††institutetext: National Institute of Science Education and Research, HBNI,
Bhubaneshwar 752050, Odisha, India

# Comparison between fluid-gravity and membrane-gravity dualities for
Einstein-Maxwell system

Milan Patra<EMAIL_ADDRESS>
###### Abstract

Derivative expansion and large-$D$ expansion are two perturbation techniques,
which are used to generate dynamical black-brane solutions to Einstein’s
equations in presence of negative cosmological constant. In this note we have
compared these two techniques and established the equivalence of the gravity
solutions generated by these two different techniques in appropriate regime of
parameter space up to first non-trivial order in both the perturbation
parameters for Einstein-Maxwell systems, generalizing the earlier works of
Bhattacharyya2019leading ; Bhattacharyya2019sunleading  for non-charged
systems. An one-to-one map between dynamical black-brane geometry and AdS
space, which also exists at finite number of dimensions, has also been
established.

##  1 Introduction :

Over the last few years, it has been demonstrated that Black hole dynamics at
large number of space-time dimensions (denoted as $D$ here) is dual to the
dynamics of a co-dimension one non-gravitational membrane propagating in a
background that solves Einstein’s equations. Emparan, Suzuki and Tanabe
Emparan:2013moa ; QNM:Emparan ; Emparan:2013xia ; Emparan:2013oza ;
Emparan:2014cia ; Emparan:2014jca  have observed that at large number of
space-time dimensions the black holes have two effective length scales; one is
the radius of the horizon $r_{H}$ and the other is the thickness of the region
over which the gravitational force is non trivial. Beyond this region the
space-time, to a good approximation, could be described the asymptotic
geometry itself (for our purpose this is going to be the AdS space). It turns
out that in the limit of large $D$, this thickness is of the order of
$\mathcal{O}\left(\frac{r_{H}}{D}\right)$ around the horizon and within this
region Einstein equations could simply be solved in a perturbation in inverse
powers of dimension.
Within this perturbation scheme, the effective dynamics of these black hole
horizons can be described by a co-dimension one massive membrane with
dynamical degrees of freedom as charge density, shape and a divergence-free
velocity field moving in the background space. This duality has been studied
for both asymptotically flat and AdS/dS background and also for Einstein-
Maxwell systems in membrane ; Chmembrane ; yogesh1 ; arbBack ;
Kundu2018chargeAdsds ; Bhattacharyya20182nd .

On the other hand ‘derivative expansion’, which is a perturbation technique in
boundary derivative expansion nonlinfluid ; arbitrarydim ; Rangamani:2009xk ;
Banerjee:2008th ; Erdmenger:2008rm ; Haack:2008cp ; Bhattacharyya:2007vs ;
Banerjee:2012iz ; Bhattacharyya:2012xi ; Hubeny:2011hd ; Bredberg:2011xw ,
generates dynamical black-brane solutions of Einstein’s equations in the bulk
with negative cosmological constant. These dynamical black brane solutions are
dual to an arbitrary fluid dynamical solutions in the boundary. In other
words, for every solution to the relativistic generalization of Navier-Stokes’
equation in the boundary, one can construct an asymptotically AdS black hole
type solution equivalent to a boundary fluid flow. These solutions thus
generated are characterized by a local temperature field, a unit normalized
velocity field and charge density living on the boundary.

The questions that we would like to address in this note are the following.

  * •

Is there any interconnection between these two perturbation techniques

  * •

Can we apply these two techniques simultaneously in any regime of parameter
space

  * •

Are the solutions generated by these two techniques equivalent in any regime
of parameter space?

These questions have been answered in Bhattacharyya2019leading ;
Bhattacharyya2019sunleading  for pure gravity systems and here we shall extend
their work to Einstein-Maxwell systems. We shall show that the gravity
solutions generated by these two perturbation techniques are equivalent also
for Einstein-Maxwell systems in an appropriate regime of parameter space.

On physical grounds this equivalence is not surprising. Since we can use the
same space-time geometry as the starting point for both the techniques and
since given a starting point both the techniques generate the solutions
uniquely, it follows that in the regime where both the techniques are
applicable, the solutions should be same. But this is not at all manifest and
it involves a series of intricate gauge and variable transformations. Just in
Bhattacharyya2019leading ; Bhattacharyya2019sunleading  here also the heart of
the paper lies in these set of subtle calculations.

###  1.1 Strategy

In this subsection we will discuss briefly the procedure we have used to show
the equivalence of the two gravity solutions and refer to
(Bhattacharyya2019leading, ; Bhattacharyya2019sunleading, ) for any logical
discussion and proof. As we know the metric $W_{AB}$ generated in large-$D$
technique are written in a split form, a background metric $\bar{W}_{AB}$ and
$W^{(rest)}_{AB}$. The metric $\bar{W}_{AB}$ is the metric of the asymptotic
geometry (in our case it is pure AdS metric) and $W^{(rest)}_{AB}$ is written
in a way such that contraction of a certain null geodesic $O^{A}\partial_{A}$
(not affinely parametrized) with it is zero to all order. But the hydrodynamic
metric is not written in such split form. So it is obvious that to compare the
solutions generated by these two techniques the first step should be to split
the hydrodynamic metric into background and ‘rest’. We will do it by the
following steps

At first we determine an affinely parametrized null geodesic field
$\bar{O}^{A}\partial_{A}$, which passes through the event horizon of the full
space-time. Then we pick up a coordinate system
$Y^{A}\equiv\\{\rho,y^{\mu}\\}$, where the background of the hydrodynamic
metric can be written in the following form

$ds^{2}_{background}=\bar{G}_{AB}dY^{A}dY^{B}=\frac{d\rho^{2}}{\rho^{2}}+\rho^{2}\eta_{\mu\nu}dy^{\mu}dy^{\nu}$
(1)

And determine the mapping function $f^{A}(X)$ that relates $Y^{A}$ to the
$X^{A}\equiv\\{r,x^{\mu}\\}$ (in which the hydrodynamic metric is written)
coordinates by the following equation

$\bar{O}^{A}{\cal G}_{AB}|_{\\{X\\}}=\bar{O}^{A}\frac{\partial f^{C}}{\partial
X^{A}}\frac{\partial f^{C^{\prime}}}{\partial
X^{B}}\bar{G}_{CC^{\prime}}|_{\\{X\\}}$ (2)

where ${\cal G}_{AB}$ is the full metric written in $X^{A}$ coordinates and
the subscripts $\\{X\\}$ refer to the fact that all the terms in the left and
right are calculated in $X^{A}$ coordinates.

However it cannot fix $f^{A}$ completely and we require some conformal type
symmetry on the background metric to fix it.

After determining the mapping function we can split the hydrodynamic metric
into background plus ‘rest’. Then we take the large-$D$ limit of the
hydrodynamic metric.

The next step is to write the large-$D$ data in terms of fluid data. The
metric and gauge field generated in large-$D$ technique are written in terms
of a smooth function $\psi$ (such that $\psi^{-D}$ is a harmonic function
w..r.t the background), a charge field $\widetilde{Q}$ and a non-affinely
parametrized null geodesic $O_{A}$. It will turn out that $O_{A}$ is related
to $\bar{O}_{A}$ (determined from the hydrodynamic metric) by an overall
normalization constant.

At first we determine $\psi$ and then $O_{A}$ and $\widetilde{Q}$ in terms of
fluid data. After that we substitute these expressions in the large-$D$ metric
and gauge field and write those in terms of fluid data. Then it is easy to
check that the metric and gauge field in large-$D$ side matches with those in
fluid side up to appropriate orders in both the perturbation parameters.

But the hydrodynamic data and large-$D$ data cannot be chosen arbitrarily,
they have to satisfy some constraint equations, named as fluid equations and
membrane equations respectively for the metric and gauge field to be a
solution of Einstein’s equations. So to show the equivalence of the gravity
solutions generated by these two techniques the final step would be to show
the equivalence of these constraint equations.

The organisation of this note is as follows.
In section §2 we review the hydrodynamic metric and gauge field up to first
order in derivative in arbitrary dimensions. In section §3 we have noted the
gravity solutions for Einstein-Maxwell systems in large-$D$ technique. In
section §4 we have rewritten the large-$D$ data in terms of fluid data and
compared the two gravity solutions. And finally in section §5 we have
concluded.

##  2 Review of Hydrodynamics from charged black-branes in arbitrary
dimensions :

In this section we will review the work on fluid-gravity correspondence for
charged black-branes Banerjee:2008th ; Erdmenger:2008rm ; articleyee  by
determining the metric and gauge field dual to charged fluid configuration up
to first order in boundary derivative expansion for all $D\geq 3$. The results
of this section were previously recorded in articleyee  in a bit different
language. As our lagrangian and notations are slightly different from the
authors of articleyee , we will redo everything with our lagrangian and
notations.

We start with the $D$ dimensional action111We will use the Latin indices
$\\{M,N,\cdots\\}$ to denote the bulk indices while the Greek indices
$\\{\mu,\nu,\cdots\\}$ refer to the boundary indices. And the
$\\{\mu,\nu,\cdots\\}$ indices are raised and lowered by the Minkowaski metric
$\eta^{\mu\nu}$. And for our case $\lambda=-1$.

$\begin{split}&\mathcal{\mathbf{S}}=\frac{1}{16\pi
G_{D}}\int\leavevmode\nobreak\ d^{D}x\leavevmode\nobreak\
\sqrt{-\text{g}}\left[R-2\Lambda-\frac{F_{MN}F^{MN}}{4}\right]\\\ \
&\text{with negative cosmological constant}\leavevmode\nobreak\
\Lambda=\frac{\left(D-1\right)\left(D-2\right)}{2}\lambda\end{split}$ (3)

By varying the above action we will get the $D$ dimensional Einstein-Maxwell
equations with negative cosmological constant

$\begin{split}&R_{AB}-\frac{1}{2}R\leavevmode\nobreak\
g_{AB}-\frac{(D-1)(D-2)}{2}g_{AB}+\frac{1}{2}\leavevmode\nobreak\
F_{AC}F^{C}_{\leavevmode\nobreak\ B}+\frac{1}{8}\leavevmode\nobreak\
g_{AB}F_{CD}F^{CD}=0\\\
&\nabla_{B}F^{AB}=0\Rightarrow\frac{1}{\sqrt{-g}}\partial_{B}\left(\sqrt{-g}\leavevmode\nobreak\
F^{AB}\right)=0\end{split}$ (4)

where $g_{AB}$ is the $D$ dimensional metric tensor and
$F_{AB}=\partial_{A}A_{B}-\partial_{B}A_{A}$.

We know that these equations (4) admit an AdS-Reisner-Nordstrom ‘boosted
black-brane solutions’, which we write in ingoing Eddington-Finkelstein
coordinates as

$\begin{split}d&s^{2}=-2u_{\mu}dx^{\mu}dr-r^{2}V\left(r,m,Q\right)u_{\mu}u_{\nu}dx^{\mu}dx^{\nu}+r^{2}P_{\mu\nu}dx^{\mu}dx^{\nu}\\\
&A=\frac{\sqrt{3}\leavevmode\nobreak\ Q}{2r^{D-3}}u_{\mu}dx^{\mu}\\\
\end{split}$ (5)

with

$\begin{split}&V(r,m,Q)=1-\frac{m}{r^{D-1}}+\frac{1}{4}\leavevmode\nobreak\
\frac{3(D-3)}{2(D-2)}\frac{Q^{2}}{r^{2(D-2)}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
P^{\mu\nu}=\eta^{\mu\nu}+u^{\mu}u^{\nu}\\\
&u^{v}=\frac{1}{\sqrt{1-\beta^{2}}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
u^{i}=\frac{\beta^{i}}{\sqrt{1-\beta^{2}}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \beta^{2}=\beta_{i}\beta^{i}\\\ \end{split}$ (6)

Like in the metric described above we will use coordinates
$X^{A}\equiv\underbrace{\\{r,v,x^{i}\\}}_{D}$ for our bulk spaces, on the
other hand coordinates $x^{\mu}\equiv\underbrace{\\{v,x^{i}\\}}_{D-1}$
parametrize our boundary and $r$ is the radial coordinate.

Now we allow the temperature, velocity and charge field in the black-brane
metric (5) to vary slowly in the boundary coordinates and determine the metric
and gauge field in boundary derivative expansion. To start we will take the
ansatz as

$\begin{split}&g_{MN}=g^{(0)}_{MN}+g^{(1)}_{MN}+g^{(2)}_{MN}+\cdots\\\
&A_{N}=A^{(0)}_{N}+A^{(1)}_{N}+A^{(2)}_{N}+\cdots\\\ \end{split}$ (7)

where the leading order ansatz $g^{(0)}_{MN}$ and $A^{(0)}_{N}$ are given by

$\begin{split}&g^{(0)}=-2u_{\mu}dx^{\mu}dr-r^{2}V(r,m,Q)u_{\mu}u_{\nu}dx^{\mu}dx^{\nu}+r^{2}P_{\mu\nu}dx^{\mu}dx^{\nu}\\\
&A^{(0)}=\frac{\sqrt{3}\leavevmode\nobreak\ Q}{2r^{D-3}}u_{\mu}dx^{\mu}\\\
\end{split}$ (8)

and $g^{(k)}_{MN}$, $A^{(k)}_{N}$, which are corrections to the bulk metric
and gauge field, are determined by solving Einstein-Maxwell equations order by
order in derivative expansion. To solve these equations by our perturbation
technique it is useful to work in a particular gauge. FollowingBanerjee:2008th
we work in the following gauge

$\begin{split}g_{rr}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\
g_{r\mu}=-u_{\mu}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\text{Tr}\left[\left(g^{(0)}\right)^{-1}g^{(k)}\right]=0\leavevmode\nobreak\
,\leavevmode\nobreak\ A_{r}=0\end{split}$ (9)

Here one should note that in the relativistic case the energy flow between
different fluid elements also leads to transport of mass and momentum. Hence
we cannot define the velocity field uniquely unless we work in a particular
frame. In this context it is useful to work in Landau frame defined by
$u_{\mu}T^{\mu\nu}_{(k)}=0$, where $T^{\mu\nu}_{(k)}$ is the $\text{k}^{th}$
order stress tensor with $k\geq 1$ and in the proper frame of a fluid element
the longitudinal component of the stress tensor to the fluid velocity give the
local energy density in the fluid.

In this section our goal is to find out the metric and gauge field up to first
order in derivative expansion. To implement our perturbation technique we set
our velocity field $u^{\mu}$ to be $\\{1,0,0,\cdots\\}$ by a boundary Lorentz
transformation at a boundary point $x^{\mu}$ and solve these equations about
this special point. Since our perturbation procedure is ultra-local we can
easily write the result thus obtained in covariant form with respect to the
boundary metric. Our velocity, temperature and charge field expanded in taylor
series about this special point $x^{\mu}$ in terms of boundary derivatives as

$\begin{split}&\beta_{i}=x^{\mu}\partial_{\mu}\beta^{(0)}_{i}+\cdots\\\
&m=m^{(0)}+x^{\mu}\partial_{\mu}m^{(0)}+\cdots\\\
&Q=Q^{(0)}+x^{\mu}\partial_{\mu}Q^{(0)}+\cdots\\\ \end{split}$ (10)

Using the expressions written above if we expand the $0^{th}$ order ansatz up
to first order in derivative we have

$\begin{split}ds^{2}_{(0)}=&2\leavevmode\nobreak\ dv\leavevmode\nobreak\
dr-r^{2}V^{(0)}(r)\leavevmode\nobreak\ dv^{2}+r^{2}dx^{i}\leavevmode\nobreak\
dx_{i}\\\
&-2r^{2}\left(1-V^{(0)}(r)\right)x^{\mu}\partial_{\mu}\beta^{(0)}_{i}\leavevmode\nobreak\
dx^{i}\leavevmode\nobreak\
dv-2x^{\mu}\partial_{\mu}\beta^{(0)}_{i}\leavevmode\nobreak\ dx^{i}dr\\\
&-\left(-\frac{x^{\mu}\partial_{\mu}m^{(0)}}{r^{D-3}}+\frac{1}{4}\frac{3(D-3)}{2(D-2)}\frac{2Q^{(0)}x^{\mu}\partial_{\mu}Q^{(0)}}{r^{2(D-3)}}\right)dv^{2}\\\
A=&-\frac{\sqrt{3}}{2r^{D-3}}\left[\left(Q^{(0)}+x^{\mu}\partial_{\mu}Q^{(0)}\right)dv-Q^{(0)}x^{\mu}\partial_{\mu}\beta^{(0)}_{i}dx^{i}\right]\\\
\text{with}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&V^{(0)}(r)=1-\frac{m^{(0)}}{r^{D-1}}+\frac{1}{4}\leavevmode\nobreak\
\frac{3(D-3)}{2(D-2)}\frac{{Q^{(0)}}^{2}}{r^{2(D-2)}}\end{split}$ (11)

Obviously this metric and gauge field of equation (11) will not solve the
Einstein-Maxwell’s equations up to the order we are interested. We need to add
corrections containing first order derivatives to solve these equations. While
solving these equations we find that the bulk Einstein-Maxwell equations
decompose into constraint equations and dynamical equations. The constraint
equations are equivalent to the conservation of boundary stress energy-
momentum tensor and conservation of boundary current density. On the other
hand the dynamical equations are inhomogeneous differential equations on the
unknown parameters added to the metric and gauge field. By solving these
differential equations and imposing regularity at the future event horizon and
appropriate fall off at infinity, we can determine the unknown parameters.
Thus we get an unique solution to the Einstein-Maxwell equations parametrized
by the charge, temperature and $(D-1)$ dimensional velocity fields.

Since the background metric have the $SO(D-2)$ symmetry, the Einstein-Maxwell
equations will split into scalar, vector, and traceless symmetric two tensor
sector and we can solve each sector separately.

###  2.1 Scalars at first order

The scalar components of the added correction of the first order metric and
gauge field are parametrized by $h_{1}(r),k_{1}(r)$ and $w_{1}(r)$ and we can
write these as

$\begin{split}&g^{(1)}_{vv}(r)=\frac{k_{1}(r)}{r^{2}}\\\
&g^{(1)}_{vr}(r)=-\frac{\left(D-2\right)}{2}h_{1}(r)\\\
&\sum_{i}\leavevmode\nobreak\ g^{(1)}_{ii}(r)=(D-2)\leavevmode\nobreak\
r^{2}\leavevmode\nobreak\ h_{1}(r)\\\
&A^{(1)}_{v}(r)=-\frac{\sqrt{3}\leavevmode\nobreak\
w_{1}(r)}{2\leavevmode\nobreak\ r^{D-3}}\\\ \end{split}$ (12)

Here one should note that the metric corrections $g^{(1)}_{ii}$ and
$g^{(1)}_{vr}$ are related to each other by the gauge choice
$\text{Tr}\left[\left(g^{(0)}\right)^{-1}g^{(1)}\right]=0$. At first we
determine the constraint equations. These equations are determined by taking
dot product of Einstein-Maxwell equations with the vector dual to the one form
$dr$. We have these equations as follows

$\begin{split}\frac{\partial_{v}m^{(0)}}{m^{(0)}}+\frac{(D-1)}{(D-2)}\partial_{i}\beta^{(0)}_{i}=0\\\
\end{split}$ (13)

which is identical to the conservation of the stress tensor in the boundary

$\begin{split}\partial_{\mu}T^{\mu\nu}_{(0)}=0\end{split}$ (14)

The other constraint equation is given by

$\begin{split}\frac{\partial_{v}Q^{(0)}}{Q^{(0)}}+\partial_{i}\beta^{(0)}_{i}=0\\\
\end{split}$ (15)

which is equivalent to the conservation of boundary current density.

$\begin{split}\partial_{\mu}J^{\mu}_{(0)}=0\\\ \end{split}$ (16)

In the scalar sector we have 6 differential equations, the $rr,vv,rv$
component of the Einstein tensor along with the trace over the boundary
spatial part and the $r$ and $v$ components of the Maxwell equations. Among
these 6 equations we have to use only 3 equations to determine the three
unknown parameters $h_{1},k_{1}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ w_{1}$. The solutions thus obtained should
satisfy the rest equations. Solving these equations and demanding the
appropriate normalizability at infinity we have the following solutions

$\begin{split}&h_{1}(r)=0\\\ &w_{1}(r)=0\\\ &k_{1}(r)=2\leavevmode\nobreak\
r^{3}\frac{\partial_{i}\beta^{(0)}_{i}}{D-2}\end{split}$ (17)

Finally we have the first order metric and gauge field in the following form

$\begin{split}&g^{(1)}_{vv}(r)=2r\left(\frac{\partial_{i}\beta^{(0)}_{i}}{D-2}\right)\\\
&g^{(1)}_{vr}(r)=0\\\ &\sum_{i}\leavevmode\nobreak\ g^{(1)}_{ii}(r)=0\\\
&A^{(1)}_{v}(r)=0\\\ \end{split}$ (18)

###  2.2 Vectors at first order

The vector components of first order metric and gauge field are parametrized
by $g^{(1)}_{i}(r)$ and $j^{(1)}_{i}(r)$ as

$\begin{split}&g^{(1)}_{vi}(r)=r^{2}\left(1-V(r)\right)j^{(1)}_{i}(r)\\\
&A^{(1)}_{i}(r)=-\frac{\sqrt{3}Q^{(0)}}{2r^{D-3}}j^{(1)}_{i}(r)+g^{(1)}_{i}(r)\\\
\end{split}$ (19)

Here also at first we determine the constraint equation which is given by

$\begin{split}\frac{\partial_{i}m^{(0)}}{m^{(0)}}+(D-1)\partial_{v}\beta^{(0)}_{i}=0\end{split}$
(20)

This follows from the conservation of boundary stress tensor.

The dynamical equations in the vector sector are the $ri,vi$ components of the
Einstein tensor and $i$ th component of the Maxwell equation. Solving these
equations with appropriate boundary conditions (regularity at the future event
horizon and appropriate fall off at boundary) we have the following
expressions for $j^{(1)}(r)$ and $g^{(1)}(r)$.

$\begin{split}&j^{(1)}_{i}(r)=\frac{r\leavevmode\nobreak\
\partial_{v}\beta^{(0)}_{i}}{r^{2}\left(1-V(r)\right)}+\frac{3(D-3)\leavevmode\nobreak\
r^{2}\leavevmode\nobreak\ Q^{(0)}\leavevmode\nobreak\
\bigg{(}\partial_{i}Q^{(0)}+(D-2)\leavevmode\nobreak\
Q^{(0)}\leavevmode\nobreak\
\partial_{v}\beta^{(0)}_{i}\bigg{)}}{R^{2(D-1)}\leavevmode\nobreak\
r^{2}\left(1-V(r)\right)}R\leavevmode\nobreak\ F_{1}(\rho,M)\\\
&g^{(1)}_{i}(r)=\frac{4}{2(D-3)\sqrt{3}Q^{(0)}}\left(-r^{D-2}\partial_{v}\beta^{(0)}_{i}+(D-1)m^{(0)}j^{(1)}_{i}(r)+\left(\frac{1}{4}\frac{3(D-3)}{2(D-2)}\frac{{Q^{(0)}}^{2}}{r^{D-4}}-m^{(0)}r\right)\frac{dj^{(1)}_{i}(r)}{dr}\right)\\\
\end{split}$ (21)

Now plugging these $g^{(1)}_{i}(r)$ and $j^{(1)}_{i}(r)$ in the (19) we have
the corrected first order metric and gauge field in the vector sector as

$\begin{split}&g^{(1)}_{vi}(r)=r\leavevmode\nobreak\
\partial_{v}\beta^{(0)}_{i}+\frac{3(D-3)\leavevmode\nobreak\
r^{2}\leavevmode\nobreak\ Q^{(0)}\leavevmode\nobreak\
\bigg{(}\partial_{i}Q^{(0)}+(D-2)\leavevmode\nobreak\
Q^{(0)}\leavevmode\nobreak\
\partial_{v}\beta^{(0)}_{i}\bigg{)}}{R^{2(D-1)}}R\leavevmode\nobreak\
F_{1}(\rho,M)\\\
&A^{(1)}_{i}(r)=-{2}{\sqrt{3}}\frac{r^{D}}{R^{2(D-1)}}\bigg{(}\partial_{i}Q^{(0)}+(D-2)\leavevmode\nobreak\
Q^{(0)}\leavevmode\nobreak\
\partial_{v}\beta^{(0)}_{i}\bigg{)}F_{1}^{(1,0)}(\rho,M)\\\ \end{split}$ (22)

where $F_{1}$ is given by

$\begin{split}&F_{1}(\rho,M)=\frac{1}{4(D-2)}\left(1+\frac{1}{4}\frac{3(D-3)}{2(D-2)}\frac{Q_{1}^{2}}{\rho^{2(D-2)}}-\frac{M}{\rho^{D-1}}\right)F_{3}(\rho,M)\\\
\ \text{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&F_{3}(\rho,M)=\int_{\rho}^{\infty}dp\frac{1}{\left(1+\frac{1}{4}\frac{3(D-3)}{2(D-2)}\frac{Q_{1}^{2}}{p^{2(D-2)}}-\frac{M}{p^{D-1}}\right)^{2}}\Big{(}\frac{1}{p^{2(D-1)}}-\frac{c_{1}}{p^{2D-3}}\Big{)}\\\
\end{split}$ (23)

with

$c_{1}=\frac{D-2}{D-1}\left(1+\frac{2}{M(D-3)}\right)$ (24)

Where we have used the following rescaled variables

$\begin{split}&\rho=\frac{r}{R}\leavevmode\nobreak\ \leavevmode\nobreak\
,M=\frac{m}{R^{D-1}}\leavevmode\nobreak\ \leavevmode\nobreak\
,Q_{1}=\frac{Q}{R^{D-2}},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ Q_{1}^{2}=4\leavevmode\nobreak\
\frac{2(D-2)}{3(D-3)}(M-1)\\\ \end{split}$ (25)

And then the Hawking temperature is given by

$\begin{split}T=\frac{(D-1)R}{4\pi}\left(1-\left(\frac{D-3}{D-1}\right)\left(M-1\right)\right)\end{split}$
(26)

where $R$ is the radius of the outer horizon. In terms of these rescaled
variables the outer horizon is given by $\rho_{+}\equiv 1$.

###  2.3 Tensors at first order

The tensor component of the metric at first order can be parametrized by
$\pi^{(1)}_{ij}(r)$ as

$\begin{split}g^{(1)}_{ij}=r^{2}\pi^{(1)}_{ij}(r)\end{split}$ (27)

This unknown parameter $\pi^{(1)}_{ij}$ can be determined by solving the
dynamical equation obtained from the $ij$ component of Einstein equation,
which is given by

$\frac{d}{dr}\left(r^{D}V(r,m,Q)\frac{d\pi_{ij}^{(1)}(r)}{dr}\right)=-2(D-2)r^{D-3}\sigma_{ij}^{(0)}$
(28)

where

$\begin{split}\sigma_{ij}^{(0)}=\frac{\partial_{i}\beta^{(0)}_{j}+\partial_{j}\beta^{(0)}_{i}}{2}-\frac{\partial_{k}\beta^{(0)}_{k}}{D-2}\delta_{ij}\end{split}$
(29)

Demanding regularity at the future event horizon and appropriate fall off at
boundary the solution to the equation (28) is given by

$\begin{split}\pi^{(1)}_{ij}(r)=\frac{2}{R}\sigma_{ij}\leavevmode\nobreak\
F_{2}\left(\rho,M\right)\end{split}$ (30)

where

$\begin{split}&F_{2}\left(\rho,M\right)=\int_{\rho}^{\infty}\frac{\rho^{D}\left(\rho^{D}-\rho^{2}\right)}{\rho^{2}\left(\rho^{2D}-M\rho^{D+1}+(M-1)\rho^{4}\right)}d\rho\end{split}$
(31)

###  2.4 The global metric and gauge field at first order

We have done our computation about a special point $x^{\mu}=0$ in the
boundary. However, our perturbation procedure is ultralocal and we could set
any arbitrary velocity $u^{\mu}$ to be $\\{1,0,\cdots\\}$ by a boundary
coordinate transformation. So we could do our computation about any arbitrary
point on the boundary. Hence the results recorded in the previous subsections
contain enough information to write down the metric and gauge field in
covariant form w.r.t the boundary metric. we have the following covariant form
of the metric and gauge field

$\begin{split}ds^{2}=&g_{AB}dx^{A}dx^{B}\\\
=&-2u_{\mu}dx^{\mu}dr-r^{2}V(r,m,Q)u_{\mu}u_{\nu}dx^{\mu}dx^{\nu}+r^{2}P_{\mu\nu}dx^{\mu}dx^{\nu}\\\
&-2u_{\mu}dx^{\mu}\leavevmode\nobreak\
r\left[u^{\lambda}\partial_{\lambda}u_{\nu}-\frac{\partial_{\lambda}u^{\lambda}}{D-2}u_{\nu}\right]dx^{\nu}+\frac{2r^{2}}{R}F_{2}(\rho,M)\sigma_{\mu\nu}dx^{\mu}dx^{\nu}\\\
&-2u_{\mu}dx^{\mu}\left[\frac{3\leavevmode\nobreak\ (D-3)\leavevmode\nobreak\
Q\leavevmode\nobreak\ r^{2}}{R^{2D-3}}{\cal
P}^{\lambda}_{\nu}\left(\mathcal{D}_{\lambda}Q\right)F_{1}(\rho,M)\right]dx^{\nu}+\cdots\\\
\\\ &A=\left[\frac{\sqrt{3}Q}{2\leavevmode\nobreak\
r^{D-3}}u_{\mu}-\frac{2\sqrt{3}r^{D}}{R^{2(D-1)}}{\cal
P}^{\lambda}_{\mu}\left(\mathcal{D}_{\lambda}Q\right)F_{1}^{(1,0)}(\rho,M)\right]dx^{\mu}+\cdots\\\
\end{split}$ (32)

where

${\cal P}^{\lambda}_{\mu}\left(\mathcal{D}_{\lambda}Q\right)={\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q\right)+(D-2)\left(u^{\lambda}\partial_{\lambda}u_{\mu}\right)Q$
(33)

and

$\sigma_{\mu\nu}={\cal P}_{\mu}^{\alpha}{\cal
P}_{\nu}^{\beta}\left[\frac{\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}}{2}-{\eta}_{\alpha\beta}\left(\frac{\Theta}{D-2}\right)\right]$
(34)

And the constraint equations can be written in covariant form as

$\begin{split}&\frac{\left(u\cdot\partial\right)Q}{Q}+\partial\cdot u=0\\\
&\frac{\left(u\cdot\partial\right)m}{m}+(D-1)\frac{\partial\cdot u}{D-2}=0\\\
&\frac{{{\cal P}^{\alpha}_{\mu}\leavevmode\nobreak\
\partial_{\alpha}}m}{m}+(D-1)u^{\lambda}\partial_{\lambda}u_{\mu}=0\\\
\end{split}$ (35)

###  2.5 The boundary stress tensor and the charge current

In this section we write down the expressions for the stress tensor and charge
current dual to the metric and gauge field up to first order in derivative
expansion. By using AdS/CFT correspondence we can determine the boundary
stress tensor from the bulk space-time metric. Here we have to add suitable
counter terms to the action to regularise the divergence arising from
integrating the full space-time volume. The expression for the boundary stress
tensor dual to the metric presented in the previous subsection can be obtained
by the prescription of Balasubramanian1999 .

To calculate this stress tensor we need to know the asymptotic expansion of
the function $F_{2}(\rho,M)$, present in the tensor sector of the metric up to
$\mathcal{O}\left(\frac{1}{r^{D}}\right)$. This expansion is given by

$\begin{split}F_{2}(r,M)=\frac{R}{r}-\frac{R^{D-1}}{(D-1)r^{D-1}}-\frac{MR^{D}}{D\leavevmode\nobreak\
r^{D}}+\cdots\end{split}$ (36)

The stress tensor for the metric up to first order in derivative expansion is
given by

$\begin{split}&T^{\mu\nu}=p\left(\eta^{\mu\nu}+(D-1)u^{\mu}u^{\nu}\right)-2\eta\leavevmode\nobreak\
\sigma^{\mu\nu}\\\ &\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ p=\frac{M\leavevmode\nobreak\
R^{D-1}}{16\leavevmode\nobreak\ \pi\leavevmode\nobreak\
G_{D}}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \eta=\frac{R^{D-2}}{16\leavevmode\nobreak\
\pi\leavevmode\nobreak\ G_{D}}\\\ \end{split}$ (37)

The charge current $J_{\mu}$ can be obtained from the gauge field by using the
following expression

$\begin{split}J_{\mu}=\lim_{r\to\infty}\frac{r^{D-3}A_{\mu}}{8\leavevmode\nobreak\
\pi\leavevmode\nobreak\ G_{D}}\end{split}$ (38)

So the charge current for the gauge field is given by

$\begin{split}J_{\mu}=n\leavevmode\nobreak\
u_{\mu}-\mathfrak{D}\leavevmode\nobreak\ {\cal
P}^{\nu}_{\mu}\mathcal{D}_{\nu}n\end{split}$ (39)

where the charge density $n$ and diffusion constant $\mathfrak{D}$ are given
by

$\begin{split}&n=\frac{\sqrt{3}Q}{16\leavevmode\nobreak\
\pi\leavevmode\nobreak\ G_{D}}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathfrak{D}=\frac{(D-3)M+2}{R\leavevmode\nobreak\ M\leavevmode\nobreak\
(D-1)(D-3)}\end{split}$ (40)

###  2.6 Hydrodynamic metric and gauge field up to first order in derivative

The metric dual to hydrodynamics in arbitrary dimension $D$ is written in
terms of fluid variable $u$, a charge field and a temperature field $T$ living
on the $(D-1)$ dimensional boundary of the space-time. The independent data in
first order is written in table (1). We will write the hydrodynamic metric and
the constraint equations in the following way

Table 1: Data at 1st order in derivative | Independent Data
---|---
Scalar | $\frac{\Theta}{D-2}\equiv\frac{\partial\cdot u}{D-2}$
Vector | $a_{\mu}\equiv(u\cdot\partial)u_{\mu}\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\cal P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}$
Tensor | $\sigma_{\mu\nu}={\cal P}_{\mu}^{\alpha}{\cal P}_{\nu}^{\beta}\left[\frac{\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}}{2}-{\eta}_{\alpha\beta}\left(\frac{\Theta}{D-2}\right)\right]$

The constraint equations can be written as

$\begin{split}&\frac{\left(u\cdot\partial\right)Q_{C}}{Q_{C}}=0\\\
&\frac{\left(u\cdot\partial\right)r_{H}}{r_{H}}+\frac{\Theta}{D-2}=0\\\
&\frac{{{\cal P}^{\alpha}_{\mu}\leavevmode\nobreak\
\partial_{\alpha}}r_{H}}{r_{H}}+a_{\mu}+f(Q_{C}){\cal
P}^{\alpha}_{\mu}\leavevmode\nobreak\ \partial_{\alpha}Q_{C}=0\\\ \end{split}$
(41)

where

$f(Q_{C})=\frac{6\frac{(D-3)}{D-1}Q_{C}}{8(D-2)+3(D-3)Q_{C}^{2}}$ (42)

We write metric and gauge field in two part as

$dS^{2}=dS_{0}^{2}+dS_{1}^{2}$ (43)

where
$0^{\text{th}}$ Order Piece:

$\begin{split}dS_{0}^{2}=&-2u_{\mu}\leavevmode\nobreak\
dx^{\mu}\leavevmode\nobreak\ dr-r^{2}V(r)\leavevmode\nobreak\
u_{\mu}u_{\nu}\leavevmode\nobreak\ dx^{\mu}\leavevmode\nobreak\
dx^{\nu}+r^{2}{\cal P}_{\mu\nu}dx^{\mu}dx^{\nu}\\\ &{\cal
P}_{\mu\nu}=\eta_{\mu\nu}+u_{\mu}u_{\nu}\end{split}$ (44)

$1^{\text{st}}$ Order Piece:

$\begin{split}&dS_{1}^{2}=-r\left(u_{\mu}B_{\nu}+u_{\nu}B_{\mu}\right)dx^{\mu}\leavevmode\nobreak\
dx^{\nu}+\frac{2r^{2}}{r_{H}}\leavevmode\nobreak\
F_{2}(\rho,M)\leavevmode\nobreak\ \sigma_{\mu\nu}\leavevmode\nobreak\
dx^{\mu}\leavevmode\nobreak\ dx^{\nu}\\\ &\text{where}\\\
&B_{\mu}=a_{\mu}-\left(\Theta\over
D-2\right)u_{\mu}+\frac{3\leavevmode\nobreak\ (D-3)\leavevmode\nobreak\
r\leavevmode\nobreak\ Q_{C}}{r_{H}^{2}}{\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\bigg{[}1-(D-2)Q_{C}\leavevmode\nobreak\
f(Q_{C})\bigg{]}F_{1}(\rho,M)\\\ \end{split}$ (45)

where

${\cal
P}^{\lambda}_{\mu}\left(\mathcal{D}_{\lambda}Q\right)=r_{H}^{(D-2)}\bigg{[}1-(D-2)Q_{C}\leavevmode\nobreak\
f(Q_{C})\bigg{]}\leavevmode\nobreak\ {\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\ $ (46)

The gauge field up to first order in derivative expansion can be written as

$A=A_{0}+A_{1}$ (47)

where
$0^{\text{th}}$ Order Piece:

$\begin{split}&A_{0}=\frac{\sqrt{3}\leavevmode\nobreak\ r\leavevmode\nobreak\
Q_{C}}{2}\left(\frac{r_{H}}{r}\right)^{D-2}u_{\mu}dx^{\mu}\\\ \end{split}$
(48)

$1^{\text{st}}$ Order Piece:

$\begin{split}&A_{1}=-2\sqrt{3}\left(\frac{r}{r_{H}}\right)^{D}\bigg{[}1-(D-2)Q_{C}\leavevmode\nobreak\
f(Q_{C})\bigg{]}F_{1}^{(1,0)}(\rho,M)\leavevmode\nobreak\ {\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)dx^{\mu}\\\
\end{split}$ (49)

where,

$V(r)=1-\left(1+\frac{1}{4}\leavevmode\nobreak\
\frac{3(D-3)}{2(D-2)}Q_{C}^{2}\right)\left(\frac{r_{H}}{r}\right)^{D-1}+\frac{1}{4}\leavevmode\nobreak\
\frac{3(D-3)}{2(D-2)}Q_{C}^{2}\left(\frac{r_{H}}{r}\right)^{2(D-2)}$ (50)

And $r=r_{H}$ is the position of the outer event horizon of the space-time.

##  3 The large $D$ metric, gauge field and membrane equations:

In this section we simply quote the results of large $D$ metric and gauge
field as in paper Kundu2018chargeAdsds .
The metric in the large $D$ expansion is written in a split form as
$\bar{W}_{AB}$ plus $W^{(rest)}_{AB}$.

$W_{AB}=\bar{W}_{AB}+W_{AB}^{(rest)}$

$\bar{W}_{AB}$ is the metric of the nonsingular asymptotic geometry. In our
case it would be the metric of just pure AdS. In general, $\bar{W}_{AB}$ is
any smooth solution (where all components of the Riemann tensor are of order
${\cal O}(1)$ in terms of large $D$ counting) to Einstein’s equations with
cosmological constant and vanishing electromagnetic field. $W^{(rest)}_{AB}$
captures the effect of black hole and singularity. Using the large $D$
technique one can determine $W_{AB}^{(rest)}$ and in this case also the gauge
field $A_{M}$ order by order in an expansion in inverse powers of $D$. In
other words the metric an the gauge field will have the following form

$\begin{split}&W^{(rest)}_{AB}=\mathcal{W}^{(0)}_{AB}+\left(\frac{1}{D}\right)\mathcal{W}^{(1)}_{AB}+\cdots\\\
&A_{M}=A^{(0)}_{M}+\left(\frac{1}{D}\right)A^{(1)}_{M}+\cdots\end{split}$ (51)

The final solution is expressed in terms of a null geodesic field
$O^{A}\partial_{A}$ and two scalar fields $\psi$ and $\tilde{Q}$. $\psi$ is a
smooth function with $\psi=1$ as event horizon of the full space-time and
harmonic w.r.t the background $\bar{W}_{AB}$. $\widetilde{Q}$ is also a smooth
function satisfying

$\begin{split}n\cdot\partial\widetilde{Q}=0\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\widetilde{Q}|_{\psi=1}=\frac{1}{\sqrt{2}}\left(U^{M}A_{M}\right)|_{\psi=1}\end{split}$
(52)

where $n_{A}$ is the unit normal to the constant $\psi$ slices and $U$ is the
membrane velocity defined by $U=n-O$.
The gauge is fixed by demanding that to all orders

$O^{A}W^{(rest)}_{AB}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ O^{M}A_{M}=0$

Now we shall present the final solutions upto the first subleading order in
${\cal O}\left(1\over D\right)$ i.e., the explicit expressions for
$\mathcal{W}^{(0)}_{AB}\leavevmode\nobreak\ $, $A^{(0)}_{M}$ and
$\mathcal{W}^{(1)}_{AB}\leavevmode\nobreak\ $, $A^{(1)}_{M}$

$\begin{split}&\mathcal{W}^{(0)}_{AB}=fO_{A}O_{B}\\\
&A^{(0)}_{M}=\sqrt{2}\leavevmode\nobreak\ \widetilde{f}O_{M}\\\
&\mathcal{W}^{(1)}_{AB}=\mathcal{Z}^{(s1)}O_{A}O_{B}+\left(\mathcal{Z}^{(v)}_{A}O_{B}+\mathcal{Z}^{(v)}_{B}O_{A}\right)+\mathcal{Z}^{(T)}_{AB}\\\
\text{and}\leavevmode\nobreak\
&A^{(1)}_{M}=\mathcal{A}^{(s)}O_{M}+\mathcal{A}^{(v)}_{M}\end{split}$ (53)

where

$\begin{split}f=\left(1+\widetilde{Q}^{2}\right)\psi^{-D}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\widetilde{f}=\widetilde{Q}\psi^{-D}\end{split}$ (54)

and

$\begin{split}&\mathcal{Z}^{(s1)}=\sum_{i=1}^{N_{S}}S^{(i)}_{1}(\zeta)\mathcal{S}^{(i)}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathcal{Z}^{(s2)}=\sum_{i=1}^{N_{S}}S^{(i)}_{2}(\zeta)\mathcal{S}^{(i)}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathcal{A}^{(s)}=\sum_{i=1}^{N_{S}}a^{(i)}_{s}(\zeta)\mathcal{S}^{(i)}\\\
&\mathcal{Z}^{(v)}_{A}=\sum_{i=1}^{N_{V}}\mathcal{V}^{(i)}(\zeta){V}^{(i)}_{A}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathcal{A}^{(v)}_{A}=\sum_{i=1}^{N_{V}}a^{(i)}_{v}(\zeta){V}^{(i)}_{A}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathcal{Z}^{(T)}_{AB}=\sum_{i=1}^{N_{T}}\mathcal{T}^{(i)}(\zeta){t}^{(i)}_{AB}\end{split}$
(55)

Here $O_{A}\leavevmode\nobreak\ dX^{A}$ is a null one-form with respect to
both the background metric $\bar{W}_{AB}$ and full metric $W_{AB}$. This is
dual to the geodesic vector field $O^{A}\partial_{A}$ mentioned before. The
function $\psi$ and $\tilde{Q}$ are already defined (see the discussion around
equation (52)).

Now both the functions $\psi$ and $\widetilde{Q}$ admit ${1\over D}$ expansion
in the ‘membrane region’ - the region where the gravitational field is non
trivial in the limit of large $D$. It follows that upto first order in
$\frac{1}{D}$, the function $f$ and $\widetilde{f}$ can be written as

$\begin{split}&f(W)=\left(1+\widetilde{Q}^{2}\right)\leavevmode\nobreak\
e^{-W}-\widetilde{Q}^{2}\leavevmode\nobreak\
e^{-2W}+\left(\frac{1}{D}\right)\\\
&\widetilde{f}(W)=\widetilde{Q}\leavevmode\nobreak\
e^{-W}+\left(\frac{1}{D}\right)\end{split}$ (56)

where the parameter $W$ is a $\mathcal{O}(1)$ variable, defined by
$W=D\left(\psi-1\right)$.

And then the solution up to first order in metric and gauge field can be
written as

$\begin{split}&\mathcal{V}^{(i)}(y)=-2\int_{y}^{\infty}dx\leavevmode\nobreak\
e^{-x}\leavevmode\nobreak\ \widetilde{Q}\leavevmode\nobreak\
a^{(i)}_{v}(x)-e^{-y}\mathcal{K}_{vector}-2\left(\frac{D}{K}\right)\int_{y}^{\infty}dx\leavevmode\nobreak\
e^{-x}\int_{0}^{x}dt\frac{e^{t}\leavevmode\nobreak\
\mathfrak{v}^{(i)}_{\text{metric}}(t)}{1-f(t)}\\\
&\text{where}\leavevmode\nobreak\
\mathcal{K}_{vector}=-2\int_{0}^{\infty}dx\leavevmode\nobreak\
e^{-x}\leavevmode\nobreak\ \widetilde{Q}\leavevmode\nobreak\
a^{(i)}_{v}(x)-2\left(\frac{D}{K}\right)\int_{0}^{\infty}dx\leavevmode\nobreak\
e^{-x}\int_{0}^{x}dt\frac{e^{t}\leavevmode\nobreak\
\mathfrak{v}^{(i)}_{\text{metric}}(t)}{1-f(t)}\end{split}$ (57)

and

$\begin{split}a^{(i)}_{v}(t)&=-e^{t}(f-\widetilde{f}^{2})\int_{t}^{\infty}dx\frac{e^{-3x}}{(1-f)(f-\widetilde{f}^{2})^{2}}\int_{0}^{x}dy\leavevmode\nobreak\
M^{(i)}(y)\\\ &+\left(2\int_{0}^{\infty}dz\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\
a^{(i)}_{v}(z)\right)e^{t}(f-\widetilde{f}^{2})\int_{t}^{\infty}dx\frac{e^{-3x}}{(1-f)(f-\widetilde{f}^{2})^{2}}\int_{0}^{x}dy\leavevmode\nobreak\
e^{y}\leavevmode\nobreak\ \widetilde{f}\leavevmode\nobreak\
(f-\widetilde{f}^{2})\\\ \end{split}$ (58)

where

$\begin{split}M^{(i)}(x)=&\int_{0}^{x}dx\leavevmode\nobreak\
e^{2x}\left(f-\widetilde{f}^{2}\right)\Bigg{(}\widetilde{f}\leavevmode\nobreak\
e^{-x}\left(-\frac{2D}{K}\right)\int_{0}^{\infty}dz\leavevmode\nobreak\
e^{-z}\int_{0}^{z}dt\frac{e^{t}\leavevmode\nobreak\
\mathfrak{v}^{(i)}_{\text{metric}}(t)}{1-f(t)}-\frac{1}{N}\mathfrak{v}^{(i)}_{\text{gauge}}(x)\\\
&+2\leavevmode\nobreak\
\widetilde{f}\left(\frac{D}{K}\right)e^{-x}\int_{0}^{x}dt\leavevmode\nobreak\
\frac{e^{t}\leavevmode\nobreak\
\mathfrak{v}^{(i)}_{\text{metric}}(t)}{1-f(t)}\Bigg{)}\end{split}$ (59)

and

$\begin{split}&\int_{0}^{\infty}dt\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\ a^{(i)}_{v}=\frac{A}{B}\\\
\text{where}\leavevmode\nobreak\ &B=1-2\int_{0}^{\infty}dz\leavevmode\nobreak\
\widetilde{Q}\leavevmode\nobreak\
(f-\widetilde{f}^{2})\int_{z}^{\infty}dx\frac{e^{-3x}}{(1-f)(f-\widetilde{f}^{2})^{2}}\int_{0}^{x}dy\leavevmode\nobreak\
e^{y}\leavevmode\nobreak\ \widetilde{f}\leavevmode\nobreak\
(f-\widetilde{f}^{2})\\\ \text{And}\leavevmode\nobreak\
&A=-\int_{0}^{\infty}dz\leavevmode\nobreak\ \widetilde{Q}\leavevmode\nobreak\
(f-\widetilde{f}^{2})\int_{z}^{\infty}dx\frac{e^{-3x}}{(1-f)(f-\widetilde{f}^{2})^{2}}\int_{0}^{x}dy\leavevmode\nobreak\
M^{(i)}(y)\end{split}$ (60)

and

$\begin{split}a^{(i)}_{s}(y)=-e^{-y}\left(\frac{1}{N}\right)\int_{0}^{\infty}\leavevmode\nobreak\
d\rho\leavevmode\nobreak\ e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\
s^{(i)}_{\text{gauge}}(\zeta)+\left(\frac{1}{N}\right)\int_{y}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{(i)}_{\text{gauge}}(\zeta)\end{split}$ (61)

and

$\begin{split}&S^{(i)}_{1}(y)=-4\int_{y}^{\infty}d\rho\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\ a^{(i)}_{s}(\rho)-e^{-y}\leavevmode\nobreak\
A_{scalar}+\left(\frac{2}{N}\right)\int_{y}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{(i)}_{\text{metric}}(\zeta)\\\
\text{where}\leavevmode\nobreak\
&A_{scalar}=-4\int_{0}^{\infty}d\rho\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\
a^{(i)}_{s}(\rho)+\left(\frac{2}{N}\right)\int_{0}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{(i)}_{\text{metric}}(\zeta)\\\ \end{split}$
(62)

and correction to the tensor sector

$\begin{split}{\cal
T}\left(W\right)=2\frac{D}{K}\text{log}\left[1-\widetilde{Q}^{2}e^{-W}\right]\end{split}$
(63)

where

$\begin{split}&\mathfrak{v}^{(1)}_{\text{metric}}=\frac{\widetilde{f}^{2}-f}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathfrak{v}^{(2)}_{\text{metric}}=\mathfrak{v}^{(4)}_{\text{metric}}=\frac{f}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ \mathfrak{v}^{(\leavevmode\nobreak\
3)}_{\text{metric}}=-\frac{\widetilde{f}^{2}}{2}\\\
&\mathfrak{v}^{(1)}_{\text{gauge}}=\widetilde{f}\leavevmode\nobreak\
,\leavevmode\nobreak\ \mathfrak{v}^{(2)}_{\leavevmode\nobreak\
\text{gauge}}=-\widetilde{f}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\mathfrak{v}^{(\leavevmode\nobreak\
4)}_{\text{gauge}}=-\widetilde{f}\end{split}$ (64)

and

$\begin{split}&{s}^{(1)}_{\text{metric}}=\frac{N}{2}\left(\dot{\cal
V}^{(2)}+\dot{\cal V}^{(3)}+\dot{\cal
V}^{(4)}\right)-\widetilde{f}^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\
{s}^{(2)}_{\text{metric}}=\frac{N}{2}\left(-\dot{\cal V}^{(2)}+\dot{\cal
V}^{(3)}\right)+\widetilde{f}^{2}\\\
&{s}^{(3)}_{\text{metric}}=\frac{f-\widetilde{f}^{2}}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ {s}^{(4)}_{\text{metric}}=\frac{N}{2}\dot{\cal
V}^{(1)}\leavevmode\nobreak\ ,\leavevmode\nobreak\
{s}^{(5)}_{\text{metric}}=\left(\widetilde{Q}\widetilde{f}-\widetilde{f}^{2}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ {s}^{(6)}_{\text{metric}}=0\\\
&{s}^{(7)}_{\text{metric}}=-\widetilde{f}^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ {s}^{(8)}_{\text{metric}}=\frac{N}{2}\left(-\dot{\cal
V}^{(2)}+\dot{\cal V}^{(3)}\right)\\\
&s^{(1)}_{\text{gauge}}=N\left(\widetilde{f}{\cal
V}^{(2)}+f\dot{a}^{(2)}_{v}+\widetilde{f}{\cal
V}^{(4)}+f\dot{a}^{(4)}_{v}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\
s^{(2)}_{\text{gauge}}=N\left(-\widetilde{f}{\cal
V}^{(2)}-f\dot{a}^{(2)}_{v}+\widetilde{f}{\cal
V}^{(3)}+f\dot{a}^{(3)}_{v}\right)\\\
&s^{(4)}_{\text{gauge}}=N\left(\widetilde{f}{\cal
V}^{(1)}+f\dot{a}^{(1)}_{v}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\
s^{(6)}_{\text{gauge}}=\widetilde{f}\leavevmode\nobreak\ ,\leavevmode\nobreak\
s^{(3)}_{\text{gauge}}=s^{(5)}_{\text{gauge}}=s^{(7)}_{\text{gauge}}=0\\\
&s^{(8)}_{\text{gauge}}=N\left(-\widetilde{f}{\cal
V}^{(2)}-f\dot{a}^{(2)}_{v}+\widetilde{f}{\cal
V}^{(3)}+f\dot{a}^{(3)}_{v}\right)+\widetilde{f}\end{split}$ (65)

Table 2: Membrane Data Scalar | Vector | Tensor
---|---|---
${\cal S}^{(1)}\equiv\frac{\left(U\cdot\nabla\right)K}{K}$ | ${V}^{(1)}_{A}\equiv P^{C}_{A}\left(\frac{\nabla_{C}K}{K}\right)$ | $t_{AB}\equiv P^{C}_{A}P^{C^{\prime}}_{B}\left[\left(\frac{\nabla_{C}O_{C^{\prime}}+\nabla_{C^{\prime}}O_{C}}{2}\right)-\frac{P_{CC^{\prime}}}{D}\left(\nabla\cdot O\right)\right]$
${\cal S}^{(2)}\equiv U\cdot K\cdot U$ | ${V}^{(2)}_{A}\equiv P^{C}_{A}\left(U\cdot\nabla\right)O_{C}$ |
${\cal S}^{(3)}\equiv\hat{\nabla}\cdot U$ | ${V}^{(3)}_{A}\equiv P^{C}_{A}\left(U\cdot\nabla\right)U_{C}$ |
${\cal S}^{(4)}\equiv\frac{\hat{\nabla}^{2}K}{K^{2}}$ | ${V}^{(4)}_{A}\equiv P^{C}_{A}\left(\frac{\hat{\nabla}^{2}U_{C}}{K}\right)$ |
${\cal S}^{(5)}\equiv U\cdot\left(\frac{\nabla\widetilde{Q}}{\widetilde{Q}}\right)$ |  |
${\cal S}^{(6)}\equiv\frac{1}{K}\nabla\cdot\left(\frac{\nabla\widetilde{Q}}{\widetilde{Q}}\right)$ |  |
${\cal S}^{(7)}\equiv\frac{K}{D}$ |  |
${\cal S}^{(8)}\equiv\frac{R_{uu}}{K}$ |  |

###  3.1 The dual system

The large-$D$ gravity solutions, described in the previous subsection, are
dual to a co-dimension one, massive and charged, membrane embedded in the
asymptotic geometry (AdS for our purpose). The membrane is characterized by a
velocity field $U$, named as ‘membrane velocity’, a charge field
$\widetilde{Q}$ and a shape function $\psi$ (the same scalar fields that
appear in the bulk metric and gauge fields). The function $\psi$ is a harmonic
function with respect to the background geometry. Just like in fluid gravity
correspondence, here also the velocity field $U$ and the shape function $\psi$
cannot be chosen arbitrarily. They have to satisfy some constraint equations,
which we shall refer to as membrane equations. For every solution to these
membrane equations, we have one solution to Einstein’s equations. The membrane
equations are given by

$\begin{split}&P^{A}_{C}\left[\frac{\widehat{\nabla}^{2}U_{A}}{K}-\left(1+\widetilde{Q}^{2}\right)\frac{\widehat{\nabla}_{A}K}{K}+U^{B}K_{BA}-\left(1+\widetilde{Q}^{2}\right)\left(U\cdot\widehat{\nabla}U_{A}\right)\right]=\mathcal{O}\left(\frac{1}{D}\right)\\\
&\hat{\nabla}\cdot U=\mathcal{O}\left(\frac{1}{D}\right)\\\
&\frac{\widehat{\nabla}^{2}\widetilde{Q}}{K}-U\cdot\widehat{\nabla}\widetilde{Q}-\widetilde{Q}\left[\frac{U\cdot\widehat{\nabla}K}{K}-U\cdot
K\cdot
U-\frac{R_{uu}}{K}\right]=\mathcal{O}\left(\frac{1}{D}\right)\end{split}$ (66)

where

$\begin{split}P_{AB}=\bar{W}_{AB}-n_{A}n_{B}+U_{A}U_{B}\leavevmode\nobreak\
,\leavevmode\nobreak\ {R}_{uu}=U^{A}\bar{R}_{AB}U^{B}\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\bar{R}_{AB}=\left(D-1\right)\lambda\bar{W}_{AB}\end{split}$ (67)

##  4 Comparing fluid-gravity and membrane-gravity dualities :

In this section we compare the two perturbation techniques, ‘derivative
expansion’ and ‘large-$D$ expansion’ which are used to generate dynamical
black-brane solutions to Einstein equations.
‘Derivative expansion’ is used to solve Einstein’s equations with negative
cosmological constant, whereas large-$D$ expansion technique is used to solve
Einstein equations with or without cosmological constant. ‘Derivative
expansion’ generate gravity solutions that are dual to the relativistic
Navier-Stokes equations of fluid dynamics. On the other hand large-$D$
expansion techniques generate solutions that are dual to a co-dimension one
dynamical membrane embedded in some background space. Like in the previous
papers Bhattacharyya2019leading ; Bhattacharyya2019sunleading , here also our
goal is to compare these two gravity solutions along with their dual systems
for the charged case. We will show that in appropriate regime of parameter
space there exists an overlap regime between these two different looking
gravity solutions generated by two different perturbation techniques, which we
can see after a coordinate transformation.

###  4.1 The split of the hydrodynamic metric

As we have mentioned earlier, the metric generated in large-$D$ expansion
technique are written in a split form, background plus ‘rest’. Here we have a
null geodesic, which when contracted with the ‘rest’ part vanishes to all
order in $\frac{1}{D}$. This is not the case for hydrodynamic metric.
So to compare these two solutions, the first step would be to split the
hydrodynamic metric into background plus ‘rest’.

We shall do it in the following way.

First we find out an affinely parametrized null geodesic
$\bar{O}^{A}\partial_{A}$ w.r.t the full space-time metric which passes
thorough the event horizon of the space-time. This null geodesic vector is
normalized in a way such that $O^{A}n_{A}=1$ ($O_{A}$ is related to
$\bar{O}_{A}$ by an overall normalization constant), everywhere in the
background, where $n_{A}$ is the unit normal to the constant $\psi$
hypersurfaces. Now the hydrodynamic metric is written in a gauge where
$\mathcal{G}_{rr}=0,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\mathcal{G}_{r\mu}=-u_{\mu}$ to all order in derivative expansion. In this
gauge $k^{A}\partial_{A}=\zeta(x)\partial_{r}$ is an affinely parametrized
null geodesic to all order in derivative expansion. However we have to
normalize this null geodesic and hence we can set $\zeta(x)$ to be one.
Ultimately we will find that $\bar{O}^{A}\partial_{A}=\partial_{r}$ is the
null geodesic which split the hydrodynamic metric as we want. After that we
will choose a coordinate system $Y^{A}\equiv\left(\rho,y^{\mu}\right)$ where
the background of hydrodynamic metric take the following form

$\begin{split}dS^{2}_{\text{background}}=\bar{G}_{AB}dY^{A}dY^{B}=\frac{d\rho^{2}}{\rho^{2}}+\rho^{2}\eta_{\mu\nu}\end{split}$
(68)

The $Y^{A}$ coordinates are related to $X^{A}\equiv\left(r,x^{\mu}\right)$
coordinate by the mapping $f$

$\begin{split}Y^{A}=f^{A}\left(X\right)\end{split}$ (69)

We can determine this mapping function by the following equation

$\begin{split}\bar{O}^{A}\mathcal{G}_{AB}|_{\\{X\\}}=\bar{O}^{A}\left(\frac{\partial
f^{C}}{\partial X^{A}}\right)\left(\frac{\partial f^{{C^{\prime}}}}{\partial
X^{B}}\right)\bar{G}_{CC^{\prime}}|_{\\{X\\}}\end{split}$ (70)

If we use the fact that $\bar{O}^{A}\partial_{A}=\partial_{r}$, then (70) can
be written as

$\begin{split}\mathcal{G}^{(rest)}_{rB}=0\end{split}$ (71)

where
$\mathcal{G}^{(rest)}_{AB}=\left(\mathcal{G}_{AB}-\mathcal{\bar{G}}_{AB}\right)$,
all written in $\\{X^{A}\\}$ coordinates. As previously noted the hydrodynamic
metric metric is written in a particular gauge
$\mathcal{G}_{rr}=0,\text{and}\leavevmode\nobreak\
\mathcal{G}_{r\mu}=-u_{\mu}$ to all order in derivative expansion. We will
find that coordinate transformation of the form

$\begin{split}\rho=r+\chi(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
y^{\mu}=x^{\mu}+\frac{u^{\mu}}{r+\chi(x)}+\zeta^{\mu}(x)\end{split}$ (72)

where $u_{\mu}\partial_{\nu}\zeta^{\mu}=0$, keep the hydrodynamic metric in
this required gauge. Further it will turn out that for the exact matching of
the two metric and gauge field we should have $\zeta^{\mu}=0$ and
$\chi(x)=-\frac{\Theta}{D-2}$. So finally we have222For a detail discussion
see Bhattacharyya2019sunleading .

$\begin{split}\rho=r-\frac{\Theta}{D-2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
y^{\mu}=x^{\mu}+\frac{u^{\mu}}{r-\frac{\Theta}{D-2}}\end{split}$ (73)

If we apply these coordinate transformations, the background metric in
$\\{X^{A}\\}$ coordinate can be written as333The inverse of the background
metric and the christoffel symbols w.r.t background metric are give in
Appendix B

$\begin{split}\mathcal{\bar{G}}_{rr}&=0\\\ \mathcal{\bar{G}}_{\mu
r}&=-u_{\mu}\\\ \mathcal{\bar{G}}_{\mu\nu}&=r^{2}\left({\cal
P}_{\mu\nu}-u_{\mu}u_{\nu}\right)-r\left(u_{\mu}a_{\nu}+u_{\nu}a_{\mu}\right)+2r\sigma_{\mu\nu}+2r\frac{\Theta}{D-2}u_{\mu}u_{\nu}+\mathcal{O}\left(\partial^{2}\right)\\\
\end{split}$ (74)

Once we know the background metric in $\\{X^{A}\\}$ coordinates, by
subtracting it from the full metric we can determine
$\mathcal{G}^{(rest)}_{AB}$. Now by our construction
$\mathcal{G}^{(rest)}_{rr}$ and $\mathcal{G}^{(rest)}_{r\mu}$ are identically
zero to all order in derivative expansion and the
$\mathcal{G}^{(rest)}_{\mu\nu}$ component can be written as

$\begin{split}\mathcal{G}^{\left(rest\right)}_{\mu\nu}&=\mathcal{G}^{(S1)}u_{\mu}u_{\nu}+\mathcal{G}^{(S2)}{\cal
P}_{\mu\nu}+\left(\mathcal{G}^{(V)}_{\mu}\leavevmode\nobreak\
u_{\nu}+\mathcal{G}^{(V)}_{\nu}\leavevmode\nobreak\
u_{\mu}\right)+\mathcal{G}^{(T)}_{\mu\nu}\\\ \text{where,}\\\
&\mathcal{G}^{(S1)}=r^{2}\bigg{(}1-V(r)\bigg{)}\\\
&\mathcal{G}^{(S2)}=\mathcal{O}\left(\partial^{2}\right)\\\
&\mathcal{G}^{(V)}_{\mu}=-\frac{3(D-3)\leavevmode\nobreak\
r^{2}\leavevmode\nobreak\
Q_{C}}{r_{H}^{2}}\bigg{[}1-(D-2)Q_{C}f(Q_{C})\bigg{]}F_{1}(\rho,M)\leavevmode\nobreak\
{\cal P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\
&\mathcal{G}^{(T)}_{\mu\nu}=2r\left(\frac{r}{r_{H}}F_{2}\left(\rho,M\right)-1\right)\sigma_{\mu\nu}\end{split}$
(75)

###  4.2 Membrane data in terms of fluid data

In derivative expansion the solutions are characterized by a velocity filed
$u$, called fluid velocity, a temperature field $T$ and a charge field $Q$,
whereas in large-$D$ expansion characterising data are the shape function
$\psi$, the charge $\tilde{Q}$ and the membrane velocity $U$. The number of
data does match on both sides, as it should be. But these variables are not
the same and we need to rewrite one in terms of the other, to perform a
comparison. In this subsection we rewrite the characterising data of the
membrane in terms of the fluid variables.

####  4.2.1 Determining $\psi$

As we have described before, $\psi$ is a scalar function, harmonic with
respect to the background geometry. The hypersurface $\psi=1$ is identified
with the dynamical horizon of the black brane solution. So we have to solve
the differential equation $\nabla^{2}\psi^{-D}=0$ in this background geometry
order by order order in both the perturbation parameters.

After solving we find the following expression for the function $\psi$ (see
appendix-B of (Bhattacharyya2019leading, ) for the details of the calculation)

$\psi\left(r,x^{\mu}\right)=1+\left(1-\frac{1}{D}\right)\left(\frac{r}{r_{H}}-1\right)+\mathcal{O}\left(\partial^{2},\frac{1}{D^{3}}\right)$
(76)

####  4.2.2 Determining $U^{A}$

Once we have the $\psi$ field everywhere, we could compute the unit normal to
the constant $\psi$ surfaces.

$\begin{split}&n_{r}=\frac{1}{r}+\mathcal{O}\left(\partial^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
n_{\mu}=-\frac{\Theta}{D-2}u_{\mu}+a_{\mu}+f(Q_{C}){\cal{P}}^{\alpha}_{\mu}\partial_{\alpha}Q_{C}+\mathcal{O}\left(\partial^{2}\right)\\\
&n^{r}=r-\frac{\Theta}{D-2}+\mathcal{O}\left(\partial^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
n^{\mu}=\frac{u^{\mu}}{r}+\frac{f(Q_{C})}{r^{2}}\leavevmode\nobreak\ {\cal
P}^{\mu\alpha}\leavevmode\nobreak\
\partial_{\alpha}Q_{C}+\mathcal{O}\left(\partial^{2}\right)\\\ \end{split}$
(77)

Now $U^{A}$ is defined as follows

$U_{A}=n_{A}-O_{A}$ (78)

After properly normalizing our null geodesic field by
$O^{A}=\frac{\bar{O}^{A}}{\left(n_{A}\cdot\bar{O}^{A}\right)}$ , we have

$\begin{split}&O_{r}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
O_{\mu}=-ru_{\mu}+\mathcal{O}\left(\partial^{2}\right)\\\
&O^{r}=r+\mathcal{O}\left(\partial^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ O^{\mu}=0\\\ \end{split}$ (79)

Then we can find out the membrane velocity $U_{A}$ as

$\begin{split}&U_{r}=\frac{1}{r}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
U_{\mu}=ru_{\mu}-\frac{\Theta}{D-2}u_{\mu}+a_{\mu}+f(Q_{C}){\cal{P}}^{\alpha}_{\mu}\partial_{\alpha}Q_{C}+\mathcal{O}\left(\partial^{2}\right)\\\
&U^{r}=-\frac{\Theta}{D-2}+\mathcal{O}\left(\partial^{2}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
U^{\mu}=\frac{u^{\mu}}{r}+\frac{f(Q_{C})}{r^{2}}\leavevmode\nobreak\ {\cal
P}^{\mu\alpha}\leavevmode\nobreak\
\partial_{\alpha}Q_{C}+\mathcal{O}\left(\partial^{2}\right)\\\ \end{split}$
(80)

####  4.2.3 Determining $\widetilde{Q}$

Next our goal is to write the smooth function $\widetilde{Q}$ present in the
large $D$ metric and gauge field in terms of fluid data. This function
satisfies the subsidiary condition $\left(n\cdot\nabla\right)\widetilde{Q}=0$.
The boundary condition which fix it completely is given by

$\begin{split}\widetilde{Q}|_{\psi=1}&=\frac{1}{\sqrt{2}}\left(U^{M}A_{M}\right)|_{\psi=1}\\\
&=-\frac{1}{\sqrt{2}}\leavevmode\nobreak\
\frac{\sqrt{3}Q}{2\leavevmode\nobreak\ r^{D-2}}|_{\psi=1}\\\
&=-\frac{\sqrt{3}Q}{2\sqrt{2}\leavevmode\nobreak\ r_{H}^{D-2}}\\\ \end{split}$
(81)

We want to solve $\widetilde{Q}$ such that
$\left(n\cdot\partial\right)\widetilde{Q}=0$ and for that we will take the
following expansion in $\widetilde{Q}$.

$\begin{split}&\widetilde{Q}=\widetilde{Q}_{0}+\widetilde{Q}_{1}\leavevmode\nobreak\
\left(r-r_{H}\right)+\cdots\\\ \text{where}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
&\widetilde{Q}_{0}=-\frac{\sqrt{3}Q}{2\sqrt{2}\leavevmode\nobreak\
r_{H}^{D-2}}\\\ \end{split}$ (82)

Collecting coefficients of $\left(r-r_{H}\right)^{0}$ after applying
$\left(n\cdot\partial\right)\widetilde{Q}=0$ and using the fact that
$\left(n\cdot\partial\right)\widetilde{Q}_{0}=\frac{\sqrt{3}Q}{2\sqrt{2}\leavevmode\nobreak\
r\leavevmode\nobreak\
r_{H}^{D-2}}\left(\frac{\left(u\cdot\partial\right)Q}{Q}+\Theta\right)=0$, we
have $\widetilde{Q}_{1}=0$. So we have

$\widetilde{Q}=-\frac{\sqrt{3}Q}{2\sqrt{2}\leavevmode\nobreak\
r_{H}^{D-2}}+\mathcal{O}\left(\partial^{2}\right)$

and finally we have

$\widetilde{Q}=-\frac{\sqrt{3}\leavevmode\nobreak\
Q_{C}}{2\sqrt{2}}+\mathcal{O}\left(\partial^{2}\right)$ (83)

####  4.2.4 Relevant derivatives of the basic data

Large-$D$ metric is determined in terms of the basic functions $\psi$,
$\widetilde{Q}$ and $U^{A}$ and their derivatives with respect to the induced
coordinates on the membrane. In this subsection we shall convert these
‘membrane derivatives’ of the basic ‘membrane data’ in terms of the fluid
data.

One of the key structure that arises repeatedly in large-$D$ construction is
the extrinsic curvature of the membrane, viewed as a hypersurface embedded in
the background. The expressions for extrinsic curvature can be re-expressed in
terms of fluid variables as

$\begin{split}&K_{rr}=-\frac{1}{r^{2}}+\mathcal{O}\left(\partial^{2}\right)\\\
&K_{r\mu}=-u_{\mu}+\frac{1}{r}\left(\frac{\Theta}{D-2}u_{\mu}-a_{\mu}\right)+\mathcal{O}\left(\partial^{2}\right)\\\
&K_{\mu\nu}=r^{2}\left({\cal
P}_{\mu\nu}-u_{\mu}u_{\nu}\right)+2\leavevmode\nobreak\
r\left(\frac{\Theta}{D-2}u_{\mu}u_{\nu}-\frac{u_{\mu}a_{\nu}+u_{\nu}a_{\mu}}{2}+\sigma_{\mu\nu}\right)+\mathcal{O}\left(\partial^{2}\right)\\\
&K=\left(D-1\right)+\mathcal{O}\left(\partial^{2}\right)\end{split}$ (84)

where $K_{AB}$ is defined as

$\begin{split}&K_{AB}=\Pi_{A}^{C}\leavevmode\nobreak\
\nabla_{C}\leavevmode\nobreak\ n_{B}\\\ \text{with}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&\Pi_{AB}=\bar{\mathcal{G}}_{AB}-n_{A}\leavevmode\nobreak\ n_{B}\\\
\end{split}$ (85)

The rest of the data that are relevant for our purpose are presented in the
tables -(3), (4) and (5). Here $\hat{\nabla}$ is defined for a general $n$
index tensor $X_{A_{1}A_{2}\cdots A_{n}}$ as

$\hat{\nabla}_{A}X_{A_{1}A_{2}\cdots
A_{n}}=\Pi^{C}_{A}\Pi^{C_{1}}_{A_{1}}\Pi^{C_{2}}_{A_{2}}\cdots\Pi^{C_{n}}_{A_{n}}{\nabla}_{C}X_{C_{1}C_{2}\cdots
C_{n}}$ (86)

Table 3: Scalar large-$D$ Data in terms of fluid Data Large-$D$ Data | Corresponding Fluid Data
---|---
${\mathcal{S}}^{(1)}\equiv\frac{\left(U\cdot\nabla\right)K}{K}$ | $=\leavevmode\nobreak\ 0$
${\mathcal{S}}^{(2)}\equiv U\cdot K\cdot U$ | $=\leavevmode\nobreak\ -1$
${\mathcal{S}}^{(3)}\equiv\hat{\nabla}\cdot U=\Pi^{AB}(\nabla_{A}U_{B})$ | $=\leavevmode\nobreak\ 0$
${\mathcal{S}}^{(4)}\equiv\frac{\hat{\nabla}^{2}K}{K^{2}}$ | $=\leavevmode\nobreak\ 0$
${\mathcal{S}}^{(5)}\equiv U\cdot\left(\frac{\nabla\widetilde{Q}}{\widetilde{Q}}\right)$ | $=\leavevmode\nobreak\ 0$
${\mathcal{S}}^{(6)}\equiv\frac{1}{K}\nabla\cdot\left(\frac{\nabla\widetilde{Q}}{\widetilde{Q}}\right)$ | $=\leavevmode\nobreak\ 0$
${\mathcal{S}}^{(7)}\equiv\frac{K}{D}$ | $=\leavevmode\nobreak\ 1-\frac{1}{D}$
${\mathcal{S}}^{(8)}\equiv\frac{R_{uu}}{K}$ | $=\leavevmode\nobreak\ -\lambda$
Table 4: Vector large-$D$ Data in terms of fluid Data Large-$D$ Data | Corresponding Fluid Data
---|---
${V}^{(1)}_{A}\leavevmode\nobreak\ dX^{A}\equiv P^{C}_{A}\left(\frac{\nabla_{C}K}{K}\right)$ | $=\leavevmode\nobreak\ 0$
${V}^{(2)}_{A}\leavevmode\nobreak\ dX^{A}\equiv P^{C}_{A}\left(U\cdot\nabla\right)O_{C}$ | $=\leavevmode\nobreak\ f\left(Q_{C}\right){\cal P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\leavevmode\nobreak\ dx^{\mu}$
${V}^{(3)}_{A}\leavevmode\nobreak\ dX^{A}\equiv P^{C}_{A}\left(U\cdot\nabla\right)U_{C}$ | $=\leavevmode\nobreak\ -f\left(Q_{C}\right){\cal P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\leavevmode\nobreak\ dx^{\mu}$
${V}^{(4)}_{A}\leavevmode\nobreak\ dX^{A}\equiv P^{C}_{A}\left(\frac{\hat{\nabla}^{2}U_{C}}{K}\right)$ | $=\leavevmode\nobreak\ -f\left(Q_{C}\right){\cal P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\leavevmode\nobreak\ dx^{\mu}$
Table 5: Tensor large-$D$ Data in terms of fluid Data Large-$D$ Data | Corresponding Fluid Data
---|---
$t_{AB}\equiv P^{C}_{A}P^{C^{\prime}}_{B}\left[\left(\frac{\nabla_{C}O_{C^{\prime}}+\nabla_{C^{\prime}}O_{C}}{2}\right)-\frac{P_{CC^{\prime}}}{D}\left(\nabla\cdot O\right)\right]$ | $=\leavevmode\nobreak\ \begin{aligned} &-r\sigma_{\mu\nu}+\frac{1}{D}\Big{(}r^{2}{\cal{P}_{\mu\nu}}+2r\sigma_{\mu\nu}\\\ &+rf\left(Q_{C}\right)\left(u_{\mu}{\cal{P}^{\lambda}_{\nu}}\partial_{\lambda}Q_{C}+u_{\nu}{\cal{P}^{\lambda}_{\mu}}\partial_{\lambda}Q_{C}\right)\Big{)}\end{aligned}$

###  4.3 Comparing the metrics and gauge fields

In this subsection we shall take the large $D$ limit of the fluid metric and
gauge field and match with the metric and gauge field in large $D$ side after
expressing them in terms of fluid data.

####  4.3.1 Comparing the gauge fields

At first we decompose the fluid gauge field into scalar and vector components.
As from the gauge condition $A_{r}$ component of the gauge field is zero to
all order in derivative, we write only the components in the boundary
directions.

$\begin{split}A_{\mu}^{\text{(fluid)}}=\mathcal{B}^{(S)}\leavevmode\nobreak\
u_{\mu}+\mathcal{B}^{(V)}_{\mu}\end{split}$ (87)

where

$\begin{split}&\mathcal{B}^{(S)}=\frac{\sqrt{3}\leavevmode\nobreak\
r\leavevmode\nobreak\ Q_{C}}{2}\left(\frac{r_{H}}{r}\right)^{D-2}\\\
&\mathcal{B}^{(V)}_{\mu}=-2\sqrt{3}\left(\frac{r}{r_{H}}\right)^{D}\bigg{[}1-(D-2)Q_{C}\leavevmode\nobreak\
f(Q_{C})\bigg{]}F_{1}^{(1,0)}(\rho,M)\leavevmode\nobreak\ {\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\ \end{split}$ (88)

The gauge field in large-$D$ side is given as

$\begin{split}&A_{M}=\sqrt{2}\left[\widetilde{f}\leavevmode\nobreak\
O_{M}+\frac{1}{D}A^{(1)}_{M}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\right]\\\
&\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{A}^{(1)}_{M}=\mathcal{A}^{(s)}O_{M}+\mathcal{A}^{(v)}_{M}\end{split}$
(89)

Also in this case the radial component of the gauge field vanishes and only
the components in boundary directions are non-zero. We decompose this into
scalar and vector components as follows

$\begin{split}A_{\mu}^{\text{(D)}}=\mathcal{Y}^{(S)}\leavevmode\nobreak\
u_{\mu}+\mathcal{Y}^{(V)}_{\mu}\end{split}$ (90)

where

$\begin{split}&\mathcal{Y}^{(S)}=-\sqrt{2}\leavevmode\nobreak\
r\left(\widetilde{f}+\frac{1}{D}\mathcal{A}^{(s)}\right)+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{Y}^{(V)}_{\mu}=\frac{1}{D}\sqrt{2}\leavevmode\nobreak\
\mathcal{A}^{(v)}_{\mu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
\end{split}$ (91)

$\mathcal{A}^{(s)}$ is defined as follows

$\begin{split}\mathcal{A}^{(s)}&=\sum_{i=1}^{N_{s}}a^{(i)}_{s}(Y)\leavevmode\nobreak\
\mathcal{S}^{(i)}\\\ &=\left(a^{(2)}_{s}(Y)\leavevmode\nobreak\
\mathcal{S}^{(2)}+a^{(7)}_{s}(Y)\leavevmode\nobreak\
\mathcal{S}^{(7)}+K\leavevmode\nobreak\
a^{(R_{uu})}_{s}(Y)\leavevmode\nobreak\ \mathcal{S}^{(8)}\right)\\\
&=\left(-a^{(2)}_{s}(Y)+a^{(7)}_{s}(Y)+K\leavevmode\nobreak\
a^{(R_{uu})}_{s}(Y)\right)\\\ &=a^{\text{(total)}}_{s}(Y)\end{split}$ (92)

where $Y=D(\psi-1)$.444Here one should note that the integrations in the
large-$D$ side are parametrized by $W=D(\psi-1)$. On the other hand we can
define another parameter $\widetilde{R}=D\left(\frac{r}{r_{H}}-1\right)$ to
expand other functions in inverse power of dimensions. But it is easy to check
that $W=\widetilde{R}+\mathcal{O}\left(\frac{1}{D}\right)$. And hence up to
the order we are interested the two parameters are just equal and we simply
denote both of them by the $\mathcal{O}(1)$ parameter $Y$ without any further
confusion.
where

$\begin{split}a^{\text{(total)}}_{s}(Y)&=-e^{-Y}\left(\frac{1}{N}\right)\int_{0}^{\infty}\leavevmode\nobreak\
d\rho\leavevmode\nobreak\ e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\
s^{\text{(total)}}_{\text{gauge}}(\zeta)+\left(\frac{1}{N}\right)\int_{Y}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{\text{(total)}}_{\text{gauge}}(\zeta)\\\
\end{split}$ (93)

And $s^{\text{(total)}}_{\text{gauge}}$ is defined as

$\begin{split}s^{\text{(total)}}_{\text{gauge}}&=-s^{(2)}_{\text{gauge}}+s^{(7)}_{\text{gauge}}+s^{(R_{uu})}_{\text{gauge}}\\\
&=\widetilde{f}\\\ &=\widetilde{Q}\leavevmode\nobreak\
e^{-Y}+\mathcal{O}\left(\frac{1}{D}\right)\end{split}$ (94)

Substituting (94) into (93) we have

$\begin{split}&a^{\text{(total)}}_{s}(Y)=Y\leavevmode\nobreak\
\widetilde{Q}\leavevmode\nobreak\ e^{-Y}\end{split}$ (95)

Putting this in (92) we have

$\begin{split}\mathcal{A}^{(s)}=Y\leavevmode\nobreak\
\widetilde{Q}\leavevmode\nobreak\ e^{-Y}\end{split}$ (96)

The vector component calculated up to leading order in large-$D$ as

$\begin{split}\mathcal{A}^{(v)}_{\mu}&=\sum_{i=1}^{N_{s}}a^{(i)}_{v}(Y)\leavevmode\nobreak\
{V}^{(i)}_{\mu}\\\
&=f\left(Q_{C}\right)\left(a^{(2)}_{v}(Y)-a^{(3)}_{v}(Y)-a^{(4)}_{v}(Y)\right){\cal
P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\
&=\mathcal{O}\left(\frac{1}{D}\right)\end{split}$ (97)

So finally we have

$\begin{split}&\mathcal{Y}^{(S)}=-\sqrt{2}\leavevmode\nobreak\
r\left(\widetilde{f}+\frac{1}{D}Y\leavevmode\nobreak\
\widetilde{Q}\leavevmode\nobreak\
e^{-Y}\right)+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{Y}^{(V)}_{\mu}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
\end{split}$ (98)

Now we have from A.1

$\begin{split}F_{1}^{(1,0)}\left(1+\frac{Y}{D},M\right)=\left(\frac{1}{D}\right)^{2}\end{split}$
(99)

Now if we expand both large-$D$ and fluid gauge field up to
$\mathcal{O}\left(\frac{1}{D}\right)$ by substituting
$r=r_{H}\left(1+\frac{Y}{D}\right)$, we find

$\begin{split}&\mathcal{Y}^{(S)}-\mathcal{B}^{(S)}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{Y}^{(V)}_{\mu}-\mathcal{B}^{(V)}_{\mu}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
\end{split}$ (100)

So within the membrane region, the two gauge fields are equivalent to one
another.

####  4.3.2 Comparing the metric

As we have discussed earlier we can determine
$\mathcal{G}^{\left(rest\right)}_{\mu\nu}$, by simply subtracting the
background piece from the full hydrodynamic metric once we know the background
metric in $X^{A}$ coordinates. After decomposing
$\mathcal{G}^{\left(rest\right)}_{\mu\nu}$ into scalar, vector and tensor
components, we can write it as follows

$\begin{split}\mathcal{G}^{\left(rest\right)}_{\mu\nu}&=\mathcal{G}^{(S1)}u_{\mu}u_{\nu}+\mathcal{G}^{(S2)}{\cal
P}_{\mu\nu}+\left(\mathcal{G}^{(V)}_{\mu}\leavevmode\nobreak\
u_{\nu}+\mathcal{G}^{(V)}_{\nu}\leavevmode\nobreak\
u_{\mu}\right)+\mathcal{G}^{(T)}_{\mu\nu}\\\ \end{split}$ (101)

where

$\begin{split}&\mathcal{G}^{(S1)}=r^{2}\bigg{(}1-V(r)\bigg{)}\\\
&\mathcal{G}^{(S2)}=\mathcal{O}\left(\partial^{2}\right)\\\
&\mathcal{G}^{(V)}_{\mu}=-\frac{3(D-3)\leavevmode\nobreak\
r^{2}\leavevmode\nobreak\
Q_{C}}{r_{H}^{2}}\bigg{[}1-(D-2)Q_{C}f(Q_{C})\bigg{]}F_{1}(\rho,M)\leavevmode\nobreak\
{\cal P}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\
&\mathcal{G}^{(T)}_{\mu\nu}=2r\left(\frac{r}{r_{H}}F_{2}\left(\rho,M\right)-1\right)\sigma_{\mu\nu}\end{split}$
(102)

Now we will write the large $D$ metric in terms of fluid data. The large $D$
metric up to first order in $\frac{1}{D}$ can be written as

$\begin{split}&{W}_{AB}=\bar{{W}}_{AB}+fO_{A}O_{B}+\frac{1}{D}\mathcal{W}^{(1)}_{AB}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{W}^{(1)}_{AB}=\mathcal{Z}^{(s1)}O_{A}O_{B}+\left(\mathcal{Z}^{(v)}_{A}O_{B}+\mathcal{Z}^{(v)}_{B}O_{A}\right)+\mathcal{Z}^{(T)}_{AB}\end{split}$
(103)

Subtracting the background part from this full large-$D$ metric, we will get,
${W}^{\left(rest\right)}_{AB}$. Now from the construction of this metric
${W}^{\left(rest\right)}_{rr}$ and ${W}^{\left(rest\right)}_{r\mu}$ vanishes,
and we only have ${W}^{\left(rest\right)}_{\mu\nu}$, which we decompose into
scalar, vector and tensor components as follows

$\begin{split}{W}^{\left(rest\right)}_{\mu\nu}&=\mathcal{W}^{(S1)}u_{\mu}u_{\nu}+\mathcal{W}^{(S2)}{\cal
P}_{\mu\nu}+\left(\mathcal{W}^{(V)}_{\mu}\leavevmode\nobreak\
u_{\nu}+\mathcal{W}^{(V)}_{\nu}\leavevmode\nobreak\
u_{\mu}\right)+\mathcal{W}^{(T)}_{\mu\nu}\\\ \end{split}$ (104)

where

$\begin{split}&\mathcal{W}^{(S1)}=r^{2}\left(f+\frac{1}{D}\mathcal{Z}^{(s1)}\right)+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(S2)}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(V)}_{\mu}=-\frac{1}{D}\leavevmode\nobreak\
r\leavevmode\nobreak\
\mathcal{Z}^{(v)}_{\mu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(T)}_{\mu\nu}=\frac{1}{D}\mathcal{Z}^{(T)}_{\mu\nu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\end{split}$
(105)

So at first we determine $\mathcal{Z}^{(s1)}$ in terms of fluid variables. We
write the leading order expressions in large $D$ expansion by substituting
$r=r_{H}\left(1+\frac{Y}{D}\right)$.

$\begin{split}\mathcal{Z}^{(s1)}&=\sum_{i=1}^{N_{s}}S^{(i)}_{1}(Y)\leavevmode\nobreak\
\mathcal{S}^{(i)}\\\ &=\left(S^{(2)}_{1}(Y)\leavevmode\nobreak\
\mathcal{S}^{(2)}+S^{(7)}_{1}(Y)\leavevmode\nobreak\
\mathcal{S}^{(7)}+K\leavevmode\nobreak\
S^{(R_{uu})}_{1}(Y)\leavevmode\nobreak\ \mathcal{S}^{(8)}\right)\\\
&=\left(-S^{(2)}_{1}(Y)+S^{(7)}_{1}(Y)+K\leavevmode\nobreak\
S^{(8)}_{1}(Y)\right)\\\ &=S^{\text{(total)}}_{1s}(Y)\end{split}$ (106)

where

$\begin{split}&S^{\text{(total)}}_{1s}(y)=-4\int_{y}^{\infty}d\rho\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\
a^{\text{(total)}}_{s}(\rho)-e^{-y}\leavevmode\nobreak\
A_{\text{scalar}}^{\text{(total)}}+\left(\frac{2}{N}\right)\int_{y}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{\text{(total)}}_{\text{metric}}(\zeta)\\\
\text{where}\leavevmode\nobreak\
&A_{\text{scalar}}^{\text{(total)}}=-4\int_{0}^{\infty}d\rho\leavevmode\nobreak\
\widetilde{f}\leavevmode\nobreak\
a^{\text{(total)}}_{s}(\rho)+\left(\frac{2}{N}\right)\int_{0}^{\infty}d\rho\leavevmode\nobreak\
e^{-\rho}\int_{0}^{\rho}d\zeta\leavevmode\nobreak\
e^{\zeta}\leavevmode\nobreak\ s^{\text{(total)}}_{\text{metric}}(\zeta)\\\
\end{split}$ (107)

and $s^{\text{(total)}}_{\text{metric}}$ is defined as

$\begin{split}s^{\text{(total)}}_{\text{metric}}&=-s^{(2)}_{\text{metric}}+s^{(7)}_{\text{metric}}+s^{(R_{uu})}_{\text{metric}}\\\
&=-2\widetilde{f}^{2}\\\ &=-2\widetilde{Q}^{2}\leavevmode\nobreak\
e^{-2\leavevmode\nobreak\ Y}+\mathcal{O}\left(\frac{1}{D}\right)\end{split}$
(108)

Substituting (108) into (107), we have

$\begin{split}&A_{\text{scalar}}^{\text{(total)}}=-3\widetilde{Q}^{2}\\\
&S^{\text{(total)}}_{1s}(Y)=\widetilde{Q}^{2}e^{-2Y}\left(1-2Y-e^{Y}\right)\end{split}$
(109)

And hence we have

$\begin{split}\mathcal{Z}^{(s1)}&=\widetilde{Q}^{2}e^{-2Y}\left(1-2Y-e^{Y}\right)+\mathcal{O}\left(\frac{1}{D}\right)\end{split}$
(110)

Now we will calculate $\mathcal{Z}^{(V)}_{\mu}$ component

$\begin{split}\mathcal{Z}^{(v)}_{\mu}&=f\left(Q_{C}\right)\left(\mathcal{V}^{(2)}-\mathcal{V}^{(3)}-\mathcal{V}^{(4)}\right){\cal{P}}^{\lambda}_{\mu}\left({\partial}_{\lambda}Q_{C}\right)\\\
&=\mathcal{O}\left(\frac{1}{D}\right)\end{split}$ (111)

And the tensor component $\mathcal{Z}^{(T)}_{\mu\nu}$ is given by

$\begin{split}\mathcal{Z}^{(T)}_{\mu\nu}&=-2r\left(\frac{D}{K}\right)\text{log}\left(1-\widetilde{Q}^{2}e^{-Y}\right)\sigma_{\mu\nu}+\mathcal{O}\left(\frac{1}{D}\right)\end{split}$
(112)

So finally we have

$\begin{split}&\mathcal{W}^{(S1)}=r^{2}\left[f+\frac{1}{D}\left(\widetilde{Q}^{2}e^{-2Y}\left(1-2Y-e^{Y}\right)\right)\right]+\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(S2)}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(V)}_{\mu}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{W}^{(T)}_{\mu\nu}=-r\left(\frac{2}{K}\right)\text{log}\left(1-\widetilde{Q}^{2}e^{-Y}\right)\sigma_{\mu\nu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\end{split}$
(113)

From A.1, we have

$\begin{split}F_{1}\left(1+\frac{Y}{D},M\right)=\mathcal{O}\left(\frac{1}{D}\right)^{3}\end{split}$
(114)

hence

$\begin{split}\mathcal{G}^{(V)}_{\mu}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\end{split}$
(115)

and from A.2, we have

$\begin{split}\mathcal{G}^{(T)}_{\mu\nu}=-r\left(\frac{2}{D}\right)\text{log}\left(1-\widetilde{Q}^{2}e^{-Y}\right)\sigma_{\mu\nu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\end{split}$
(116)

So finally we have

$\begin{split}&\mathcal{G}^{(S1)}=r^{2}\bigg{(}1-V(r)\bigg{)}\\\
&\mathcal{G}^{(S2)}=\mathcal{O}\left(\partial^{2}\right)\\\
&\mathcal{G}^{(V)}_{\mu}=\mathcal{O}\left(\frac{1}{D}\right)^{2}\\\
&\mathcal{G}^{(T)}_{\mu\nu}=-r\left(\frac{2}{D}\right)\text{log}\left(1-\widetilde{Q}^{2}e^{-Y}\right)\sigma_{\mu\nu}+\mathcal{O}\left(\frac{1}{D}\right)^{2}\end{split}$
(117)

So, now if we subtract the fluid metric from the large-$D$ metric both
expanded up to $\mathcal{O}\left(\frac{1}{D}\right)^{2}$, we will find that in
the membrane region the two metric matches.

###  4.4 Comparing the evolution of two sets of data

In the previous subsection we have seen that the metric and gauge fields of
both the perturbation techniques are equivalent in their overlap regime.
However, our intention is to show that the solutions generated by these two
perturbation techniques are equivalent. Hence we also have to show the
equivalence of the differential equations that govern the time evolution of
the defining data of both the systems. But the defining data of hydrodynamic
metric and large-$D$ metric are constrained by two different looking sets of
differential equations. For hydrodynamic metric these equations are given by
the equations in (41), on the other hand the constraint equations for the
large-$D$ case are given by equations in (66). To show the equivalence of
these two gravity solutions we have to show that once ‘membrane equations’ are
satisfied, the constraint equations in hydrodynamics are also satisfied. One
of these equations is given by555It is well known that fluid equations can be
written as conservation equations of a stress tensor and charge current living
on $(D-1)$ dimensional flat space. Also from paper radiation ; Biswas:2019xip
we know that we can define a stress tensor and charge current on the membrane
order by order in inverse power of dimensions such that the membrane equations
is simply the conservation equations of this stress tensor and charge current.
So it would be interesting and easier to show that the conservation of
membrane stress tensor and charge current follows from the conservation
equations of fluid stress tensor and charge current. But unfortunately at this
time we do not have the expressions for the stress tensor and charge current
for a charged membrane propagating in AdS space. Hence we are forced to check
the equivalence of the two different looking sets of constraint equations,
namely membrane equations and fluid equations.

$P^{A}_{C}\left[\frac{\widehat{\nabla}^{2}U_{A}}{K}-\left(1+\widetilde{Q}^{2}\right)\frac{\widehat{\nabla}_{A}K}{K}+U^{B}K_{BA}-\left(1+\widetilde{Q}^{2}\right)\left(U\cdot\widehat{\nabla}U_{A}\right)\right]=\mathcal{O}\left(\frac{1}{D}\right)$
(118)

We have calculated the different components of these equations in terms of
fluid data as follows

$\begin{split}&P^{A}_{C}\frac{\widehat{\nabla}^{2}U_{A}}{K}\leavevmode\nobreak\
=\leavevmode\nobreak\ -f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\\\
&P^{A}_{C}\frac{\widehat{\nabla}_{A}K}{K}\leavevmode\nobreak\
=\leavevmode\nobreak\ 0\\\
&P^{A}_{C}U^{B}K_{BA}=\mathcal{O}\left(\partial^{2}\right)\\\
&P^{A}_{C}\left(U\cdot\widehat{\nabla}U_{A}\right)\leavevmode\nobreak\
=\leavevmode\nobreak\ -f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\\\ \end{split}$ (119)

Substituting these in the L.H.S of equation (118) evaluate to

$\begin{split}&=-f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}-\left(1+\widetilde{Q}^{2}\right)\left(-f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\right)\\\
&=\widetilde{Q}^{2}\leavevmode\nobreak\ f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\\\ &=\frac{\leavevmode\nobreak\
3}{8}Q_{C}^{2}\leavevmode\nobreak\ f\left(Q_{C}\right){\cal
P}^{\lambda}_{\mu}\partial_{\lambda}Q_{C}\\\
&=\mathcal{O}\left(\frac{1}{D}\right)\end{split}$ (120)

As $f\left(Q_{C}\right)=\mathcal{O}\left(\frac{1}{D}\right)$, so our membrane
velocity and shape function satisfies (118).

The second constraint equation is

$\hat{\nabla}\cdot U=\mathcal{O}\left(\frac{1}{D}\right)$ (121)

now $\hat{\nabla}\cdot U=\mathcal{O}\left(\partial^{2}\right)$. So up to the
order we are interested our membrane velocity satisfies this equation.

And the third equation is given by

$\frac{\widehat{\nabla}^{2}\widetilde{Q}}{K}-U\cdot\widehat{\nabla}\widetilde{Q}-\widetilde{Q}\left[\frac{U\cdot\widehat{\nabla}K}{K}-U\cdot
K\cdot U-\frac{R_{uu}}{K}\right]=\mathcal{O}\left(\frac{1}{D}\right)$ (122)

The large-$D$ structures appeared in this equation can be calculated in terms
of fluid data as

$\begin{split}&\frac{\widehat{\nabla}^{2}\widetilde{Q}}{K}=0\\\
&U\cdot\widehat{\nabla}\widetilde{Q}=0\\\
&\frac{U\cdot\widehat{\nabla}K}{K}=0\\\ &U\cdot K\cdot U=-1\\\
&\frac{R_{uu}}{K}=-\lambda\end{split}$ (123)

Substituting these the L.H.S of (122) we can check that (122) is also
satisfied. So up to the order we are interested our membrane velocity
satisfies this equation. So we have shown that our membrane equations follow
as a consequence of fluid constraint equations.

##  5 Conclusons and future directions :

In this note, we have compared two different perturbation techniques - namely
derivative expansion and expansion in inverse powers of dimension, in the
regime where both techniques are applicable.
We have considered the case, when these techniques are used to generate
asymptotically AdS, dynamical black hole type solutions of Einstein-Maxwell
systems. We have shown that in the appropriate regime of the parameter space,
the two solutions are equivalent to one another upto the first non-trivial
order in both the perturbation parameters. It turns out that after a series of
gauge transformation and field redefinitions, the metrics and the gauge fields
generated by these two different techniques are exactly same upto the order
the solutions are known on both sides.
This work could be extended to many directions. Below we are listing few of
them.

As mentioned before, this equivalence is very much expected on physical
grounds. Still we believe it is important to chart out the subtle details of
the redefinitions and transformation it involves. If we know how and when the
two perturbation techniques generate the same solution, it will help us to
find out when they are really different where one is generating new set of
dynamical black hole solutions that could not be generated from the other. So
once we identified and studied the overlapping regime of the parameter space,
it would be interesting to look at the non-overlapping regimes.

Because of the very generic nature of the physical intuition that asserts this
equivalence, we believe that it exists not only for pure Einstein systems or
Einstein-Maxwell systems but also for Einstein-dilaton systems, higher-
derivative gravity theory Bhattacharyya:2008ji ; Armas:2016mes ; Chen:2017rxa
; articleDutta  or any other systems where we can apply both the perturbation
techniques. However, our explicit calculations are very much system-specific,
which somehow obscures this genericity. It would be interesting to set these
calculations more physically or abstractly, without using too much details of
a given theory.

In some sense, this note is also describing a duality between the dynamics of
a $(D-1)$ dimensional charged and massive membrane, embedded in $D$
dimensional AdS space and that of charged fluid, living on the boundary of the
AdS space. The basic variables of charged fluids are temperature ($T$),
velocity ($u^{\mu}$) and charge density ($Q$); whereas any charged dynamical
membrane would be characterized by the embedding function ($\psi$), the charge
density field ($\tilde{Q}$) and the velocity field ($U^{A}$)666Just like the
velocity field in fluids, the membrane velocity $U^{A}$ captures the charge or
mass redistribution within the membrane.. The statement of the duality could
be as follows.
If it is possible to have an all order completion for both the membrane
equations and the relativistic fluid equations, then they are actually the
same equations, just written in terms of two different set of variables.
In this note, we have worked out this variable redefinition (see equations
(76),(79) and (80)), needed to show the equivalence, upto the order the
equations are known on both sides.
As it is clear from all previous discussions, the key reason for this
equivalence is simply the fact that both the systems are dual to the same
gravity solution in the overlap regime of the two perturbation parameters.
Still it might be possible to formulate the duality, removing the gravity
altogether, because once we know the appropriate variable redefinition, that
is enough to show the equivalence of the membrane and fluid equations.
However, we should emphasize that at this stage it is a duality between a very
specific membrane and a very specific fluid, the ones that could have gravity
duals. It would be really interesting to see if we can extend such duality to
more generic fluids and membranes. It will provide new ways to analyze the
unsolved problems in both sets of equations.

## Acknowledgement

I am extremely grateful to Sayantani Bhattacharyya for suggesting me this
problem and providing guidance at every stage of this project. I am also
thankful to her for going through the draft of this note several times and for
very useful suggestions. I would like to thank Parthajit Biswas and Anirban
Dinda for collaboration at the initial stage of this project. I would also
like to thank Parthajit Biswas, Anirban Dinda, Suman Kundu, Sarthak Duary and
Shuvayu Roy for useful discussion. Finally, I would like to acknowledge my
gratitude to the people of India for their support to the research in basic
sciences.

##  Appendix A The Large-$D$ limit of the integrations appearing in
hydrodynamic metric

In this section we will try to do the integrations appearing in the fluid
metric and we will do those integrals in $\frac{1}{D}$ expansion.

###  A.1 Analysis of the integral in the function $F_{1}$

The integral appearing in $F_{1}$ is given by

$\begin{split}F_{3}(\rho,M)&=\int_{\rho}^{\infty}dp\frac{1}{\left(1+\frac{1}{4}\frac{3(D-3)}{2(D-2)}\frac{Q_{1}^{2}}{p^{2(D-2)}}-\frac{M}{p^{D-1}}\right)^{2}}\Big{(}\frac{1}{p^{2(D-1)}}-\frac{c_{1}}{p^{2D-3}}\Big{)}\\\
&=\int_{\rho}^{\infty}dp\frac{1}{\left(1+\frac{\left(M-1\right)}{p^{2(D-2)}}-\frac{M}{p^{D-1}}\right)^{2}}\Big{(}\frac{1}{p^{2(D-1)}}-\frac{c_{1}}{p^{2D-3}}\Big{)}\end{split}$
(124)

Though it is difficult to do the integral for general $M$, it is easier to do
the integral for $M=1$. Hence we have done the integration by expanding $M$
around $1$ as $M=1+\delta M$ in a power series of $\delta M$. It is easy to do
the integrals appearing in each coefficient in the power series of $\delta M$
and if we do those integrals and expand each of the results of those integrals
in $\frac{1}{D}$ as $\rho=1+\frac{Y}{D}$ we will find that

$\begin{split}F_{3}(1+\frac{Y}{D},M)=\left(\frac{1}{D}\right)^{2}\end{split}$
(125)

###  A.2 Analysis of the integral in the function $F_{2}$

The integral appearing in $F_{2}$ is given by

$\begin{split}&F_{2}(\rho,M)=\int_{\rho}^{\infty}\frac{p^{D}\left(p^{D}-p^{2}\right)}{p^{2}\left(p^{2D}-Mp^{D+1}+(M-1)p^{4}\right)}dp\\\
\end{split}$ (126)

This integration is also difficult to do other than $M=1$ and hence by
pursuing the previous method at first we expand the integrand around $M=1$ by
putting $M=1+\delta M$. Then the integrand becomes

$\begin{split}\frac{p^{D}\left(p^{D}-p^{2}\right)}{p^{2}\left(p^{2D}-Mp^{D+1}+(M-1)p^{4}\right)}&=\frac{p^{D-2}-1}{p\left(p^{D-1}-1\right)}+\frac{\left(p^{D}-p^{3}\right)\left(p^{D}-p^{2}\right)}{p^{D+1}\left(p^{D}-p\right)^{2}}{\delta
M}\\\
&+\frac{\left(p^{D}-p^{3}\right)^{2}\left(p^{D}-p^{2}\right)}{p^{2D}\left(p^{D}-p\right)^{3}}{\delta
M}^{2}+\cdots\end{split}$ (127)

If we do each of these integrals for each of the coefficients of $\delta
M,\leavevmode\nobreak\ {\delta M}^{2},\cdots$ and then expand those results in
$\frac{1}{D}$ by replacing $\rho=1+\frac{Y}{D}$, we will get terms like
$e^{-Y}\left(\frac{\delta M}{D}\right)$, $e^{-2Y}\left(\frac{{\delta
M}^{2}}{2D}\right),e^{-3Y}\left(\frac{{\delta M}^{3}}{3D}\right),\cdots$. If
we combine all those terms for power of $\delta M$ $\geq 1$ we will get
$-\frac{1}{D}\text{log}[1-\widetilde{Q}^{2}e^{-Y}]$ and for the
$\text{0}^{th}$ order term in $\delta M$ following the appendix A of
Bhattacharyya2019leading  we have $1+\left(\frac{1}{D}\right)^{2}$.

##  Appendix B The inverse of the background metric and christoffel symbols
w.r.t background metric

We can find out the inverse of the background metric order by order in
derivative expansion. These expressions are given by

$\begin{split}\bar{\mathcal{G}}^{rr}&=r^{2}-2r\frac{\Theta}{D-2}+\mathcal{O}\left(\partial^{2}\right)\\\
\bar{\mathcal{G}}^{\mu
r}&=u^{\mu}-\frac{a^{\mu}}{r}+\mathcal{O}\left(\partial^{2}\right)\\\
\bar{\mathcal{G}}^{\mu\nu}&=\frac{1}{r^{2}}{\cal
P}^{\mu\nu}-\frac{2}{r^{3}}\sigma^{\mu\nu}+\mathcal{O}\left(\partial^{2}\right)\\\
\end{split}$ (128)

Having these expression for the background metric we have the christoffel
symbols for the background metric as

$\begin{split}&\bar{\Gamma}^{r}_{rr}=0\\\ &\bar{\Gamma}^{\alpha}_{rr}=0\\\
&\bar{\Gamma}^{r}_{\mu r}=r\leavevmode\nobreak\
u_{\mu}-\frac{\Theta}{D-2}u_{\mu}\\\ &\bar{\Gamma}^{\alpha}_{\mu
r}=\frac{1}{r}{\cal P}^{\alpha}_{\mu}-\frac{1}{2\leavevmode\nobreak\
r^{2}}\left(u_{\mu}a^{\alpha}-a_{\mu}u^{\alpha}\right)+\frac{1}{2\leavevmode\nobreak\
r^{2}}\left(\partial^{\alpha}u_{\mu}-\partial_{\mu}u^{\alpha}\right)-\frac{1}{r^{2}}\sigma^{\alpha}_{\mu}\\\
&\bar{\Gamma}^{r}_{\alpha\mu}=-r^{3}\left({\cal
P}_{\alpha\mu}-u_{\alpha}u_{\mu}\right)+{r^{2}}\left(u_{\alpha}a_{\mu}+u_{\mu}a_{\alpha}\right)-2\leavevmode\nobreak\
r^{2}\sigma_{\alpha\mu}+r^{2}\frac{\Theta}{D-2}{\cal
P}_{\alpha\mu}-3r^{2}\frac{\Theta}{D-2}u_{\alpha}u_{\mu}\\\
&\bar{\Gamma}^{\beta}_{\alpha\mu}=u^{\beta}\left(-r\left({\cal
P}_{\alpha\mu}-u_{\alpha}u_{\mu}\right)+\left(u_{\alpha}a_{\mu}+u_{\mu}a_{\alpha}\right)-2\sigma_{\alpha\mu}-\frac{\Theta}{D-2}u_{\alpha}u_{\mu}-\frac{\Theta}{D-2}{\cal
P}_{\alpha\mu}\right)+a^{\beta}\left({\cal
P}_{\alpha\mu}-u_{\alpha}u_{\mu}\right)\\\ \end{split}$ (129)

##  Appendix C Notation

In this section we write down the various symbols we have used throughout this
note.

$\displaystyle\bar{G}_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Background metric in
}Y^{A}\equiv\\{\rho,y^{\mu}\\}\leavevmode\nobreak\ \text{coordinates}$
$\displaystyle\bar{\mathcal{G}}_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Background metric in
}X^{A}\equiv\\{r,x^{\mu}\\}\leavevmode\nobreak\ \text{coordinates}$
$\displaystyle{\mathcal{G}}_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Full metric in
}X^{A}\equiv\\{r,x^{\mu}\\}\leavevmode\nobreak\ \text{coordinates}$
$\displaystyle\bar{W}_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Background metric in arbitrary
coordinates}$ $\displaystyle{W}_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Full metric in arbitrary
coordinates}$ $\displaystyle{\eta}_{\mu\nu}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{The boundary metric}$
$\displaystyle{u}_{\mu}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Fluid velocity}$
$\displaystyle{U}_{A}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Membrane velocity}$ $\displaystyle
Q\leavevmode\nobreak\ $ $\displaystyle:\leavevmode\nobreak\ \text{Charge field
in hydrodynamic metric}$ $\displaystyle\widetilde{Q}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Charge field in large-$D$ metric}$
$\displaystyle\Pi_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Projector perpendicular to $n_{A}$}$
$\displaystyle P_{AB}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Projector perpendicular to both
$n_{A}$ and $U_{A}$}$ $\displaystyle{\cal P}_{\mu\nu}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Projector perpendicular to
$u_{\mu}$}$ $\displaystyle\nabla_{A}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Covarient derivative w.r.t
background metric}$ $\displaystyle\hat{\nabla}\leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ \text{Covarient derivative projected along
the membrane}$

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