)\,.$
(20)

In order for (15) to hold, the right-hand-side of (20) must vanish under the
equations of motion. Let us consider each such term separately. To ensure that
$W^{\alpha}{}_{\beta}\bar{\tau}^{\beta}$ vanishes, it is convenient to choose
$\bar{\tau}^{\mu}\propto u^{\mu}$ so that
$W^{\alpha}{}_{\beta}\bar{\tau}^{\beta}=0$ as a result of
$\Omega_{\alpha\beta}u^{\beta}=0$. The simplest method by which we can set
$\tau_{\alpha}W^{\alpha}{}_{\beta}=0$ is to use a non invertible $g^{\mu\nu}$
and then choose $\tau_{\alpha}$ as the (appropriately normalized) eigenvector
of $g^{\mu\nu}$ with zero eigenvalue, $g^{\mu\nu}\tau_{\nu}=0$. If we are
insistent on choosing an invertible metric $g^{\mu\nu}$ then setting
$\tau_{\alpha}=g_{\alpha\nu}u^{\nu}$ will ensure that
${\tau}_{\alpha}W^{\alpha}{}_{\beta}=0$. Such a construction is viable only if

$g_{\mu\nu}u^{\mu}u^{\nu}=\bar{h}_{\mu\nu}u^{\mu}u^{\nu}-Nn_{\mu}n_{\nu}u^{\mu}u^{\nu}$
(21)

has definite sign. Otherwise, some modifications to the normalization
$\tau_{\alpha}\bar{\tau}^{\alpha}=-1$ may be needed. Either way, (20) reduces
to

$\Omega^{\mu\nu}\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=-S\Omega^{2}+\Theta\left(1-\frac{2}{d}\right)\Omega^{2}+2\Sigma^{\alpha}{}_{\beta}W^{\beta}{}_{\alpha}\,.$
(22)

The second term on the right of (22) vanishes in $2+1$ spacetime dimensions
($d=2$). The last term also vanishes for the following reason. Consider the
tensor $\Omega^{\alpha}{}_{\beta}=g^{\alpha\mu}\Omega_{\mu\beta}$. By
construction it has one zero eigenvalue associated with the eigenvector
$\bar{\tau}^{\beta}$. Let us denote another eigenvector by $p^{\alpha}$. Skew
symmetric matrices have either zero or imaginary eigenvalues and their rank is
even. Thus, since $\Omega_{\mu\nu}$ is non vanishing, we must have
$\Omega^{\alpha}{}_{\beta}p^{\beta}=icp^{\alpha}$ with $c\neq 0$ and
$c\in\mathbb{R}$, and the third eigenvector of $\Omega^{\alpha}{}_{\beta}$ is
$(p^{\beta})^{*}$. Evaluating $\Omega^{\alpha}{}_{\beta}$ and then
$W^{\alpha}{}_{\beta}$ in terms of this orthogonal basis, it is
straightforward to show that $W^{\alpha}{}_{\beta}$ is proportional to
$P_{(\tau)}^{\alpha}{}_{\beta}$. It follows that

$\Sigma^{\alpha}{}_{\beta}W^{\beta}{}_{\alpha}=0$ (23)

on account of tracelessness of $\Sigma^{\alpha}{}_{\beta}$. Note that (23) is
referred to as the absence of vortex stretching in the context of
incompressible fluid flow. See, e.g., [12].

Using (18), the coefficient of the first term on the right-hand-side of (22)
evaluates to

$S=\bar{\tau}^{\mu}{\tau}_{\alpha}\nabla_{\mu}u^{\alpha}\,.$ (24)

There are several instances under which $S$ vanishes. If we use an invertible
$g^{\alpha\beta}$ (so that $\tau_{\alpha}\propto g_{\alpha\beta}u^{\beta}$)
and we also have $u^{\mu}g_{\mu\nu}u^{\nu}=c_{0}$ with $u^{\mu}$ (and the
constant, $c_{0}$) real, then we obtain $S=0$. Otherwise, we end up with the
equivalent constraints

$u^{\mu}{\tau}_{\alpha}\nabla_{\mu}u^{\alpha}=0\qquad\hbox{or}\qquad
u^{\mu}u^{\alpha}\nabla_{\mu}{\tau}_{\alpha}=0\,.$ (25)

Thus, the right-hand-side of (22) vanishes when the geometry is such that
${\tau}_{\alpha}$ is covariantly constant, or when we choose a geometry which
is compatible with the dynamics of $u^{\alpha}$ in such a way that
${\tau}_{\alpha}\nabla_{\mu}u^{\alpha}=0$. Notice that, in a flat background,
$n_{\mu}=-\delta_{\mu}^{0}$ in Cartesian coordinates, which is covariantly
constant.

To summarize, we have shown that equation (16) is sufficient to ensure (15) as
long as $g^{\mu\nu}$ is invertible and we can choose
$u^{\mu}g_{\mu\nu}u^{\nu}=c_{0}$ with real $u^{\mu}$. Or, one of (25) is
satisfied. It now remains to compute under which conditions (16) is satisfied.
We will solve this condition in two instances. In section 3 we will look for
the most general (first order in derivatives) $\Omega_{\mu\nu}$ and equation
of state such that (16) is satisfied. In section 4 we will take the
incompressible (low Mach number) limit of the fluid equations and look for a
solution which does not rely on a particular class of equations of state.

##  3 Solving $i_{u}\Omega=0$ by constraining the equation of state

Our task is now reduced to constructing an $\Omega$ satisfying (16) at least
under the equations of motion. We will be interested in solutions to the
equations of motion in a flat background where $S$ in (24) vanishes and in the
absence of any external forces. In such a background, the equations of motion
of an inviscid fluid in the absence of any type of boost symmetry are given by
[6],

$\displaystyle\begin{split}0&=\partial_{0}v_{i}+v^{k}\partial_{k}v_{i}+\frac{1}{\rho}\partial_{i}P+\frac{v_{i}}{\rho}\left(\partial_{0}\rho+\partial_{k}\left(v^{k}\rho\right)\right)\,,\\\
0&=\partial_{0}s+\partial_{i}\left(sv^{i}\right)\,,\\\
0&=\partial_{0}n+\partial_{i}\left(nv^{i}\right)\,.\end{split}$ (26)

Here $v^{i}$ is a velocity field as seen in the lab frame, $s$, $n$ and $\rho$
are the entropy density, a conserved $U(1)$ charge density, and the kinetic
mass density respectively. The latter is related to the momentum density
covector $P_{i}$ via $P_{i}=\rho\delta_{ij}v^{j}$. The entropy density $s$ is
conjugate to the temperature $T$ and the charge density $n$ is conjugate to a
chemical potential $\mu$. The pressure, $P$, is related to the remaining
hydrodynamic variables via

$dP=sdT+nd\mu+\frac{1}{2}\rho dv^{2}\,.$ (27)

We emphasize that viscous terms are absent from (26). An analysis of viscous
corrections to (26) can be found in [7, 8, 9]. Since we are interested in an
approximate conservation law for the enstrophy, valid when viscous terms are
negligible, it is sufficient to consider (26). A full derivation of (26) can
be found in [6]. In brief, translation invariance implies the existence of a
stress tensor. Insisting that in thermodynamic equilibrium the energy flux,
momentum flux and charge flux move with the same velocity, and that the free
energy can be identified with the pressure leads to (26). It is interesting to
contrast (26) with the equations of motion in [10] valid under the same
symmetry group but in the absence of thermodynamic equilibrium.

It is convenient to define the velocity vectors $v^{\mu}$ and $\bar{v}_{\mu}$
satisfying

$\bar{v}_{\mu}=\bar{h}_{\mu\nu}v^{\nu}\,,\qquad v^{\mu}n_{\mu}=-1\,,$ (28)

(so that

$v^{\mu}=(1,\,v^{i})\,,\qquad\bar{v}_{\mu}=(0,\,v_{i})\,,$ (29)

in the coordinate system (7)) and

$\Theta=\nabla_{\mu}v^{\mu}\,,\qquad
a_{\mu}=v^{\alpha}\nabla_{\alpha}\bar{v}_{\mu}\,,\qquad\tilde{n}=n/s\,,$ (30)

and split the derivatives into transverse and parallel components such that

$\nabla_{\nu}=D^{\bot}_{\nu}-n_{\nu}D\,,\qquad\bar{n}^{\mu}D^{\bot}_{\mu}=0\,.$
(31)

In this notation the equations (26) take the form

$\displaystyle\begin{split}0&=a_{\mu}+\frac{1}{\rho}D^{\bot}_{\mu}P+\frac{\bar{v}_{\mu}}{\rho}\left(\rho\Theta+v^{\nu}\partial_{\nu}\rho\right)\,,\\\
0&=D\tilde{n}+v^{\mu}D^{\bot}_{\mu}\tilde{n}\,,\\\
0&=Ds+s\Theta+v^{\mu}D^{\bot}_{\mu}s\,.\end{split}$ (32)

Note that $v^{\mu}a_{\mu}=\frac{1}{2}v^{\nu}\partial_{\nu}v^{2}$ with
$v^{2}=v^{\mu}v^{\nu}\bar{h}_{\mu\nu}$ and that $\bar{n}^{\mu}a_{\mu}=0$.

We have made a distinction between the velocity $v^{\mu}$ which we have
introduced in (29) and the velocity $u^{\mu}$ that appears in, say, (12). In
the later part of this work we will find that the natural velocity field
$u^{\mu}$ which appears in thermodynamic quantities, as in, e.g., (12), is
proportional to, if not equal to, $v^{\mu}$. (For instance, in a Lorentz
invariant system $u^{\mu}=v^{\mu}/\sqrt{1-v^{2}}$ while in a Galilean
invariant system it is natural to use $u^{\mu}=v^{\mu}$.)

Recall that our goal is to find an $\Omega$ that satisfies (16) under the
equations of motion. To this end, let us start with the ansatz

$\Omega=d(f\bar{v})+d(gn)\,,$ (33)

where $\bar{v}=\bar{v}_{\mu}dx^{\mu}$, $n=n_{\mu}dx^{\mu}$, and $f$ and $g$
are arbitrary functions of $s$, $\tilde{n}=n/s$ and $v^{2}$.222It is somewhat
unfortunate that we have used both $n=n_{\mu}dx^{\mu}$ and $n$ the charge
density in the same sentence. We hope that this notation does not confuse the
reader and that the appropriate choice of $n$ is clear from context. In the
presence of an external gauge field we could add to (33) its associated field
strength. Equation (33) ensures that $\Omega$ is closed. It remains to find
the constraints on the pressure and on $f$ and $g$ so that
$\Omega_{\mu\nu}u^{\nu}=0$ under the equations of motion. Under the assumption
that $u^{\mu}\propto v^{\mu}$ this amounts to solving
$\Omega_{\mu\nu}v^{\nu}=0$.

The computation we wish to carry out is now straightforward though somewhat
technical. In what follows we will highlight its salient features. First, we
found it convenient to replace the pressure $P$ with the potential $G$,

$G+P=sT+\mu n$ (34)

in which case we have

$dG=Tds+nd\mu-\frac{1}{2}\rho dv^{2}\,.$ (35)

Next, we note that the equation of motion for $a_{\mu}$ in (32) can be dotted
with $v^{\mu}$ to obtain

$\Theta
v^{2}+\frac{1}{2}Dv^{2}+\frac{v^{2}}{\rho}D\rho+\frac{1}{\rho}v^{\alpha}D^{\bot}_{\alpha}P+\frac{1}{2}v^{\alpha}D^{\bot}_{\alpha}v^{2}+\frac{v^{2}}{\rho}v^{\alpha}D^{\bot}_{\alpha}\rho=0.$
(36)

Thus, we can use the equations of motion (32) to get rid of $a_{\mu}$,
$Dv^{2}$, $D\tilde{n}$ and $Ds$. Since $Dv^{2}$ drops out of (36) if
$\partial_{v^{2}}G+2v^{2}\partial^{2}_{v^{2}}G=0$, we will focus, for now, on
generic expressions for $G$ and treat
$\partial_{v^{2}}G+2v^{2}\partial^{2}_{v^{2}}G=0$ as a special case.

After some algebra, it is possible to show that, under the equations of motion

$\Omega_{\mu\nu}v^{\nu}=\sum_{a}B_{\mu}^{a}\beta_{a}$ (37)

where

$\displaystyle B_{\mu}^{1}$ $\displaystyle=(v^{2}n_{\mu}+\bar{v}_{\mu})v\cdot
D^{\bot}s\,,$ $\displaystyle B_{\mu}^{2}$
$\displaystyle=(v^{2}n_{\mu}+\bar{v}_{\mu})v\cdot D^{\bot}\tilde{n}\,,$
$\displaystyle B_{\mu}^{3}$ $\displaystyle=(v^{2}n_{\mu}+\bar{v}_{\mu})v\cdot
D^{\bot}v^{2}\,,$ $\displaystyle B_{\mu}^{4}$
$\displaystyle=(v^{2}n_{\mu}+\bar{v}_{\mu})\nabla\cdot v\,,$ $\displaystyle
B_{\mu}^{5}$ $\displaystyle=n_{\mu}v\cdot D^{\bot}s+D^{\bot}_{\mu}s\,,$
$\displaystyle B_{\mu}^{6}$ $\displaystyle=n_{\mu}v\cdot
D^{\bot}\tilde{n}+D^{\bot}_{\mu}\tilde{n}\,,$ $\displaystyle B_{\mu}^{7}$
$\displaystyle=n_{\mu}v\cdot D^{\bot}v^{2}+D^{\bot}_{\mu}v^{2}\,.$ (38)

Thus, $\Omega_{\mu\nu}v^{\mu}=0$ reduces to $\beta_{a}=0$. Our strategy for
solving these equations is as follows. First we solve $\beta_{a}=0$ with
$a=5,\ldots,7$ algebraically for $\partial_{s}g$, $\partial_{\tilde{n}}g$ and
$\partial_{v^{2}}g$. We then construct from the above solutions additional
equations $\kappa_{1}=0$, $\kappa_{2}=0$ and $\kappa_{3}=0$ by requiring that
mixed partial derivatives of $g$ are compatible.

Assuming that $f\neq 0$ (since if $f=0$ implies that $g$ is constant in which
case we get the trivial solution on account of the torsionless condition
$\partial_{\mu}n_{\nu}-\partial_{\nu}n_{\mu}=0$) we find from $\beta_{1}=0$
that

$f=f_{1}(s,\tilde{n})\partial_{v^{2}}G\\\ \qquad\hbox{or}\qquad
G=G_{1}(s,\tilde{n})+sG_{2}(\tilde{n},v^{2})\,.$ (39)

We refer to these two branches of solutions as branch A and branch B
respectively. For branch A one finds that the above solution automatically
sets $\beta_{2}=\beta_{3}=0$. One can then solve $\beta_{4}=0$ which gives

$f_{1}=\frac{1}{s\xi(\tilde{n})}\,,$ (40)

with $\xi$ an arbitrary function of $\tilde{n}$. The remaining non trivial
equations are $\kappa_{2}=0$ and $\kappa_{3}=0$. The solution to the former is

$G=H\left(s\xi(\tilde{n}),v^{2}\right)+\eta(s,\tilde{n})\,,$ (41)

with $H$ an arbitrary function of its two variables. The solution to the
remaining $\kappa_{3}=0$ is

$\eta=\eta_{1}\left(s\xi(\tilde{n})\right)+s\eta_{2}(\tilde{n})\,.$ (42)

It is then straightforward to go back and solve $\beta_{a}=0$ for
$a=5,\ldots,7$ for $g$ and obtain (after some relabeling)

$f=\frac{\partial_{v^{2}}H}{s\xi}\,,\qquad
g=\frac{v^{2}}{s\xi}\partial_{v^{2}}H-\frac{1}{2}\partial_{s\xi}H\,,\qquad
G=H+s\eta\,,$ (43)

where the arguments of the various functions are

$H=H(s\xi,\,v^{2})\,,\qquad\xi=\xi(\tilde{n})\,,\qquad\eta=\eta(\tilde{n})\,,$
(44)

and we have removed a constant, $g_{0}$, from $g$ since it does not contribute
to $\Omega$. Notice that in this case we have

$P=-H+s\xi\partial_{s\xi}H\,.$ (45)

Since $s\eta_{2}$ is not absolutely convex it does not contribute to the
pressure.

The strategy for solving the B branch is similar to that of the A branch. One
finds that the B branch splits into two branches. One of them is a special
case of the A branch solution. The other is given by

$f=f(v^{2}),\,\qquad g=v^{2}f(v^{2})-\frac{1}{2}\int^{v^{2}}f(x)dx,\,\qquad
G=-P_{0}+sJ(\tilde{n},v^{2})\,.$ (46)

In this case we find that

$P=P_{0}\,.$ (47)

It remains to treat the special case
$\partial_{v^{2}}G+2v^{2}\partial^{2}_{v^{2}}G=0$ in which case $Dv^{2}$
becomes an independent variable. This case can be solved by the same method as
the generic case and one finds that it leads to several branches of solutions
all of which coincide with or are a special case of (43). At the end of the
day, we find that (43) is valid for any $H$ as long as $\xi\neq 0$ and
$\partial_{v^{2}}H\neq 0$ and (46) is valid whenever
$\partial_{v^{2}}G+2v^{2}\partial^{2}_{v^{2}}G\neq 0$.

In the absence of additional symmetries we have from (26) and (12) that
$u^{\mu}=v^{\mu}$. In this case, for general values of $N$ in (9a) we find

$u^{\mu}g_{\mu\nu}u^{\nu}=N-v^{2}\,.$ (48)

Since the right-hand-side of (48) is not sign definite we can not enforce (25)
and then the resulting enstrophy current can not be conserved.333 Note that we
may also attempt the following: in place of (12) we can use $S^{\mu}=\sigma
u^{\mu}$ where $u^{\mu}$ is chosen such that $u^{\mu}\propto v^{\mu}$ with a
proportionality constant such that $u^{\mu}g_{\mu\nu}u^{\nu}=c_{0}$. With this
choice of $u^{\mu}$ we will get $S=0$ in (24) and so, should be able to
construct an enstrophy current as in (4) with $s$ replaced by $\sigma$ (and
$n$ replaced by $\nu$ with $N^{\mu}=\nu u^{\mu}$). The problem with this
construction is that there may exist solutions where locally $v^{2}>N$ and
also $v^{2}<N$. For such configurations, if we normalize the velocity
$u^{\mu}$ to take a constant value then we will end up with a complex valued
velocity field, c.f., (48). When we will discuss Lorentzian symmetry we will
see that the a construction of the type described in this footnote is viable.
Thus, we are forced to set $N=0$ in order to obtain a non trivial enstrophy
current. The resulting enstrophy current is then given by (4) with

$\Omega^{2}=\Omega_{\mu\nu}\Omega_{\rho\sigma}{h}^{\mu\rho}{h}^{\nu\sigma}\,,$
(49)

and $u^{\mu}=v^{\mu}$. Note that the contributions to $\Omega$ coming from $g$
in (33) drop off from the expression for $\Omega^{2}$ due to the non torsion
condition $dn=0$ and $h^{\mu\nu}n_{\nu}=0$. We have summarized our results in
tables 1 and 2.

While our result for the enstrophy current is very general, it is interesting
to study its behavior in several limiting cases where the Aristotelian
symmetry is lifted to one with some boost invariance. In particular, following
[6], we will be interested in equations of state where Lorentz invariance,
Galilean invariance, Carrollian invariance or Lifshitz invariance are present.
We now turn our attention to these non generic cases.

###  3.1 Recovering the Galilean invariant solution

We can use our generic results (43) or (46) to construct a Galilean covariant
enstrophy current so long as we restrict the above solutions to those which
possess Galilean symmetry and also ensure that the resulting expression for
the enstrophy current transforms covariantly under Galilean boosts.

Recall that the Galilean group is generated by the (massive) Bargmann algebra.
To enhance the generic fluid equations of motion to those of a Galilean
invariant fluid [13, 14, 15, 16, 17, 18], one has to relate the kinematic mass
term $\rho$ to the mass density of the fluid [6]

$\rho=mn$ (50)

where $n$ the particle number density and $m$ is the mass.

For the solution in (43) the identification given in (50) implies

$-2\partial_{v}^{2}H(s\xi,\,v^{2})=mn\,.$ (51)

This is a differential equation for both $H$ and $\xi$ and can be solved by
integrating over $v^{2}$ and expanding in a power series in $s$ and
$\tilde{n}$. We find

$H=H_{G}(s\tilde{n})-\frac{1}{2}ms\tilde{n}v^{2}\,,\qquad\xi=-\frac{m\tilde{n}}{2f_{0}}\,,$
(52)

which gives us

$f=f_{0}\,,\qquad
g=\frac{1}{2}f_{0}v^{2}+\frac{f_{0}}{m}H_{G}^{\prime}\,,\qquad
G=H_{G}(s\tilde{n})+s\eta(\tilde{n})-\frac{1}{2}ms\tilde{n}v^{2}\,,$ (53)

and a barotropic pressure term

$P(n)=-H_{G}(n)+nH_{G}^{\prime}(n)\,.$ (54)

The result (54) leads to the known approximately conserved enstrophy current
which exists in compressible barotropic flow.

Solving (50) for (46) yields

$f=f(v^{2})\,,\qquad g=v^{2}f(v^{2})-\frac{1}{2}\int^{v^{2}}f(x)dx\,,\qquad
G=-P_{0}+sJ_{G}(\tilde{n})-\frac{1}{2}mnv^{2}\,.$ (55)

We will see shortly that while (55) solves (51) and (16) it does not allow for
a Galilean covariant enstrophy current.

In order to construct a Galilean covariant enstrophy current we need to
identify a Galilean covariant velocity field, $u_{G}^{\mu}$, and ensure that
$\Omega^{2}$ is a scalar under Galilean boosts. Let us start with the former.
The natural velocity field to use in a Galilean invariant theory is one which
transforms covariantly under a change of reference frame. By this we mean the
following: if $u^{\mu}_{G}(\vec{v})$ specifies the velocity of a particle
moving with velocity $\vec{v}$, then it must be the case that when we
transform to a coordinate system moving at constant velocity $\vec{v}_{0}$
relative to the first,

$G^{\mu}{}_{\nu}(\vec{v}_{0})u^{\nu}_{G}(\vec{v})=u^{\nu}_{G}(\vec{v}+\vec{v}_{0})\,,$
(56)

with $G^{\mu}{}_{\nu}(\vec{v}_{0})$ representing a Galilean boost to the
reference frame moving at velocity $\vec{v}_{0}$. Equation (56) implies that

$u_{G}^{\mu}=v^{\mu}\,.$ (57)

Of course, (57) could have been obtained by considering the change in the
particles coordinates $X^{\mu}(\tau)$ relative to the Galilean invariant
proper time, or by taking the small velocity limit of a relativistic velocity
field.

Next, consider
$\Omega^{2}=h^{\mu\alpha}h^{\nu\beta}\Omega_{\mu\nu}\Omega_{\alpha\beta}$. If
$\Omega^{2}$ is a scalar it should be invariant under Galilean
transformations. Recall that a Galilean transformation on dynamical fields is
a coordinate transformation of the type (106) while a Galilean transformation
on the background fields $\bar{h}_{\mu\nu}$ and $\bar{n}^{\mu}$ involves, in
addition, a Milne transformation. The latter ensures that, as opposed to the
situation in an Aristotelian geometry, there is an equivalence class of time
directions: $\bar{n}_{1}^{\mu}\sim\bar{n}_{2}^{\mu}$, if
$n_{\mu}\bar{n}_{1}^{\mu}=n_{\mu}\bar{n}_{2}^{\mu}=-1$. Thus, in particular,
for a flat background geometry, we find that

$\bar{h}_{\mu\nu}\xrightarrow[Galilean]{}\bar{h}_{\mu\nu}+2\lambda^{2}n_{\mu}n_{\nu}$
(58)

where $\lambda_{\mu}$ is the boost parameter (e.g., in Cartesian coordinates
$\lambda_{\mu}=(0,\vec{v}_{0})$) and
$\lambda^{2}=\lambda_{\mu}\lambda_{\nu}h^{\mu\nu}$. See appendix A for a
concise summary or [19, 20, 21, 22, 23, 14, 24, 25] for an extended
discussion. Thus, while $u_{G}^{\mu}=v^{\mu}$ transforms covariantly under
Galilean boosts, $\bar{u}_{G\,\mu}=\bar{v}_{\mu}=\bar{h}_{\mu\nu}v^{\nu}$ does
not, implying that $\Omega_{\mu\nu}$ is not Galilean covariant.

A resolution to a problem of this type can be found in [14]. In Galilean
invariant theories the Christoffel connection is not the unique, symmetric,
metric compatible one. Rather, in order to define such a connection one needs,
at the very least, to introduce an additional one form $A^{G}_{\mu}dx^{\mu}$.
This one form is often identified with the gauge field associated with the
inherent $U(1)$ symmetry which leads to mass conservation in Galilean
theories. Metric compatibility then implies that $A_{\mu}^{G}$ does not
transform covariantly under Galilean transformations. Despite that, in [14] it
was shown that gauge invariant combinations constructed out of
$A_{\mu}^{G}+\bar{h}_{\mu\nu}u_{G}^{\nu}+\frac{1}{2}n_{\mu}v^{2}$ are Galilean
covariant.

So far, we have considered vanishing external sources for both the stress
tensor and conserved $U(1)$ currents, and we may continue to do so by choosing
a flat connection. The discussion in the previous paragraph suggests that in a
Galilean invariant theory, (33) must be replaced by

$\Omega=d\left(f^{G}\left(\bar{v}+A^{G}+\frac{1}{2}nv^{2}\right)\right)+d({g}^{G}n)\,,$
(59)

which forces $f^{G}=f_{0}$, a constant, on account of gauge invariance.
(Recall that $f^{G}$ and $g^{G}$ are functions of the entropy density, charge
density and velocity field and that $n=n_{\alpha}dx^{\alpha}$.) Since we can
always choose a gauge where $A^{G}=0$, we are free to use our generic results
as long as we restrict $f$ to be a constant. Surprisingly, this is precisely
the solution given in (53) with $g=\frac{1}{2}f_{0}v^{2}+{g}^{G}$ and
${g}^{G}=f_{0}H^{\prime}_{G}/m$. One can also choose $f$ to be a constant in
(55) in which case (55) becomes a special case of (53).

Let us summarize our findings. A Galilean equation of state implies that (53)
or (55) are valid expressions for constructing $\Omega_{\mu\nu}$. In order for
$\Omega_{\mu\nu}$ to be Galilean covariant we must use $f=f_{0}$ in which case
(55) becomes a special case of (53). In order to construct $\Omega^{2}$ we use
the Galilean invariant metric $h^{\mu\nu}$ (or $g_{A}^{\mu\nu}$ with $N=0$)
which yields (49). $\Omega^{2}$ is then a scalar on account that the
transformation of $\Omega_{\mu\nu}$ under Galilean boosts is proportional to
$n_{\mu}$, which therefore vanishes when contracted with $h^{\mu\nu}$. Finally
Galilean invariance also enforces that the velocity field in (4) is given by
$u_{G}^{\mu}$ with (57).

###  3.2 Recovering the Lorentz invariant solution

Following [6], the fluid equations given in (26) are Lorentz covariant
whenever

$\rho=\frac{sT+\mu n}{1-v^{2}}\,.$ (60)

Inserting the first branch of solutions, (43), into (60) we find that (43) is
restricted to take the form

${H}=H_{L}\left(\frac{s\xi}{\gamma}\right)-s\xi,\,\qquad\eta=\xi\,,$ (61)

with $\gamma=1/\sqrt{1-v^{2}}$. In terms of $f$, $g$ and $G$ the first branch
of solutions becomes (after some relabeling)

$f=-\frac{1}{2}\gamma H_{L}^{\prime}(s_{L}\xi(\tilde{n}))\,,\qquad
g=f\,,\qquad G=H_{L}\left(s_{L}\xi(\tilde{n})\right)\,,$ (62)

where we have defined

$s_{L}=s/\gamma$ (63)

and we note that we may write

$\tilde{n}=\frac{n_{L}}{s_{L}}\,.$ (64)

Indeed, as shown in [6] and as we will see shortly, $s_{L}$ and
$n_{L}=n/\gamma$ are the relativistic expressions for the entropy density and
$U(1)$ charge.

Going to the second branch of solutions, (46), we find that it takes the form

$f=f(v^{2})\,,\qquad g=v^{2}f(v^{2})-\frac{1}{2}\int^{v^{2}}f(x)dx\,,\qquad
G=-P_{0}+s_{L}J_{L}(\tilde{n})\,,$ (65)

under (60).

The Lorentz invariant metric is the Minkowski metric, or $g_{\mu\nu}$ with
$N=1$. The natural velocity field for a Lorentz invariant theory is given by

$u_{L}^{\mu}=\gamma v^{\mu}$ (66)

in which case the inviscid entropy current and charge current take the form
$S^{\mu}=s_{L}u_{L}^{\mu}$ and $N^{\mu}=n_{L}u_{L}^{\mu}$ respectively.
Clearly

$\bar{u}_{L\,\mu}=g_{\mu\nu}u_{L}^{\mu}=\gamma\bar{v}_{\mu}+\gamma n_{\mu}$
(67)

is also Lorentz covariant. Thus, the expression for $\Omega_{\mu\nu}$ given in
(33), should reduce to

$\Omega=d\left(f_{L}\bar{u}_{L}\right)$ (68)

with

${f_{L}}=f/\gamma,\,$ (69)

(and $\bar{u}_{L}=\bar{u}_{L\,\mu}dx^{\mu}$) if it is to be Lorentz covariant.
The second branch of solutions, (65), does not meet this criterion (unless
$f_{L}$ is constant) but the first branch of solutions, (62), does. Thus,
$\Omega_{\mu\nu}$ takes the form given in (68) with (69) and (62). To
construct (4) we use

$\Omega^{2}=\Omega_{\mu\nu}\Omega_{\rho\sigma}g_{L}^{\mu\rho}g_{L}^{\nu\sigma}$
(70)

with $g_{L}^{\mu\nu}$ given in (9b) and $u^{\mu}=u_{L}^{\mu}$ given in (66).
In this case, $g_{\mu\nu}u^{\mu}_{L}u^{\nu}_{L}=-1$ and therefore $S$ in (24)
clearly vanishes.

In a relativistic setting it is often convenient to work with temperature and
chemical potential instead of entropy density and charge density,

$T\gamma={T}_{L}=\frac{\partial
G}{\partial{s_{L}}}\,,\qquad\mu\gamma={\mu}_{L}=\frac{\partial G}{\partial
n_{L}}\,.$ (71)

It is straightforward to show that

$\frac{{T_{L}}}{\mu_{L}}=-\tilde{n}+\frac{\xi(\tilde{n})}{\xi^{\prime}(\tilde{n})}\,,$
(72)

implying that $\tilde{n}$ is a function of $T_{L}/\mu_{L}$. Further, if we
decompose

${f}_{L}={T_{L}}f_{r}\,,$ (73)

then

$f_{r}=-\frac{1}{2\xi-\tilde{n}\xi^{\prime}}$ (74)

is also a function of $T_{L}/\mu_{L}$.

To obtain a simple expression for the pressure consider

$H_{L}(x)=\int^{x}\frac{P(Q(-2y))}{y^{2}}dy\,,$ (75)

where $Q(P^{\prime}(x))=x$. Note that
$P=-G+{s_{L}}{T_{L}}+{\mu_{L}}{s_{L}}\tilde{n}$ is the pressure. In these
variables we find that

$Q(s\xi)={T}_{L}f_{r}$ (76)

implying

$P=P\left(T_{L}f_{r}\left(\frac{T_{L}}{\mu_{L}}\right)\right)$ (77)

which matches earlier results obtained for relativistic fluids [2, 5].

###  3.3 Carrollian symmetry

Carrollian invariance [26, 27] is perhaps the least familiar form of boost
invariance. The Carrollian algebra can be obtained by taking the $c\to 0$
limit of the Lorentzian algebra. That is, it describes dynamics when the
lightcone degenerates to a line. The most natural way to interpret this limit
is by considering the limiting case of Lorentz transformations whose velocity
parameter is much larger than the speed of light. To be explicit, consider

$t\to
t^{\prime}=\frac{t-\frac{\vec{\beta}\cdot\vec{x}}{c}}{\sqrt{1-\beta^{2}}}\,,\\\
\qquad\vec{x}\to\vec{x}^{\prime}=\frac{\vec{x}-\vec{\beta}ct}{\sqrt{1-\beta^{2}}}\,.$
(78)

Often, we attribute these transformations to the dynamics of a massive
particle: By identifying the velocity of the particle with the boost parameter
required to bring it to a reference frame where it is stationary (in space) we
obtain $\vec{\beta}=\vec{v}/c$. Following [27], we may use the same technique
to identify $\vec{\beta}=c\vec{v}/{v^{2}}$ for tachyonic particles by equating
the velocity of the tachyon with the boost parameter required to bring it to a
reference frame where it is stationary in time. Taking the $c/|\vec{v}|\to 0$
limit of these transformations leads to Carrollian boosts

$t^{\prime}=t-\frac{\vec{v}}{v^{2}}\cdot\vec{x}\,,\qquad\vec{x}^{\prime}=\vec{x}\,.$
(79)

We comment that it is also possible to take the $c\to 0$ limit of the Lorentz
transformations associated with subluminal velocities by scaling the velocity
with $c^{2}$ ($\vec{v}\to{c^{2}\vec{v}}/{v^{2}}$) as $c$ is taken to zero
resulting also in (79). See, e.g., [28, 29]. This limit is potentially
associated with the dynamics of massive particles trapped inside the lightcone
that has shrunk to the $t$ axis. We also note that, curiously, the Carrollian
algebra allows for a particular type of central extension in $2+1$ dimensions,
[30], whose implications on hydrodynamics has not been worked out, at least as
far as we know. It would be interesting to see whether this central charge
relates to enstrophy.444We thank W. Sybesma for pointing this out to us. We
leave this direction to future work.

In order to have Carrollian covariant fluid equations we must set (see [6])

$\rho=-\frac{sT+\mu n}{v^{2}}\,.$ (80)

The first branch of solutions (43) now reads

$G=H_{C}({s_{C}}\xi(\tilde{n}))\,,\qquad
f=\frac{1}{2}\frac{H_{C}^{\prime}({s_{C}}\xi(\tilde{n}))}{\sqrt{v^{2}}}\,,\qquad
g=0\,,$ (81)

and the second branch of solutions (46) is given by

$G=-P_{0}+s_{C}J_{C}(\tilde{n})\,,\qquad f=f(v^{2})\,,\qquad
g=v^{2}f(v^{2})-\frac{1}{2}\int^{v^{2}}f(x)dx\,.$ (82)

where $s_{C}=\sqrt{v^{2}}s$. (Note that (82) satisfies
$\rho+2v^{2}\partial_{v^{2}}\rho=0$ even though we assumed that it should not
vanish. Nevertheless, it is still a solution to $i_{u}\Omega=0$.)

As was the case for the Galilean theory, to ensure that the enstrophy current
is Carrollian covariant we must identify a Carrollian covariant velocity field
$u^{\mu}_{C}$ and show that $\Omega^{2}$ behaves as a scalar under Carrollian
transformations. There are many equivalent ways of constructing a Carrollian
covariant velocity field. Following the discussion for the Galilean covariant
theory, we will determine $u^{\mu}_{C}$ by requiring that it is compatible
with Carrollian addition of velocities.

$\vec{v}^{\prime}=\frac{\vec{v}}{1-\vec{v}\cdot\frac{\vec{v}_{0}}{|\vec{v}_{0}^{2}|}}\,,$
(83)

where $\vec{v}_{0}$ is the boost parameter of the Carrollian transformation.
It is now straightforward to show that

${u}_{C}^{\mu}(v_{i})=\frac{1}{\sqrt{v^{2}}}\left(1,v^{i}\right)=\frac{v^{\mu}}{\sqrt{v^{2}}}$
(84)

is the unique vector that satisfies

$C^{\mu}{}_{\nu}(\vec{v}_{0})u_{C}^{\nu}(\vec{v})=u_{C}^{\mu}(\vec{v}^{\prime})\,,$
(85)

with $C^{\mu}{}_{\nu}$ a Carrollian transformation with boost parameter
$\vec{v}_{0}$. (The same result can be obtained by considering the $c\to 0$
limit of the Lorentz invariant $u_{L}^{\mu}=\frac{\partial
X^{\mu}}{\partial\tau}$ where $d\tau=\sqrt{-c^{2}dt^{2}+|d\vec{x}|^{2}}$). It
follows that

$\bar{u}_{C\,\mu}=\bar{h}_{\mu\nu}u^{\nu}_{C}=\left(0,\,\frac{v_{i}}{\sqrt{v^{2}}}\right)=\frac{\bar{v}_{\mu}}{\sqrt{v^{2}}}$
(86)

transforms covariantly under Carrollian boosts.

To determine whether $\Omega^{2}$ is a Carrollian scalar we must first
determine the Carrollian transformation laws of the geometric data
$\bar{h}_{\mu\nu}$, $\bar{n}^{\mu}$, $h^{\mu\nu}$ and $n_{\mu}$. Similar to
the situation in a Newton-Cartan geometry, the geometry associated with
Carrollian invariant theories is determined by the set $\bar{h}_{\mu\nu}$,
$\bar{n}^{\mu}$ and $n_{\mu}$ where all $n_{\mu}$’s satisfying
$n_{\mu}\bar{n}^{\mu}=-1$ are equivalent. This equivalence is made manifest by
introducing a Carrollian version of the Milne transformation, [11], which we
will refer to as a C-Milne transformation for short.555 What we refer to as a
Carrollian version of the Milne transformation was termed a local Carrollian
transformations in [11]. Our construction of these transformations is somewhat
different from that of [11].  In analogy to the situation in Newton-Cartan
geometries, a Carrollian boost is a combination of a coordinate transformation
and a C-Milne transformation. Likewise, $\bar{h}_{\mu\nu}$ (and
$\bar{n}^{\mu}$) are Carrollian covariant while $h^{\mu\nu}$ (and $n_{\mu}$)
are not. We conclude that if we construct $\Omega^{2}$ from the inverse metric
$h^{\mu\nu}$, it is bound to behave non covariantly under Carrollian
transformations. We refer the reader to Appendix A for a concise summary of
Carrollian geometry or to [11] for an extended discussion.

While $h^{\mu\nu}$ and $n_{\mu}$ are not Carrollian covariant, we may
construct a modified inverse metric
$\tilde{h}^{\mu\nu}=h^{\mu\nu}-M_{C}^{\mu}\bar{n}^{\nu}-M_{C}^{\nu}\bar{n}^{\mu}+M_{C}^{2}\bar{n}^{\mu}\bar{n}^{\nu}$
and a modified one-form $\tilde{n}_{\mu}=n_{\mu}-\bar{h}_{\mu\nu}M^{\nu}_{C}$
which are Carrollian covariant. Here $M_{C}^{\mu}$ is an additional vector
field available in geometries associated with Carrollian symmetry, similar to
the gauge field $A^{G}_{\mu}$ which appears in Newton-Cartan geometries. It
originates in the ambiguity in defining a symmetric, metric compatible
connection. The transformation laws for this additional field, $M_{C}^{\mu}$,
are given in (132) and (133). Using these transformation laws it is
straightforward to check that, indeed, $\tilde{h}^{\mu\nu}$ and
$\tilde{n}_{\mu}$ are covariant tensors under Carrollian transformations. By
replacing ${h}^{\mu\nu}$ with $\tilde{h}^{\mu\nu}$ and $n_{\mu}$ with
$\tilde{n}_{\mu}$ throughout this section, (and by replacing the connection
(10) with an $\tilde{h}^{\mu\nu}$ compatible one), $\Omega^{2}$ will be a
Carrollian scalar.

Going back to (33) we find that $f$ and $g$ must be such that (33) takes the
form

$\Omega=d\left(f_{C}\bar{u}_{C}\right)+d\left(g_{C}(n-M_{C})\right)$ (87)

where $f_{C}=f(s_{C},\tilde{n})\sqrt{v^{2}}$ and
$\bar{u}_{C}=u_{C\,\mu}dx^{\mu}$. This is naturally satisfied by the first
branch of solutions, (81), and also by the second branch of solutions, (82),
once we set $f=f_{0}/\sqrt{v^{2}}$. Note that by doing so, the second branch
of solutions is a special case of the first. One can now follow the same
analysis as in the relativistic case to obtain
$f_{C}={T_{C}}f_{c}({T_{C}}/{\mu_{C}})$ and $P=P({T_{C}}f_{c}(T_{C}/\mu_{C}))$
with ${T}_{C}=T/\sqrt{v^{2}}$ and ${\mu}_{C}=\mu/\sqrt{v^{2}}$.

Let us summarize. The Carrollian covariant 2-form
$\Omega=\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$ satisfying (16) is given by (87). If
we raise its indices using the Carrollian covariant metric
$\tilde{h}^{\mu\nu}$ then $\Omega^{2}$ is invariant under Carrollian boosts.

###  3.4 Lifshitz symmetry

Apart from enhancing the (spacetime) translation and (spatial) rotation
invariant dynamics to a boost invariant one, it is also possible to add a
scaling symmetry. Lifshitz symmetry is a scaling symmetry whereby the time and
space coordinates are scaled differently, $t\to\lambda^{z}t$,
$\vec{x}\to\lambda\vec{x}$. Once again, following [6], Lifshitz scale symmetry
implies that

$dP=zG+(z-1)\rho v^{2}$ (88)

with $d$ the number of space dimensions ($d=2$ in this work). Equation (88)
amounts to

$G=s^{\frac{2+z}{2}}G_{0}\left(\tilde{n},\,v^{2}s^{1-z}\right)\,.$ (89)

This implies that in a Lifshitz invariant theory the solution to (16) takes
the form (43) with

$H=-s\xi(\tilde{n})+\left(s\xi(\tilde{n})\right)^{\frac{2+z}{2}}h\left(v^{2}\left(s\xi(\tilde{n})\right)^{1-z}\right)\,,\qquad\eta=\xi\,,$
(90)

or (46) with

$J=(v^{2})^{\frac{z}{2(z-1)}}j(\tilde{n})\,,\qquad P_{0}=0\,,$ (91)

for $z\neq 1$. When $z=1$ the solution (46) is trivial. (Curiously, if we set
$z=-2$ then $P_{0}\neq 0$ is allowed.)

One can now impose Lifshitz invariance in addition to boost invariance.
Lorentz invariance is compatible only with $z=1$ scaling leading to

$f_{L}=-\frac{3}{4}\sqrt{s_{L}\xi(\tilde{n})}H_{0}\,,\qquad
g/\gamma=f_{L}\,,\qquad G=(s_{L}\xi(\tilde{n}))^{3/2}H_{0}\,,$ (92)

where $H_{0}$ is a constant and we have omitted constant terms in $g$ which do
not contribute to the enstrophy current. Galilean invariant fluids are
compatible only with $z=2$ (this follows by requiring both (88) and (50) as
has been pointed out in [6, 31]) leading to

$f=f_{0}\,,\qquad g=\frac{1}{2}f_{0}v^{2}+\frac{2f_{0}H_{0}}{m}n\,,\qquad
G=n^{2}H_{0}-\frac{1}{2}mnv^{2}\,.$ (93)

Finally, equations (16) are satisfied by Carrollian invariant Lifshitz fluids
if

$f_{C}=\frac{1}{2}\frac{(2+z)}{(1+z)}\left(s_{C}\xi(\tilde{n})\right)^{\frac{1}{1+z}}H_{0}\,,\qquad
g=0\,,\qquad G=\left(s_{C}\xi(\tilde{n})\right)^{\frac{2+z}{1+z}}H_{0}\,.$
(94)

A summary of these results can be found in tables 1 and 2.

##  4 Solving $i_{u}\Omega=0$ for incompressible flow

Strictly speaking, all fluids are compressible. Yet, most day-to-day fluid
flows, from the stream of water in a garden hose to automotive aerodynamics,
are described by the incompressible Navier-Stokes equations. Indeed, under the
assumption of subsonic or low Mach number flow, the Galilean invariant
compressible fluid equations of motion reduce to the incompressible Navier-
Stokes equations. This limiting behavior makes the latter a robust and well
studied approximation of a wide variety of commonplace physical phenomena.

Of particular relevance to the current work is that incompressible Galilean
fluid flow supports an enstrophy current regardless of the equation of state.
As we will see shortly, the low Mach number limit of non frame invariant
fluids also leads to incompressible flow which also supports an enstrophy
current independent on the equation of state.

To start, let us consider the fluid equations (32) with the rescaling

$v^{i}\to V_{0}\hat{v}^{i}\,,\qquad t\to\frac{L_{0}}{V_{0}}\hat{t}\,,\qquad
x\to L_{0}\hat{x}\,,$ (95)

where hatted quantities are dimensionless. Inserting (95) into (32) we find

$\displaystyle\begin{split}0&=\hat{a}_{\mu}+\frac{1}{V_{0}^{2}}\frac{1}{\rho}\left(\left(\frac{\partial{P}}{\partial{s}}\right)_{{n},\,v^{2}}\hat{D}^{\bot}_{\mu}{s}+\left(\frac{\partial{P}}{\partial{n}}\right)_{s,\,v^{2}}\hat{D}^{\bot}_{\mu}{n}\right)+\frac{1}{\rho}\left(\frac{\partial{P}}{\partial{v^{2}}}\right)_{s,\,n}\hat{D}^{\bot}_{\mu}{\hat{v}^{2}}+\frac{\hat{\bar{v}}_{\mu}}{\rho}\left(\rho\hat{\Theta}+\hat{v}^{\nu}\hat{\partial}_{\nu}\rho\right)\\\
0&=\hat{D}n+n\hat{\Theta}+\hat{v}^{\mu}\hat{D}^{\bot}_{\mu}n\\\
0&=\hat{D}s+s\hat{\Theta}+\hat{v}^{\mu}\hat{D}^{\bot}_{\mu}s\,.\end{split}$
(96)

where hatted quantities are dimensionless versions of their unhatted
counterparts, viz., $\hat{\Theta}=\hat{\nabla}\cdot\hat{v}$.

The speed of sound of the fluid may be computed by considering linearized
perturbations of a uniform, equilibrated, configuration (see [6]). At low
velocities it is given by

$V_{s}^{2}=\frac{1}{\rho}\left(\left(\frac{\partial P}{\partial
s}\right)_{s}s+\left(\frac{\partial P}{\partial n}\right)_{s}n\right)\,.$ (97)

Expanding the equations of motion and dynamical variables, $s$, $n$ and
$\hat{v}^{i}$ around small $M=V_{0}/V_{s}$ we find

$\displaystyle\begin{split}0&=\left(\frac{\partial P^{(0)}}{\partial
s^{(0)}}\right)_{n}^{(0)}\hat{D}_{\mu}^{\bot}s^{(0)}+\left(\frac{\partial
P^{(0)}}{\partial n^{(0)}}\right)_{s}^{(0)}\hat{D}_{\mu}^{\bot}n^{(0)}\\\
0&=\hat{D}n^{(0)}+n^{(0)}\hat{\Theta}^{(0)}+\hat{v}^{(0)\mu}\hat{D}_{\mu}^{\bot}n^{(0)}\\\
0&=\hat{D}s^{(0)}+s^{(0)}\hat{\Theta}^{(0)}+\hat{v}^{(0)\mu}\hat{D}_{\mu}^{\bot}s^{(0)}\\\
\end{split}$ (98)

where we have defined,

$s=s^{(0)}+M^{2}s^{(2)}+\ldots,\,\qquad
n=n^{(0)}+M^{2}n^{(2)}+\ldots,\,\qquad\hat{v}^{2}=V_{0}^{2}\left(\hat{v}^{(0)}\right)^{2}+\ldots,\,$
(99)

and

$P=P^{(0)}\left(s^{(0)},\,n^{(0)}\right)+M^{2}\left(P^{(2)}\left(s^{(0)},\,n^{(0)};\,s^{(2)},\,n^{(2)}\right)+V_{s}^{2}P_{v^{2}}^{(2)}\left(s^{(0)},\,n^{(0)}\right)\left(\hat{v}^{(0)}\right)^{2}\right)+\ldots\,.$
(100)

Further assuming that the particle number is constant to leading order in $M$,

$\hat{D}n^{(0)}=0\,,\qquad\hat{D}_{\mu}^{\bot}n^{(0)}=0\,,$ (101)

implies that

$\hat{\Theta}^{(0)}=0\,,\qquad\hat{D}_{\mu}^{\bot}s^{(0)}=0\,,\qquad\hat{D}s^{(0)}=0\,,$
(102)

and therefore that $\rho^{(0)}$ and $P^{(0)}$ are constant as well. The
leading order equations for the velocity field now become

$\displaystyle\begin{split}0&=\hat{a}^{(0)}_{\mu}+\frac{1}{{\rho}^{(0)}}\left(\hat{D}_{\mu}^{\bot}{P}^{(2)}\right)+\frac{P_{v^{2}}^{(2)}}{\rho^{(0)}}\hat{D}_{\mu}^{\bot}\left(\hat{v}^{(0)}\right)^{2}\,,\\\
0&=\hat{\Theta}^{(0)}\,.\end{split}$ (103)

Note that (103) is a set of $3$ equations for three unknowns: $\hat{v}^{(0)i}$
and $\hat{P}^{(2)}$.

We can now go through an analysis similar to that of the previous section, in
order to obtain an enstrophy current which is independent of the equation of
state. That is, look for an
$f(\hat{s}^{(2)},\,\hat{n}^{(2)},\left(\hat{v}^{(0)}\right)^{2})$ and a
$g(\hat{s}^{(2)},\,\hat{n}^{(2)},\left(\hat{v}^{(0)}\right)^{2})$, defined in
(33), which solve (16b) under the equations of motion (103). We find

$f=f(v^{2})\,,\qquad
g=f(v^{2})v^{2}+\left(\frac{P_{v^{2}}^{(2)}}{\rho^{(0)}}-\frac{1}{2}\right)\int
f(v^{2})dv^{2}\,.$ (104)

The result presented in (104) is inline with the known behavior of
incompressible Galilean invariant fluids once we enforce $f=f_{0}$ a constant
due to Galilean invariance. The velocity field of a relativistic fluid flowing
subsonically is usually too low to exhibit relativistic effects and so it is
less interesting from a physical standpoint. It is even less clear how
subsonic Carrollian flow would manifest.

##  5 Summary

Our main result in this work has been to provide an operative technique to
compute a putative enstrophy current in $2+1$ dimensional flow with varying
amounts of symmetry. We used our technique to compute the enstrophy current of
a non frame invariant fluid both for a generic flow, in which case the
enstrophy current exists only for special equations of state, and for
incompressible flow where the equation of state is unconstrained. By taking
various limits of this result we managed to recover or discover how the
enstrophy current behaves in Galilean, Lorentz and Carrollian invariant fluids
and in fluids with an additional Lifshitz scale symmetry. Our results are
summarized in tables 1 and 2.

Flow type | $f$ | $\bar{u}_{\mu}$
---|---|---
Non frame invariant (I) | $\frac{1}{s\xi}\left(\frac{\partial G}{\partial v^{2}}\right)$ | $(0,v_{i})$
Non frame invariant (II) | $f(v^{2})$ | $(0,v_{i})$
Galilean | $f_{0}$ | $(0,v_{i})$
Lorentzian | $-\frac{1}{2}\left(\frac{\partial G}{\partial x}\right)\Big{|}_{x=s\xi}$ | $\frac{1}{\sqrt{1-v^{2}}}(-1,v_{i})$
Carrollian | $\frac{1}{2}\left(\frac{\partial G}{\partial x}\right)\Big{|}_{x=s\xi}$ | $\frac{1}{\sqrt{v^{2}}}(0,v_{i})$
Incompressible | $f(v^{2})$ | $(0,v_{i})$
Table 1:  Values of $f$ and $\bar{u}_{\mu}$ which determine $\Omega_{\mu\nu}=\partial_{\mu}(f\bar{u}_{\nu})-\partial_{\nu}\left(f\bar{u}_{\mu}\right)$ from which a conserved enstrophy current can be constructed and the equation number where the value of $f$ was determined. Here, $G$ is a free energy related to the pressure, $P$, via $G+P=sT+n\mu$ with $s$ and $n$ the entropy density and charge density, and $T$ and $\mu$ the temperature and chemical potential. The explicit expression for $G$ in each of the above cases can be found in table 2 and can be obtained from equations (43), (46), (53), (62), (81) and (104) respectively. Of particular relevance to us is its dependence on $s\xi$ where $\xi$ is an arbitrary function of $\tilde{n}=n/s$. Flow type | $G$ | $u^{\mu}$ | $g^{\mu\nu}$
---|---|---|---
Non frame invariant (I) | $H(s\xi(\tilde{n}),\,v^{2})+s\eta(\tilde{n})$ | ${(1,v^{i})}$ | $h^{\mu\nu}$
Non frame invariant (II) | $-P_{0}+sJ(\tilde{n},\,v^{2})^{\dagger}$ | ${(1,v^{i})}$ | $h^{\mu\nu}$
Galilean | $H_{G}(s\tilde{n})+s\eta(\tilde{n})-\frac{1}{2}ms\tilde{n}v^{2}$ | $(1,v^{i})$ | $h^{\mu\nu}$
Lorentzian | $H_{L}(s\xi(\tilde{n}))$ | $\frac{(1,v^{i})}{\sqrt{1-v^{2}}}$ | $h^{\mu\nu}-\bar{n}^{\mu}\bar{n}^{\nu}$
Carrollian | $H_{C}(s\xi(\tilde{n}))$ | $\frac{(1,v^{i})}{\sqrt{v^{2}}}$ | ${h^{\mu\nu}}^{\ddagger}$
Non frame invariant Lifshitz (I) | $\left(s\xi(\tilde{n})\right)^{\frac{2+z}{2}}h\left(v^{2}\left(s\xi(\tilde{n})\right)^{1-z}\right)$ | ${(1,v^{i})}$ | $h^{\mu\nu}$
Non frame invariant Lifshitz (II) ($z\neq 1$) | $-P_{0}+s\left(v^{2}\right)^{\frac{z}{2(z-1)}}j(\tilde{n})$ | ${(1,v^{i})}$ | $h^{\mu\nu}$
Galilean Lifshitz ($z=2$) | $n^{2}H_{0}-\frac{1}{2}mnv^{2}$ | $(1,v^{i})$ | $h^{\mu\nu}$
Lorentzian Lifshitz ($z=1$) | $\left(s\xi(\tilde{n})\right)^{\frac{3}{2}}H_{0}$ | $\frac{(1,v^{i})}{\sqrt{1-v^{2}}}$ | $h^{\mu\nu}-\bar{n}^{\mu}\bar{n}^{\nu}$
Carrollian Lifshitz | $\left(s\xi(\tilde{n})\right)^{\frac{2+z}{1+z}}H_{0}$ | $\frac{(1,v^{i})}{\sqrt{v^{2}}}$ | ${h^{\mu\nu}}^{\ddagger}$
Table 2:  Constraints on the form of the free energy $G$, the velocity field,
$u^{\mu}$ and
$\Omega^{2}=\Omega_{\mu\nu}\Omega_{\rho\sigma}g^{\mu\rho}g^{\nu\sigma}$ needed
in order to construct a conserved enstrophy current
$J^{\mu}=q\left(\frac{s}{n}\right)\frac{\left(\Omega^{2}\right)^{\alpha}}{s^{2\alpha-1}}u^{\mu}$.
Here, $G$ is a free energy related to the pressure, $P$, via $G+P=sT+n\mu$
with $s$ and $n$ the entropy density and charge density, and $T$ and $\mu$ the
temperature and chemical potential. We have also used the shorthand
$\tilde{n}=n/s$. The values of $G$ in the table can be found in (43), (46),
(53), (62), (81), (89), (46) with (91), (92), (93) and (94) respectively. We
have not included an entry for incompressible flow where the equation of state
is arbitrary and the velocity field and inverse metric take on their non frame
invariant, Lorentz invariant, Galilean invariant, or Carrollian invariant
form.
† $J$ must satisfy $\partial_{v^{2}}G+2v^{2}\partial_{v^{2}}G\neq 0$.
‡ This is the expression for the inverse metric after setting $M^{C}=0$.
Otherwise the inverse metric is given by (9d).

Carrollian invariance is perhaps the least familiar limit of Lorentz
invariance. Recall that the Carrollian invariant frame transformation is
obtained by taking the $c\to 0$ limit of Lorentz transformations for tachyonic
observers moving at superluminal velocities. Going on a slight detour, we note
that it is also possible to take the Carrollian limit, where the lightcone
collapses to the time axis, in such a way that velocities of massive particles
vanish sufficiently fast so as to retain some of their dynamics. If we take
the $c=0$ limit of Lorentz transformations associated with observers moving at
subluminal velocities and make the replacement $\vec{v}\to\vec{\nu}c^{2}$ then
we obtain a Carrollian transformation. It would be interesting to find a
dimensionless control parameter about which this limit could be expanded. In
any case, in this limit we are describing the dynamics of observers trapped
inside the lightcone. (Note that one can take the same type of limit to
describe tachyonic observers trapped on the $t=0$ plane when taking the
Galilean, $c\to\infty$, limit.)

Keeping track of factors of $c$ one finds that the velocity field associated
with particles trapped inside the shrinking lightcone is given by
$u^{\mu}=(1,0,0,0)+\mathcal{O}(c^{2})$ and $\bar{u}_{\mu}=\mathcal{O}(c^{2})$.
However, when taking the same $c\to 0$ limit of the stress tensor for
subluminal fluid motion one finds a non trivial dependence on velocity $v_{i}$
due to cancellation of factors of $c^{2}$ in the stress tensor and in
subleading components of $u^{\mu}$ and $\bar{u}_{\mu}$. The dynamics of such
gasses have been described in [28]. Since the Carrollian invariant velocity
field in this case is constant, $u^{\mu}=(1,0,0,0)$, it is not clear if there
is a sense in which there exists an enstrophy current even in this somewhat
degenerate setting.

So far, we have only considered an approximately conserved enstrophy current.
In Galilean invariant fluids the enstrophy current given by (1) has a negative
definite divergence once dissipative corrections are taken into account [12].
This property, together with energy conservation, leads to an inverse energy
cascade in 2+1 dimensional turbulence [1]. It is unclear at this point whether
one can systematically construct an Aristotelian enstrophy current with a
sign-definite divergence. If such a construction exists, it will shed light on
the role of enstrophy in 2+1 dimensional turbulent flow with varying symmetry.

The existence of a relativistic enstrophy current in $2+1$ dimensional flow,
implies the existence of a dual quantity in a holographic description of fluid
flow in $3+1$ dimensional gravity. Like entropy, the gravitational
manifestation of enstrophy may persist beyond asymptotically AdS black brane
geometries. We leave such issues for future work.

## Acknowledgments

We would like to thank A. Frishman for useful discussions and W. Sybesma for
discussions and comments on a draft of this manuscript. NPF is supported by
the European Commission through the Marie Sklodowska-Curie Action UniCHydro
(grant agreement ID: 886540). AY is supported in part by an Israeli Science
Foundation excellence center grant 2289/18 and a Binational Science Foundation
grant 2016324.

##  Appendix A Boost invariance

In this section we briefly review some salient features of Galilean and
Carrollian boost invariance paraphrasing the results of [14, 25] and [11].

###  A.1 Galilean boosts

The Lorentz group is, by definition, represented by those coordinate
transformations under which the Minkowski metric is invariant. In a similar
vein, a representation of the Galilean group may be defined as those
transformations under which the flat spatial metric and its accompanying
geometric data remain invariant. Recall that an Aristotelian geometry is given
by a degenerate inverse metric, $h^{\mu\nu}$, satisfying
$h^{\mu\nu}n_{\nu}=0$, and a preferred time direction $\bar{n}^{\mu}$
normalized such that $\bar{n}^{\mu}n_{\mu}=-1$. From these one may construct
the metric $\bar{h}_{\mu\nu}$ as discussed in section 2. A flat Aristotelian
geometry, in Cartesian coordinates, is given by (7).

A Newton-Cartan geometry is also equipped with a degenerate inverse metric
${h}^{\mu\nu}$ and its associated eigenvector $n_{\mu}$, but instead of a time
direction one considers an equivalence class of time directions
$\bar{n}^{\mu}$ where $\bar{n}_{1}^{\mu}\sim\bar{n}_{2}^{\mu}$ as long as
$\bar{n}_{1}^{\mu}n_{\mu}=\bar{n}_{2}^{\mu}n_{\mu}=-1$. In addition, Newton-
Cartan geometry possesses additional data without which a unique, symmetric,
metric compatible and Galilean covariant connection can not be specified. The
minimal requirements for defining such a connection is to introduce a covector
$M^{G}_{\mu}$ which transforms appropriately under Galilean boosts. Often,
$M^{G}_{\mu}$ is identified with the gauge field $A_{\mu}^{G}$ associated with
particle number conservation. In what follows we will elaborate on the
equivalence class associated with time directions mentioned above and on the
roles played by $M^{G}_{\mu}$ and $A_{\mu}^{G}$.

To make the equivalence $\bar{n}_{1}^{\mu}\sim\bar{n}_{2}^{\mu}$ manifest we
allow the geometric data to transform under a Milne transformation
parameterized by a covector $\psi_{\nu}$,

$\bar{n}^{\mu}\xrightarrow[G-Milne]{}\bar{n}^{\mu}+h^{\mu\nu}\psi_{\nu}\,.$
(105a) The prefactor ‘$G$’ is a reminder that we are referring to the Galilean
version of the Milne transformation, distinct from its Carrollian version to
be discussed in the next section. The transformation (105a) implies
$\bar{h}_{\mu\nu}\xrightarrow[G-Milne]{}\bar{h}_{\mu\nu}+n_{\mu}P^{\alpha}_{\nu}{\psi}_{\alpha}+n_{\nu}P^{\alpha}_{\mu}{\psi}_{\alpha}+\psi^{2}n_{\mu}n_{\nu}\,,$
(105b) with $\psi^{2}=h^{\mu\nu}\psi_{\mu}\psi_{\nu}$, and
$P^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\bar{n}^{\mu}n_{\nu}$.

The inverse metric $h^{\mu\nu}$ and its eigenvector $n_{\mu}$ are taken to be
inert under Milne transformations.

As mentioned earlier, the Galilean group can be represented by those
coordinate transformations $x^{\mu}\rightarrow x^{\prime\,\mu}(x)$ and Milne
transformations with parameter $\psi_{\nu}$ which keep the flat Cartesian
Newton-Cartan geometric data (7) invariant. The Galilean boosts are a subset
of these transformations satisfying

$x^{\prime\,\mu}=(t,\vec{x}-\vec{v}_{0}\,t)\,,\qquad\psi_{\mu}=(0,\vec{v}_{0})\,,$
(106)

with constant $\vec{v}_{0}$. In general coordinates, Galilean boosts are given
by

$G^{\mu}{}_{\nu}\equiv\partial x^{\prime\,\mu}/\partial
x^{\nu}=\delta^{\mu}_{\nu}+h^{\mu\alpha}\lambda_{\alpha}n_{\nu}\,,\qquad\psi_{\mu}=\lambda_{\mu}\,,$
(107)

where $\lambda_{\mu}$ is spacetime dependent and reduces to
$\lambda_{\mu}=(0,\vec{v}_{0})$ in a Cartesian coordinate system.

Coordinate transformations associated with $G^{\mu}{}_{\nu}$ defined in (107)
act on tensors in the standard way,

$\displaystyle\begin{split}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}&\xrightarrow[Coordinate]{}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}+\bar{\lambda}^{\mu_{1}}n_{\alpha}T^{\alpha\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}+\dots+\bar{\lambda}^{\mu_{p}}n_{\alpha}T^{\mu_{1}\dots\alpha}{}_{\nu_{1}\dots\nu_{q}}\\\
&-n_{\nu_{1}}\bar{\lambda}^{\alpha}T^{\mu_{1}\dots\mu_{p}}{}_{\alpha\dots\nu_{q}}-\dots-
n_{\nu_{q}}\bar{\lambda}^{\alpha}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\alpha}+\ldots\end{split}$
(108)

where we have defined $\bar{\lambda}^{\mu}=h^{\mu\alpha}\lambda_{\alpha}$ and
the last $\ldots$ denote nonlinear terms in $\lambda$, e.g.,

$\displaystyle\begin{split}T^{\mu\nu}&\xrightarrow[Coordinate]{}T^{\mu\nu}+\bar{\lambda}^{\mu}n_{\alpha}T^{\alpha\nu}+\bar{\lambda}^{\nu}n_{\alpha}T^{\mu\alpha}+\bar{\lambda}^{\mu}\bar{\lambda}^{\nu}(T^{\alpha\beta}n_{\alpha}n_{\beta})\\\
O_{\mu\nu}&\xrightarrow[Coordinate]{}O_{\mu\nu}-n_{\mu}\bar{\lambda}^{\alpha}O_{\alpha\nu}-n_{\nu}\bar{\lambda}^{\alpha}O_{\mu\alpha}+n_{\mu}n_{\nu}(O_{\alpha\beta}\bar{\lambda}^{\alpha}\bar{\lambda}^{\beta})\,.\end{split}$
(109)

We define a tensor to be Galilean covariant if it transforms as in (108) under
Galilean boosts, (107). Thus, $n_{\mu}$ and $h^{\mu\nu}$ are Galilean
covariant (and also invariant under Galilean transformations). Contrarily,
$\bar{n}^{\mu}$ and $\bar{h}^{\mu\nu}$ do not transform covariantly under
Galilean boosts. Instead we find

$\bar{n}^{\mu}\xrightarrow[Galilean]{}\bar{n}^{\mu}\,,\qquad\bar{h}_{\mu\nu}\xrightarrow[Galilean]{}\bar{h}_{\mu\nu}+2\lambda^{2}n_{\mu}n_{\nu}\,,$
(110)

where $\lambda^{2}=\lambda_{\alpha}\lambda_{\beta}h^{\alpha\beta}$.

As opposed to a Lorentzian geometry, in Newton-Cartan geometry, metric
compatibility (with respect to $h^{\mu\nu}$ and $n_{\mu}$) does not uniquely
specify the symmetric part of the connection $\Gamma_{\mu\nu}^{\rho}$. In the
torsionless case, a computation similar to the one carried out in [14] leads
to

$\Gamma_{G}^{\mu}{}_{\nu\rho}=-\bar{n}^{\mu}\partial_{\rho}n_{\nu}+\frac{1}{2}h^{\mu\sigma}\left(\partial_{\nu}\bar{h}_{\rho\sigma}+\partial_{\rho}\bar{h}_{\nu\sigma}-\partial_{\sigma}\bar{h}_{\nu\rho}\right)+h^{\mu\sigma}n_{(\nu}F_{\rho)\sigma}$
(111)

with $F^{G}=dM_{(G)}$. In order to ensure that the covariant derivative
associated with (111) is compatible with Galilean invariance we must associate
to $M^{G}_{\mu}$ a Milne transformation of the form

$M^{G}_{\mu}\xrightarrow[G-Milne]{}M^{G}_{\mu}+P^{\alpha}_{\mu}{\psi}_{\alpha}+\frac{1}{2}n_{\mu}\psi^{2}\,.$
(112)

We note in passing that Galilean covariance of the new connection (111) can be
made manifest by rewriting it in the form

$\Gamma_{G}^{\mu}{}_{\nu\rho}=-\tilde{n}^{\mu}\partial_{\rho}n_{\nu}+\frac{1}{2}h^{\mu\sigma}\left(\partial_{\nu}\tilde{h}_{\rho\sigma}+\partial_{\rho}\tilde{h}_{\nu\sigma}-\partial_{\sigma}\tilde{h}_{\nu\rho}\right)\,,$
(113)

where

$\tilde{n}^{\mu}=\bar{n}^{\mu}-h^{\mu\sigma}M^{G}_{\sigma}\,,\qquad\tilde{h}_{\mu\nu}=\bar{h}_{\mu\nu}-n_{\mu}P_{\nu}^{\alpha}M^{G}_{\alpha}-n_{\nu}P^{\alpha}_{\mu}M^{G}_{\alpha}+(M^{G})^{2}n_{\mu}n_{\nu}\,,$
(114)

and $(M^{G})^{2}=M^{G}_{\mu}M^{G}_{\nu}h^{\mu\nu}$. It is straightforward to
check that $\tilde{n}^{\mu}$ and $\tilde{h}_{\mu\nu}$ are Milne invariant and
therefore also Galilean covariant. For completeness we note that one may use
(114) to define the connection

$\tilde{\Gamma}_{G}^{\mu}{}_{\nu\rho}=-\tilde{n}^{\mu}\partial_{\rho}n_{\nu}+\frac{1}{2}h^{\mu\sigma}\left(\partial_{\nu}\tilde{h}_{\rho\sigma}+\partial_{\rho}\tilde{h}_{\nu\sigma}-\partial_{\sigma}\tilde{h}_{\nu\rho}\right)+\frac{1}{2}h^{\mu\sigma}n_{\rho}\pounds_{\tilde{n}}\tilde{h}_{\sigma\nu}\,,$
(115)

which generates a Galilean covariant connection compatible with $n_{\mu}$,
$h^{\mu\nu}$, $\tilde{n}^{\mu}$ and $\tilde{h}_{\mu\nu}$.

Often one considers the Galilean group associated with the massive Galilean,
or Bargmann, algebra. In that case the background geometry includes an
additional gauge field $A_{\mu}^{G}$ associated with the $U(1)$ symmetry
responsible for particle number, or mass, conservation. This gauge field
transforms as a covector under coordinate transformations associated with
Galilean boosts,

$A_{\mu}^{G}\xrightarrow{Coordinate}A_{\mu}^{G}-n_{\mu}\left(A^{G}\cdot\bar{\lambda}\right)\,,$
(116)

as a connection under $U(1)$ gauge transformations,

$A_{\mu}^{G}\xrightarrow[Gauge]{}A_{\mu}^{G}-\partial_{\mu}\Lambda\,,$ (117)

and if we use $A_{\mu}^{G}$ in place of $M^{G}_{\mu}$ to specify the
connection, as is often done in the literature, then it transforms
inhomogenously under Milne transformations,

$A_{\mu}^{G}\xrightarrow[G-Milne]{}A_{\mu}^{G}+P^{\alpha}_{\mu}{\psi}_{\alpha}+\frac{1}{2}n_{\mu}\psi^{2}\,.$
(118)

As observed in [14], $A_{\mu}^{G}=0$ is invariant under the combination of a
Galilean boost (107) and a gauge transformation (117) with parameter

$\Lambda=\int\left({\lambda}_{\mu}+\frac{1}{2}\lambda^{2}n_{\mu}\right)dx^{\mu}\,,$
(119)

where

$\partial_{\mu}\partial_{\nu}\Lambda-\partial_{\nu}\partial_{\mu}\Lambda=0\,.$
(120)

In a torsionless background, the latter condition is satisfied if
$\lambda_{\mu}$ and $n_{\mu}$ are covariantly constant.

Of particular importance to this work is the Galilean velocity field
$u^{\mu}_{G}=\partial x^{\mu}/\partial t$ which transforms as a vector under
coordinate transformations associated with Galilean boosts,

$u^{\mu}_{G}\xrightarrow[Coordinate]{}u^{\mu}_{G}+\bar{\lambda}^{\mu}\left(u_{G}\cdot
n\right)=u^{\mu}_{G}-\bar{\lambda}^{\mu}$ (121)

and is inert under Milne transformations,

$u^{\mu}_{G}\xrightarrow[G-Milne]{}u^{\mu}_{G}\,.$ (122)

While $u^{\mu}_{G}$ transforms as a vector under Galilean boosts (107),
$\bar{u}_{G\,\mu}=\bar{h}_{\mu\nu}u^{\nu}_{G}$ and $u_{G}^{2}$ do not,

$\displaystyle\begin{split}\bar{u}_{G\,\mu}&\xrightarrow[Galilean]{}\bar{u}_{G\,\mu}-{\lambda}_{\mu}-n_{\mu}\lambda^{2}\,,\\\
u_{G}^{2}&\xrightarrow[Galilean]{}u_{G}^{2}+\lambda^{2}-2(\bar{u}\cdot\bar{\lambda})\,.\end{split}$
(123)

The Galilean covariance of $\bar{u}_{G\,\mu}$ is spoiled by the nontrivial
Milne transformation properties of $\bar{h}_{\mu\nu}$ given in (105). While
$\bar{u}_{G\,\mu}$ is not a Galilean covariant vector, it is straightforward
to check that gauge invariant expressions constructed out of the combination
$A^{G}_{\mu}+\bar{u}_{G\,\mu}+\frac{1}{2}n_{\mu}u^{2}_{G}$ are Galilean
covariant (see, e.g., [14]). This is the reason that in (59) we replaced
$d(f^{G}\bar{u}_{G})$ with the Galilean covariant expression
$f_{0}d\left(A^{G}_{\mu}+\bar{u}_{G\,\mu}+\frac{1}{2}n_{\mu}u^{2}_{G}\right)$
with $f_{0}$ a constant.

###  A.2 Carrollian boosts

The Carrollian equivalent of Newton-Cartan geometry includes a degenerate
metric $\bar{h}_{\mu\nu}$ satisfying $\bar{h}_{\mu\nu}\bar{n}^{\mu}=0$, an
equivalence class of normals $n^{1}_{\mu}\sim n^{2}_{\mu}$, and an extra field
$M_{C}^{\mu}$ associated with the Carrollian connection. As was the case in
Newton-Cartan geometry, the equivalence class between normals may be made
manifest by introducing a Carrollian version of the Milne transformation which
leaves $\bar{h}_{\mu\nu}$ and $\bar{n}^{\mu}$ invariant and transforms
$n_{\mu}$ as

$n_{\mu}\xrightarrow[C-Milne]{}n_{\mu}+\bar{h}_{\mu\nu}\phi^{\nu}\,,$ (124a)
implying
$h^{\mu\nu}\xrightarrow[C-Milne]{}h^{\mu\nu}+\bar{n}^{\mu}P^{\nu}_{\alpha}\phi^{\alpha}+\bar{n}^{\nu}P^{\mu}_{\alpha}\phi^{\alpha}+\bar{n}^{\mu}\bar{n}^{\nu}\phi^{2}\,,$
(124b)

where $\phi$ is a generic spacetime dependent parameter and
$\phi^{2}=\phi^{\mu}\phi^{\nu}\bar{h}_{\mu\nu}$. The prefactor ’$C$’ in (124)
and throughout this section is used to distinguish the Carrollian version of
the Milne transformation from its Galilean counterpart.

The Carrollian group is represented by coordinate transformations $x\to
x^{\prime}(x)$ and Carrollian-Milne (C-Milne for short) transformations with
parameter $\phi^{\mu}$ which keep the flat Carrollian geometry invariant. In
general coordinates, Carrollian boosts, which are a subset Carrollian
transformations, are given by

$C^{\mu}{}_{\nu}\equiv\frac{\partial x^{\prime\mu}}{\partial
x^{\nu}}=\delta^{\mu}_{\nu}-\bar{n}^{\mu}\bar{h}_{\nu\alpha}\beta^{\alpha}\,,\qquad\phi^{\mu}=\beta^{\mu}\,,$
(125)

where $\beta^{\mu}$ is a covariantly constant parameter. In flat Cartesian
coordinates, where

$\bar{h}_{\mu\nu}=\delta_{\mu}^{i}\delta_{\nu}^{j}\,\delta_{ij}\,,\qquad
n_{\mu}=-\delta_{\mu}^{0}\,,$ (126)

the parameter $\beta^{\mu}$ reduces to a constant,
$\beta^{\mu}=\frac{1}{v_{0}^{2}}(0,\vec{v}_{0})$, and (125) reduces to

$x^{\mu}=(t,\vec{x})\xrightarrow[Coordinate]{}\left(t-\frac{\vec{v}_{0}\cdot\vec{x}}{v_{0}^{2}},\vec{x}\right)\,,\qquad\phi^{\mu}=\frac{1}{v_{0}^{2}}(0,\vec{v}_{0})\,.$
(127)

We define a tensor to be Carrollian covariant if

$\displaystyle\begin{split}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}&\xrightarrow[Carrollian]{}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}-\bar{n}^{\mu_{1}}\bar{\beta}_{\alpha}T^{\alpha\dots\mu_{p}}{}_{\nu_{1}\dots\nu_{q}}-\dots-\bar{n}^{\mu_{p}}\bar{\beta}_{\alpha}T^{\mu_{1}\dots\alpha}{}_{\nu_{1}\dots\nu_{q}}\\\
&+\bar{\beta}_{\nu_{1}}\bar{n}^{\alpha}T^{\mu_{1}\dots\mu_{p}}{}_{\alpha\dots\nu_{q}}+\dots+\bar{\beta}_{\nu_{q}}\bar{n}^{\alpha}T^{\mu_{1}\dots\mu_{p}}{}_{\nu_{1}\dots\alpha}+\ldots\end{split}$
(128)

where we have defined $\bar{\beta}_{\mu}=\bar{h}_{\mu\nu}\beta^{\nu}$ and the
last ellipses denote terms which are quadratic in $\beta^{\mu}$. Similar to
the Galilean case, $\bar{h}_{\mu\nu}$ and $\bar{n}^{\mu}$ are Carrollian
covariant but $n_{\mu}$ and $h^{\mu\nu}$ are not

$n_{\mu}\xrightarrow[Carrollian]{}n_{\mu}\,,\qquad
h^{\mu\nu}\xrightarrow[Carrollian]{}h^{\mu\nu}+2\beta^{2}\bar{n}^{\mu}\bar{n}^{\nu}\,,$
(129)

where $\beta^{2}=\bar{h}_{\mu\nu}\beta^{\mu}\beta^{\nu}$.

To obtain a Carrollian covariant connection one can go through a construction
analogous to the one which lead to (113). We will not go through it in detail
but merely quote the end result. The connection

${\Gamma}^{\mu}_{C}{}_{\nu\rho}=-\bar{n}^{\mu}\partial_{\rho}\tilde{n}_{\nu}+\frac{1}{2}\tilde{h}^{\mu\sigma}\left(\partial_{\nu}\bar{h}_{\rho\sigma}+\partial_{\rho}\bar{h}_{\nu\sigma}-\partial_{\sigma}\bar{h}_{\nu\rho}\right)\,,$
(130)

with

$\tilde{h}^{\mu\nu}=h^{\mu\nu}-P^{\mu}_{\alpha}M_{C}^{\alpha}\bar{n}^{\nu}-P^{\nu}_{\alpha}M_{C}^{\alpha}\bar{n}^{\mu}+\bar{n}^{\mu}\bar{n}^{\nu}M_{C}^{2}\,,\qquad\tilde{n}_{\mu}=n_{\mu}-\bar{h}_{\mu\alpha}M^{\alpha}_{C}$
(131)

is Carrollian invariant given that

$M_{C}^{\mu}\xrightarrow[Coordinate]{}M_{C}^{\mu}-\bar{n}^{\mu}(M_{C}\cdot\beta)\,.$
(132)

and

$M_{C}^{\mu}\xrightarrow[C-Milne]{}M_{C}^{\mu}+P^{\mu}_{\alpha}\phi^{\alpha}+\frac{1}{2}\bar{n}^{\mu}\phi^{2}\,.$
(133)

Similar to the Galilean case, the Carrollian geometry also admits a tilde’d
connection

$\tilde{\Gamma}^{\mu}_{C}{}_{\nu\rho}=-\bar{n}^{\mu}\partial_{\rho}\tilde{n}_{\nu}+\frac{1}{2}\tilde{h}^{\mu\sigma}\left(\partial_{\nu}\bar{h}_{\rho\sigma}+\partial_{\rho}\bar{h}_{\nu\sigma}-\partial_{\sigma}\bar{h}_{\nu\rho}\right)+\frac{1}{2}\tilde{h}^{\mu\sigma}\tilde{n}_{\rho}\pounds_{\bar{n}}\bar{h}_{\sigma\nu}\,,$
(134)

which is compatible with $\tilde{h}^{\mu\nu}$, $h_{\mu\nu}$, $\bar{n}^{\mu}$
and $n_{\mu}$.

The Carrollian velocity field $u^{\mu}_{C}$ is Carrollian covariant. It
transforms covariantly under Carrollian coordinate transformations (125),

$u^{\mu}_{C}\xrightarrow[Coordinate]{}u^{\mu}_{C}-\bar{n}^{\mu}(u_{C}\cdot\beta)\,,$
(135)

and is invariant under C-Milne transformations

$u^{\mu}_{C}\xrightarrow[C-Milne]{}u^{\mu}_{C}\,.$ (136)

Since $\bar{h}_{\mu\nu}$ transforms covariantly under Carrollian boosts, so
does $\bar{u}_{C\,\mu}=\bar{h}_{\mu\nu}u^{\nu}_{C}$. The covector $n_{\mu}$ is
not Carrollian covariant but the C-Milne invariant combination
$\tilde{n}_{\mu}$ defined in (131) is. This justifies the use of
$d(f_{C}\bar{u}_{C})+d(g_{C}\tilde{n})$ in the definition of a Carrollian
covariant $\Omega_{\mu\nu}$ in (87).

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