# Two theorems for the gradient expansion of relativistic hydrodynamics

Saulo Diles<EMAIL_ADDRESS>Campus Salinópolis,
Universidade Federal do Pará,
68721-000, Salinópolis, Pará, Brazil

###### Abstract

This letter is dedicated to providing proof of two statements concerning the
gradient expansion of relativistic hydrodynamics. The first statement is that
the ordering of transverse derivatives is irrelevant in the gradient expansion
of a non-conformal fluid. The second statement is that the longitudinal
projection of the Weyl covariant derivative can be eliminated in the gradient
expansion of a conformal fluid. This second statement does not apply to
curvature tensors. In the conformal case, we know that the ordering of Weyl
covariant derivatives is irrelevant in the gradient expansion.

††preprint: APS/123-QED

##  I Introduction

Hydrodynamics is one of the oldest fields of human knowledge and dates back to
the third century B.C. johnson2016handbook . Indeed, it is still a surprising
and exciting field in theoretical (and experimental) physics. The road to the
formulation of a relativistic theory of fluid dynamics brought up new
theoretical challenges. The causality problem in the relativistic version of
Navier-Stokes equations and the definition of out-of-equilibrium thermodynamic
variables eckart1940thermodynamics ; hiscock1985generic ; landau1959lifshitz ;
Bemfica:2017wps ; Kovtun:2019hdm ; Bemfica:2019knx ; Bemfica:2020zjp  are
important examples. To solve the former, higher-order derivatives have been
introduced in equations of motion, establishing the Müller-Israel-Stweart
theory Muller:1967zza ; Israel:1976tn ; Israel:1979wp . A systematic
formulation of relativistic hydrodynamics emerged as a gradient expansion
Baier:2007ix ; Bhattacharyya:2008jc ; Romatschke:2009im . The idea of a
gradient expansion was firstly established as a second-order theory and then
extended to third order Grozdanov:2015kqa ; Diles:2019uft . The definition of
out-of-equilibrium quantities require a gauge fixing Kovtun:2019hdm ;
Bemfica:2019knx  that constrains the form of the conserved currents. An in-
depth analysis of the hydrodynamic frame choice led to a recent formulation of
hydrodynamics in a general frame Kovtun:2019hdm ; Bemfica:2019knx ;
Bemfica:2020zjp ; Hoult:2020eho ; Noronha:2021syv . Relativistic fluid
dynamics is deeply connected with gravity through two different aspects: (i)
the AdS/CFT duality and the fluid/gravity correspondence Policastro:2002se ;
kovtun2003holography ; myers2009holographic ; rangamani2009gravity ;
Hubeny:2010wp ; Hubeny:2011hd ; Campoleoni:2018ltl , and (ii) the appearance
of curvature tensors in second-order hydrodynamics. Within the holographic
approach, and restricting to the linear response regime, the radius of
convergence of the gradient expansion can be obtained in particular cases,
such as the strongly coupled $\mathcal{N}=4$ SYM plasma Withers:2018srf ;
Grozdanov:2019kge ; Grozdanov:2019uhi ; Soltanpanahi:2021mys . In Ref.
Heller:2020uuy  it is shown that the convergence radius of the gradient
expansion of conformal fluids is set by a momentum scale determined by the
underlying microscopic theory. The results of Ref. Heller:2020uuy  are
independent of the existence of a holographic dual for the conformal fluid.

Gradient expansion is a methodology for establishing a theory of relativistic
fluid dynamics. The first step is to identify the global conserved currents.
In the general case, these are the energy-momentum tensor $T^{\mu\nu}$ and the
matter current $J^{\mu}$. At order $n$, the conserved currents are expressed
as a linear combination of all covariant terms that one can construct using
the degrees of freedom and its derivatives, with at least $n$ derivatives in
each term. A product of two first-order derivatives counts as a second-order
gradient, and so on. Once this is done, one obtains the $n$-th-order currents,
say $T_{n}^{\mu\nu}$ and $J_{n}^{\mu}$.

Not all gradients are independent. Dynamical equations and algebraic
identities reveal linear dependencies in the set of gradients that one can
include in a constitutive relation. For uncharged fluids, it was shown in
Grozdanov:2015kqa  that only transverse derivatives of $T,u^{\alpha}$ are
relevant in the gradient expansion. For conformal fluids, the gradient
expansion can be performed by considering only Weyl-covariant derivatives of
the degrees of freedom Bhattacharyya:2008mz ; Loganayagam:2008is ;
Hubeny:2011hd ; Diles:2017azi . Covariance plays an important role. Only
covariant derivatives can appear in the constitutive relations, ensuring the
covariant nature of the conserved currents. One uses a connection to define a
covariant derivative operator and apply this operator to the degrees of
freedom. The connections play a special role as they can define curvature
tensors from its ordinary derivatives ($\partial_{\mu}$). One cannot eliminate
the longitudinal derivatives of curvature structures.

Here we extend the discussion on equivalences in the gradient expansion of
ordinary and conformal fluids by proving two statements. This way, we
establish two theorems for the gradient expansion of relativistic
hydrodynamics. For ordinary fluids, we prove that the ordering of transverse
derivatives is not relevant. For conformal fluids, we prove that the Weyl
covariant derivative of thermodynamic and mechanic degrees of freedom is
transverse in all order of the gradient expansion. This letter is organized as
follows. In Section II, we discuss the gradient expansion in a general
situation. In section III, we discuss the case of an ordinary relativistic
fluid and prove Theorem I. In Section IV, we discuss the relativistic
conformal fluid and prove Theorem II. Section V is dedicated to discuss the
consequences of the theorems proved here.

##  II The Gradient Expansion

The idea of gradient expansion is that the energy-momentum tensor and the
matter current are given by a series expansion governed by the number of
space-time derivatives acting on fundamental degrees of freedom. The ideal
fluid establishes the zeroth-order theory:

$T_{0}^{\mu\nu}=\epsilon
u^{\mu}u^{\nu}+p\Delta^{\mu\nu},~{}~{}~{}J_{0}^{\mu}=nu^{\mu}.$ (1)

In the above expression, we consider a mostly plus metric signature and use
the transverse projector defined by
$\Delta^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$. We then proceed by adding
corrections to the ideal fluid constitutive relations. These corrections
should be obtained by differentiating the fundamental degrees of freedom. As
fundamental degrees of freedom, we consider the minimal set of independent
functions featuring the fluid dynamics. For a fluid with one conserved charge,
one can take $\\{T,n,u^{\alpha},g_{\mu\nu}\\}$ as the fundamental degrees of
freedom. The corrections we can include should preserve the covariance of the
equations of motion. Thus, $T^{\mu\nu}$ corrections are restricted to
covariant symmetric rank-2 tensors, while the corrections to $J^{\mu}$ are
restricted to covariant vectors. The large-scale symmetries of the system
define covariance. Here we discuss two specific cases: the non-conformal
relativistic fluid and the conformal relativistic fluid. Covariance also
defines the equations of motion since the divergence-less of $T^{\mu\nu}$ and
$J^{\mu}$ are defined by the appropriate covariant derivative.

The derivative order in each term is given by the total number of actions of
the partial derivative, whether they act successively on the same function or
separately on different functions. At each order, we must look for all the
different covariant structures that can be built. One general way to do that
is proposed in Ref. Grozdanov:2015kqa  for non-conformal fluids and extended
in Ref. Diles:2019uft  for conformal fluids. Following such a procedure, we
find an extensive list containing all the possible covariant terms at each
order. The $n$-th order correction in the gradient expansion is the linear
combination of all order $n$ terms in the list. Each coefficient in the linear
combination defines one transport coefficient. However, additional ingredients
restrict this long list. The first ingredient is that defining non-equilibrium
quantities requires gauge fixing, which strongly constrains the list of
allowed corrections. We assume that a gauge choice is fixed from the
beginning. The second ingredient concerns the algebraic equations relating
gradients that are distinct a priori. These equations include the zeroth-order
equations of motion of the fluid and all the algebraic and geometric tensorial
identities. Thus, one starts with the complete list of gradients allowed by
the gauge fixing. For each additional equation relating two gradients, it is
possible to replace one with the other and re-define the set of transport
coefficients (see Ref. Lahiri:2019lpk ). Additional constraints on the
transport coefficients arise when one imposes the second law of thermodynamics
by constructing an entropy current. In this case, the constraints are not
expressed as the vanishing of linear combinations of the transport
coefficients, and they do not generate equivalences. For example, at first
order, the positive divergence of the entropy current requires that $\zeta$
and $\eta$ are positive. More details concerning this discussion can be found
in Refs. Bhattacharyya:2012nq ; Banerjee:2012iz ; Jensen:2012jh ;
Bhattacharyya:2013lha ; Bhattacharyya:2014bha  and references therein.

For ordinary fluids, it has been shown in Ref. Grozdanov:2015kqa  that the
longitudinal projection of the gradients (curvature not included) is
equivalent to a transverse derivative of a different degree of freedom. For
this reason, in Ref. Grozdanov:2015kqa  the list of corrections to the energy-
momentum tensor contains only transverse gradients of $\ln s,u^{\mu}$. Here we
extend this result by proving that the ordering of the transverse derivatives
is irrelevant in the gradient expansion. For conformal fluids, a deep
discussion on equivalences of the high order corrections is performed in Ref.
Loganayagam:2008is , where the Weyl covariant derivative is established in the
context of fluid dynamics. Here we prove that the longitudinal projections of
the Weyl covariant derivative can be eliminated from the gradient expansion of
a conformal fluid. This second theorem narrows the connections between non-
conformal and conformal relativistic fluids.

##  III The non-conformal case:
$\nabla_{\perp\mu}\nabla_{\perp\nu}\simeq\nabla_{\perp\nu}\nabla_{\perp\mu}$

One convenient way to build the gradient expansion of an uncharged fluid is
using only transverse covariant derivatives of temperature (or entropy) and
velocity. Longitudinal gradients of such quantities are eliminated using the
equations of motion. In the case of charged fluids, this extends to the
concentration gradients. In addition to these mechanic and thermodynamic
gradients, we also consider corrections in the geometry using metric
derivatives. For a torsion-free space-time, the covariant derivative of the
metric vanishes, and covariant tensors involving metric derivatives come
exclusively from the Riemann tensor.

Curvature tensors play a different role in gradient expansion. They arise from
the commutation of covariant derivatives and are expressed in terms of
connections. For a non-conformal system, the Riemann tensor and its
contractions are the only curvature structures. In the gradient expansion of
ordinary fluids, one cannot eliminate the longitudinal derivatives of the
Riemann tensor. The inclusion of the Riemann tensor and their covariant
derivatives in the gradient expansion makes irrelevant the order of the
transverse derivatives of the other degrees of freedom.

We assume that the gradient expansion of an ordinary fluid contains all the
transverse gradients of $T,n,u^{\mu}$ and the covariant derivatives of the
Riemann tensor $R_{\mu\nu~{}~{}\beta}^{~{}~{}~{}\alpha}$. In the case that one
algebraic identity relates two or more of the referred tensors, say $\alpha
T^{I}(\partial^{n})+\beta F^{I}(\partial^{n})=0$, one imposes that they are
equivalent in the gradient expansion and write $T^{I}\simeq F^{I}$. Each
equivalence allows removing one of the equivalent tensors from the gradient
expansion. In this case, we say that this removed tensor is redundant and
write $T^{I}\simeq 0$ if we choose $T^{I}$ to remove. To prove the irrelevance
in the ordering of transverse derivatives, we first prove the Lemma below.

Lemma 1: For any equilibrium observable $\phi$, if the longitudinal projection
of the first-order derivative is redundant,
$u^{\alpha}\nabla_{\alpha}\phi\simeq 0$, then the second-order longitudinal
derivative is also redundant:
$u^{\mu}\nabla_{\mu}(u^{\alpha}\nabla_{\alpha}\phi)\simeq 0$.

Proof: The equations of motion of the ideal charged fluid are Muronga:2003ta ;
Lahiri:2019lpk

$\displaystyle u^{\alpha}\nabla_{\alpha}\ln s$
$\displaystyle=-\nabla_{\perp\alpha}u^{\alpha},$ (2) $\displaystyle
u^{\alpha}\nabla_{\alpha}u^{\mu}$ $\displaystyle=-\frac{1}{sT+\mu
n}\nabla_{\perp}^{~{}\mu}p,$ (3) $\displaystyle u^{\alpha}\nabla_{\alpha}\ln
n$ $\displaystyle=-\nabla_{\perp\alpha}u^{\alpha}.$ (4)

The transverse gradient is defined as
$\nabla_{\perp\alpha}=\Delta_{\alpha}^{~{}\beta}\nabla_{\beta}$. For
simplicity, write the right-hand side of eq. (3) in terms of pressure
gradients. For scalar degrees of freedom we have that
$D\phi_{\text{scalar}}=-\nabla_{\perp\alpha}u^{\alpha}\simeq 0$, where
$D\equiv u^{\alpha}\nabla_{\alpha}$ defines the longitudinal derivative. For
the second-order longitudinal derivative, one has

$\displaystyle D(D\phi_{\text{scalar}})$
$\displaystyle=-D(\nabla_{\perp\mu}u^{\mu})$
$\displaystyle=-u^{\alpha}\nabla_{\alpha}(\nabla_{\mu}u^{\mu})\simeq
u^{\alpha}\nabla_{\mu}\nabla_{\alpha}u^{\mu}$
$\displaystyle=-(\nabla_{\mu}u_{\alpha})\nabla^{\alpha}u^{\mu}+\nabla_{\mu}(Du^{\mu}).$
(5)

The first term is transverse and the second term requires more attention. We
have from eq. (3) that
$\nabla_{\mu}(Du^{\mu})\simeq\nabla_{\mu}(\nabla_{\perp}^{~{}\mu}p)$. However,
the divergence of a transverse gradient is also transverse:

$\displaystyle\nabla_{\mu}(\nabla_{\perp}^{~{}\mu}p)$
$\displaystyle=\nabla_{\mu}\nabla^{\mu}p+(\nabla_{\mu}u^{\mu})Dp$
$\displaystyle+u_{\mu}(\nabla^{\mu}u^{\alpha})\nabla_{\alpha}p+u^{\mu}u^{\alpha}\nabla_{\mu}\nabla_{\alpha}p.$
(6)

The second term in the above equation is transverse and the third therm is
redundant: it is equivalent to $\nabla_{\perp}^{\mu}p\nabla_{\perp\mu}p$. The
first and the last terms combine into a transverse Laplacian and we have

$\nabla_{\mu}(\nabla_{\perp}^{~{}\mu}p)\simeq\nabla_{\perp\mu}\nabla_{\perp}^{~{}\mu}p\simeq
0.$ (7)

Consequently,

$D(D\phi_{\text{scalar}})\simeq 0.$ (8)

For the velocity field, we have that

$\displaystyle D(Du^{\mu})$ $\displaystyle\simeq
u^{\alpha}\nabla_{\alpha}(\nabla_{\perp}^{~{}\mu}p)$ $\displaystyle\simeq
u^{\alpha}\nabla_{\alpha}(\nabla^{~{}\mu}p)+u^{\mu}u^{\alpha}D(\nabla_{\alpha}p)$
$\displaystyle=\nabla^{\mu}(Dp)-(\nabla^{\mu}u^{\alpha})\nabla_{\alpha}p$
$\displaystyle+u^{\mu}D(Dp)-u^{\mu}(Du^{\alpha})\nabla_{\alpha}p$
$\displaystyle\simeq\nabla^{\mu}(Dp)+u^{\mu}D(Dp)=\nabla_{\perp}^{\mu}(Dp)$
$\displaystyle\simeq\nabla_{\perp}^{\mu}(\nabla_{\perp\alpha}u^{\alpha})\simeq
0.$ (9)

The equivalence in the first line comes only from eliminating products of
first-order gradients, whereas in the last line eqs. (2) and (4) were also
used.

Theorem 1: The ordering of transverse derivatives is irrelevant in the
gradient expansion of a non-conformal relativistic fluid. For any equilibrium
observable $\phi$ we have that
$\nabla_{\perp\mu}\nabla_{\perp\nu}\phi\simeq\nabla_{\perp\nu}\nabla_{\perp\mu}\phi$.

Proof: To show that the ordering of transverse derivatives is irrelevant, we
compute the commutator of two transverse derivatives acting on an arbitrary
degree of freedom. Let $\phi$ be $T,n$ or $u^{\alpha}$, so that $\phi$
satisfies $u^{\mu}\nabla_{\mu}\phi\simeq 0$. Then, one can compute the
commutator $[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi$:

$\displaystyle[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi$
$\displaystyle=\Delta_{\mu}^{~{}\alpha}\nabla_{\alpha}(\Delta_{\nu}^{~{}\beta}\nabla_{\beta}\phi)-\Delta_{\nu}^{~{}\alpha}\nabla_{\alpha}(\Delta_{\mu}^{~{}\beta}\nabla_{\beta}\phi)$
$\displaystyle=[\nabla_{\mu},\nabla_{\nu}]\phi+u_{\mu}u^{\alpha}\nabla_{\alpha}(\nabla_{\nu}\phi+u_{\nu}u^{\beta}\nabla_{\beta}\phi)$
$\displaystyle-
u_{\nu}u^{\alpha}\nabla_{\alpha}(\nabla_{\mu}\phi+u_{\mu}u^{\beta}\nabla_{\beta}\phi)$
(10)

The first term in the above equation is a commutator of covariant derivatives
and is given by a linear combination of contractions with the Riemann tensor.
Since all the Riemann tensors and their contractions are already included in
the gradient expansion, one has that $[\nabla_{\mu},\nabla_{\nu}]\phi\simeq
0$. This leads to

$\displaystyle[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi$ $\displaystyle\simeq
u_{\mu}u^{\alpha}\nabla_{\alpha}(\nabla_{\nu}\phi+u_{\nu}u^{\beta}\nabla_{\beta}\phi)$
$\displaystyle-
u_{\nu}u^{\alpha}\nabla_{\alpha}(\nabla_{\mu}\phi+u_{\mu}u^{\beta}\nabla_{\beta}\phi)$
$\displaystyle=u_{\mu}u^{\alpha}\nabla_{\alpha}\nabla_{\nu}\phi+u_{\mu}u_{\nu}u^{\alpha}u^{\beta}\nabla_{\alpha}\nabla_{\beta}\phi$
$\displaystyle+u_{\mu}u^{\alpha}u^{\beta}(\nabla_{\alpha}u_{\nu})\nabla_{\beta}\phi+u_{\mu}u_{\nu}u^{\alpha}(\nabla_{\alpha}u^{\beta})\nabla_{\beta}\phi$
$\displaystyle-u_{\nu}u^{\beta}\nabla_{\beta}\nabla_{\mu}\phi-
u_{\nu}u_{\mu}u^{\beta}u^{\alpha}\nabla_{\beta}\nabla_{\alpha}\phi$
$\displaystyle-
u_{\nu}u_{\mu}u^{\beta}(\nabla_{\beta}u^{\alpha})\nabla_{\alpha}\phi-
u_{\nu}u^{\beta}u^{\alpha}(\nabla_{\beta}u_{\mu})\nabla_{\alpha}\phi$
$\displaystyle=u_{\mu}u^{\alpha}\nabla_{\alpha}\nabla_{\nu}\phi+u_{\mu}u^{\alpha}u^{\beta}(\nabla_{\alpha}u_{\nu})\nabla_{\beta}\phi$
$\displaystyle+u_{\mu}u_{\nu}u^{\alpha}(\nabla_{\alpha}u^{\beta})\nabla_{\beta}\phi$
$\displaystyle-u_{\nu}u^{\beta}\nabla_{\beta}\nabla_{\mu}\phi-
u_{\nu}u_{\mu}u^{\beta}(\nabla_{\beta}u^{\alpha})\nabla_{\alpha}\phi$
$\displaystyle-
u_{\nu}u^{\beta}u^{\alpha}(\nabla_{\beta}u_{\mu})\nabla_{\alpha}\phi.$ (11)

In the last line, it was used the fact that
$u^{\alpha}u^{\beta}[\nabla_{\alpha},\nabla_{\beta}]\phi=0$. We proceed by
analyzing the remaining terms. The second and the sixth terms are of the same
kind, so we discuss explicitly only the second one. We have that
$u_{\mu}u^{\alpha}u^{\beta}(\nabla_{\alpha}u_{\nu})\nabla_{\beta}\phi=u_{\mu}(u^{\alpha}\nabla_{\alpha}u_{\nu})u^{\beta}\nabla_{\beta}\phi\simeq
0$, since the longitudinal derivatives can be eliminated. The third and the
fifth terms are also of the same kind and we discuss explicitly the third one.
We notice first that
$(\nabla_{\alpha}u^{\beta})\nabla_{\beta}\phi=(\nabla_{\alpha}u^{\beta})\nabla_{\perp\beta}\phi$
as a consequence of $u_{\beta}u^{\beta}=-1$. Then we have that
$u_{\mu}u_{\nu}u^{\alpha}(\nabla_{\alpha}u^{\beta})\nabla_{\beta}\phi=u_{\mu}u_{\nu}(u^{\alpha}\nabla_{\alpha}u^{\beta})\nabla_{\perp\beta}\phi$.
Nevertheless the longitudinal derivatives of the velocity can be replaced by
transverse derivatives. Thus, this term is equivalent to one involving only
transverse gradients already in the list. We have that
$u_{\mu}u_{\nu}u^{\alpha}(\nabla_{\alpha}u^{\beta})\nabla_{\beta}\phi\simeq
0$.

We remain with the first and fourth terms in eq. (11), obtaining

$[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi\simeq
u_{\mu}u^{\alpha}\nabla_{\alpha}\nabla_{\nu}\phi-
u_{\nu}u^{\beta}\nabla_{\beta}\nabla_{\mu}\phi.$ (12)

The two terms are of the same kind, differing by index commutation. Thus we
analyze the first one explicitly. At this stage, it is convenient to change
the ordering of covariant derivatives, yielding

$[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi\simeq
u_{\mu}u^{\alpha}\nabla_{\nu}\nabla_{\alpha}\phi-
u_{\nu}u^{\beta}\nabla_{\mu}\nabla_{\beta}\phi.$ (13)

Finaly, we use that as $u^{\alpha}\nabla_{\alpha}\phi\simeq 0$ then
$\nabla_{\alpha}\phi\simeq\nabla_{\perp\alpha}\phi$. From Lemma 1, we have
that
$\nabla_{\nu}(\nabla_{\alpha}\phi)\simeq\nabla_{\nu}(\nabla_{\perp\alpha}\phi)$.
We can now differentiate the equality $u^{\alpha}\nabla_{\perp\alpha}\phi=0$,
obtaining

$u^{\alpha}\nabla_{\nu}(\nabla_{\perp\alpha}\phi)\simeq-(\nabla_{\perp\nu}u^{\alpha})\nabla_{\perp\alpha}\phi\simeq
0.$ (14)

Using the above equation we can conclude that
$u_{\mu}u^{\alpha}\nabla_{\nu}\nabla_{\alpha}\phi\simeq 0$. Hence

$[\nabla_{\perp\mu},\nabla_{\perp\nu}]\phi\simeq 0.$ (15)

##  IV The Weyl covariant gradient expansion

A Weyl transformation Weyl:1918ib  is a local scaling of the line element
$ds\to e^{\Phi(x)}ds$, equivalent to a redefinition of the metric tensor
$g_{\mu\nu}\to e^{2\Phi}g_{\mu\nu}$. A tensor $\mathcal{T}$ is said to be of
weight $\omega$ under Weyl scaling if it transforms as $\mathcal{T}\to
e^{-\omega\Phi}\mathcal{T}$. Scaling invariance, or Weyl invariance, is
expected to manifest in the hydrodynamic limit of a conformal field theory. As
a consequence, the constitutive relations of such a fluid should be compatible
with this (local) scaling invariance. The compatible energy-momentum tensor,
$T^{\mu\nu}$, transforms as a rank-2 tensor density of weight $d+2$ under Weyl
scaling, where $d$ is the number of space-time dimensions. The compatible
vector current, $J^{\mu}$, transforms as a contravariant vector density of
weight $d$.

The fundamental degrees of freedom in hydrodynamics are tensor densities under
Weyl scaling. Thus, we need to ensure that their derivatives will also be
tensor densities. To do so, a derivative operator that is covariant under Weyl
transformations is needed. This operator is the Weyl covariant derivative
Loganayagam:2008is . The Weyl covariant derivative of a field $\psi$ with
weight $\omega$, denoted here by $\mathcal{D}_{\alpha}\psi$, is obtained by
replacing everywhere
$\partial_{\alpha}\psi\to(\partial_{\alpha}+\omega\mathcal{A}_{\alpha})\psi$,
where $\mathcal{A}_{\alpha}$ is the Weyl connection Diles:2017azi . Let $\Phi$
be a scalar with weight $\omega_{\Phi}$ and $\gamma^{\alpha}$ a vector with
weight $\omega_{\gamma}$. The associated Weyl covariant derivatives are given
respectively by

$\displaystyle\mathcal{D}_{\alpha}\Phi$
$\displaystyle=(\partial_{\alpha}+\omega_{\Phi}\mathcal{A}_{\alpha})\Phi,$
$\displaystyle\mathcal{D}_{\beta}\gamma^{\alpha}$
$\displaystyle=(\partial_{\beta}+\omega_{\gamma}\mathcal{A}_{\beta})\gamma^{\alpha}+\bar{\Gamma}^{~{}\alpha}_{\beta\mu}\gamma^{\mu},$
(16)

where

$\displaystyle\bar{\Gamma}^{~{}\alpha}_{\beta\mu}$
$\displaystyle=\frac{1}{2}g^{\alpha\nu}[(\partial_{\beta}-2\mathcal{A}_{\beta})g_{\nu\mu}$
$\displaystyle+(\partial_{\mu}-2\mathcal{A}_{\mu})g_{\nu\beta}-(\partial_{\nu}-2\mathcal{A}_{\nu})g_{\beta\mu}]$
(17)

is obtained by performing $\partial
g_{\alpha\beta}\to(\partial-2\mathcal{A})g_{\alpha\beta}$ in the Christoffel
symbol.

There is a unique way to define the Weyl connection in hydrodynamics, which is
requiring it to be transverse and trace-less when acting on the velocity
field. These conditions are expressed as

$\mathcal{D}_{\alpha}u^{\alpha}=0,~{}u^{\alpha}\mathcal{D}_{\alpha}u^{\beta}=0.$
(18)

The former leads to
$u^{\alpha}\mathcal{A}_{\alpha}=-\frac{\nabla_{\alpha}u^{\alpha}}{(d-1)}$,
while the latter gives
$\mathcal{A}^{\alpha}=u^{\beta}\nabla_{\beta}u^{\alpha}+u^{\beta}\mathcal{A}_{\beta}u^{\alpha}$.
We then obtain the unique Weyl connection satisfying both requirements:

$\mathcal{A}_{\alpha}=u^{\beta}\nabla_{\beta}u_{\alpha}-\frac{(\nabla_{\beta}u^{\beta})}{d-1}u_{\alpha}.$
(19)

The above definition of the Weyl connection ensures that eq. (18) holds even
in the off-shell case. For a conformal fluid, transversality of the Weyl
covariant derivative of the velocity field is not an equivalence but an
equality. Due to the trace-less condition, we have for the scalars degrees of
freedom that

$u^{\alpha}\mathcal{D}_{\alpha}T=0,~{}~{}u^{\alpha}\mathcal{D}_{\alpha}\ln
n=0.$ (20)

In the conformal case, we also find that the longitudinal derivative
redundancy does not apply to curvature tensors. The condition that the
covariant derivative of the metric vanishes is not affected by the Weyl
connection. We have

$\mathcal{D}_{\alpha}g_{\mu\nu}=0.$ (21)

This does not mean that metric derivatives are excluded from the gradient
expansion. Metric derivatives and Weyl connection derivatives are encoded in
the curvature tensors $\mathcal{R}_{\alpha\beta~{}\sigma}^{~{}~{}~{}\mu}$ and
$\mathcal{F}_{\alpha\beta}$. The former is the conformal Riemann tensor, while
the latter is the field strength for the Weyl connection. They appear in the
commutator of Weyl covariant derivatives, which acts on a vector field as

$[\mathcal{D}_{\alpha},\mathcal{D}_{\beta}]\xi^{\mu}=-\omega_{\xi}\mathcal{F}_{\alpha\beta}\xi^{\mu}-\mathcal{R}_{\alpha\beta~{}\sigma}^{~{}~{}~{}\mu}\xi^{\sigma}.$
(22)

In the following, we show that transversality of the Weyl covariant derivative
extends to higher-order terms in the gradient expansion.

Theorem 2: At any order of gradient expansion of a conformal fluid, the
longitudinal projection of the Weyl covariant derivative of entropy,
concentration, and velocity are redundant.

Proof: Let $\phi_{i}$ be either temperature, concentration, or a velocity
component. It holds that

$u^{\alpha}\mathcal{D}_{\alpha}\phi_{i}=0\to\mathcal{D}_{\alpha}\phi_{i}=\mathcal{D}_{\perp\alpha}\phi_{a},$
(23)

where
$\mathcal{D}_{\perp\alpha}\equiv\Delta_{\alpha}^{~{}\beta}\mathcal{D}_{\alpha}$.
We now consider the second order derivative:

$\mathcal{D}_{\beta}\mathcal{D}_{\alpha}\phi_{i}=\mathcal{D}_{\beta}(\mathcal{D}_{\perp\alpha}\phi_{i}).$
(24)

We first note that by differentiating
$u^{\alpha}\mathcal{D}_{\perp\alpha}\phi_{i}=0$, we get

$u^{\alpha}\mathcal{D}_{\beta}\mathcal{D}_{\alpha}\phi_{i}=-(\mathcal{D}_{\beta}u^{\alpha})\mathcal{D}_{\alpha}\phi_{i}\simeq
0.$ (25)

From eq. (22) we obtain that
$u^{\alpha}\mathcal{D}_{\beta}\mathcal{D}_{\alpha}\phi_{i}\simeq
u^{\alpha}\mathcal{D}_{\alpha}\mathcal{D}_{\beta}\phi_{i}=u^{\alpha}\mathcal{D}_{\alpha}\mathcal{D}_{\perp\beta}\phi_{i}$.
We now use eq. (25) to obtain

$u^{\alpha}\mathcal{D}_{\alpha}\mathcal{D}_{\beta}\phi_{i}\simeq-(\mathcal{D}_{\beta}u^{\alpha})\mathcal{D}_{\alpha}\phi_{i}\simeq
0.$ (26)

In the gradient expansion of a conformal fluid, if the first derivative is
transverse, the second will also be transverse. The arguments above are
extended to higher-order derivatives by taking $\phi$ as a lower-order
transverse gradient. Consequently, in any order of gradient expansion, only
the transverse derivatives of the degrees of freedom will be present and give
rise to new transport coefficients. In other words, the Weyl covariant
derivative of any hydrodynamic degree of freedom is always transverse, despite
the order of gradient expansion.

##  V Discussion

Modern understanding of relativistic hydrodynamics employs a gradient
expansion. In this context, a systematic way of generating arbitrarily high
order corrections of the constitutive relations is needed. The present work
paves one more step in establishing the rules to fix constitutive relations
and the number of transport coefficients of a viscous fluid. Our present
results hold in the general, non-linear case.

The statements proved here lead to consequences in the building of the
gradient expansion of relativistic fluids. It is important to mention that in
Ref. Diles:2019uft , the statement of Theorem 1 has been conjectured to hold
in the general case and proved to hold only in some particular cases. The
consequences of Theorem 1 have already been explored in that reference: it
leads to a reduction of 10 in the number of transport coefficients for third-
order hydrodynamics of uncharged fluids. We also remark that in the inspiring
work presented in Ref. Grozdanov:2015kqa , it is proved that only the
transverse derivatives are relevant in the gradient expansion. However, the
authors of that reference did not raise the possibility that commuting
transverse derivatives would give rise only to redundant structures. The
effects of the irrelevance in the ordering of transverse derivatives in
higher-order constitutive relations is to decrease the number of transport
coefficients.

The consequences of Theorem 2 have not been explored in higher-order
hydrodynamics. This theorem states that the Weyl covariant derivative acts on
mechanic and thermodynamic degrees of freedom of the fluid as a transverse
operator in the context of gradient expansion. Consequently, it suggests that
therms like $u^{\alpha}\mathcal{D}_{\alpha}\sigma^{\mu\nu}$ or
$u^{\alpha}\mathcal{D}_{\alpha}\Omega^{\mu\nu}$ can, in principle, be
eliminated from the gradient expansion and its transport coefficients absorbed
into the remaining ones. Here the tensors $\sigma^{\mu\nu}$ and
$\Omega^{\mu\nu}$ correpond to the symmetric and anti-symmetric parts of
$\mathcal{D}^{\mu}u^{\nu}$, respectively. However, it is not sufficient to
state the redundance of these structures. What happens is that the equality

$\displaystyle u^{\alpha}\mathcal{D}_{\alpha}\sigma^{\langle\mu\nu\rangle}=$
$\displaystyle-\sigma^{\alpha\langle\mu}\sigma_{~{}~{}\alpha}^{\nu\rangle}-\Omega^{\alpha\langle\mu}\Omega_{~{}~{}\alpha}^{\nu\rangle}$
$\displaystyle+u^{\alpha}u^{\beta}\mathcal{R}_{\alpha~{}~{}~{}~{}\beta}^{~{}\langle\mu\nu\rangle},$
(27)

is satisfied. Notice that in the expression above
$\mathcal{R}_{\alpha~{}~{}\beta}^{~{}\mu\nu}$ corresponds to the curvature
tensor for the Weyl-covariant derivative defined in eq. (22) and
$T^{\langle\mu\nu\rangle}=\frac{1}{2}\Delta^{\mu\alpha}\Delta^{\nu\beta}(T_{\alpha\beta}+T_{\beta\alpha})-\frac{2}{d-1}\Delta^{\mu\nu}\Delta_{\alpha\beta}T^{\alpha\beta}$
is the transverse, symmetric, and traceless projection. Thus, the longitudinal
derivative $u^{\alpha}\mathcal{D}_{\alpha}\sigma^{\langle\mu\nu\rangle}$ will
be redundant only if eq. (27) is linearly independent of all other algebraic
relations used to reduce the list of second-order gradients. On the other
hand, $u^{\alpha}\mathcal{D}_{\alpha}\Omega^{\langle\mu\nu\rangle}$ vanishes
identically due to the anti-symmetry of $\Omega^{\mu\nu}$.

It is important to remark that an old result of the fluid/gravity
correspondence Bhattacharyya:2008jc ; Haack:2008cp ; Bhattacharyya:2008mz
states that the coefficient $\lambda_{3}$ associated with the tensor
$\Omega^{\alpha\langle\mu}\Omega_{~{}~{}\alpha}^{\nu\rangle}$ vanishes for
$\mathcal{N}=4$ SYM conformal fluids in the limit of large $N_{c}$ and large
t’Hooft coupling $\lambda$. However, finite $\lambda$ corrections result in a
non-vanishing $\lambda_{3}$, as shown in Ref. Saremi:2011nh ;
Grozdanov:2014kva . The results of Ref. Saremi:2011nh ; Grozdanov:2014kva
hold for a specific case and even suggesting that there are, in fact, five
independent second-order conformal gradients, that is not enough to conclude
that this is valid in the general case. A detailed discussion on the
possibility of reducing the list of second-order conformal gradients is needed
and will be carried over in future work.

###### Acknowledgements.

The author thanks Alex S. Miranda, the first who noticed that changing the
ordering of transverse derivatives does not generate new structures. This
observation led to Theorem 1 as a conjecture and motivated the search for its
proof. The author thanks the Campus Salinopolis of the Universidade Federal do
Pará for the release of work hours for research.

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