#  Kerr Black Holes within the Membrane Paradigm

A.M<EMAIL_ADDRESS>and A.J.
<EMAIL_ADDRESS>

$\,{}^{\vardiamondsuit}$ Department of Physics & Technology, Karazin Kharkiv
National University,
4 Svobody Sq., Kharkiv 61022, Ukraine
$\,{}^{\spadesuit}$ Akhiezer Institute for Theoretical Physics of NSC KIPT
1 Akademicheskaya St., Kharkiv 61108, Ukraine
$\,{}^{\varheartsuit}$ Usikov Institute of Radiophysics and Electronics
12 Ak. Proskury, Kharkiv 61085, Ukraine

###### Abstract

We consider the membrane viewpoint a là Parikh-Wilczek on the Kerr solution
for a rotating black hole. Computing the stress-energy tensor of a close-to-
the-horizon stretched membrane and comparing it to the stress-tensor of a
viscous fluid, we recover transport coefficients in terms of the Kerr
geometry. Viscosities of the dual fluid remain constant, while the rest of the
transport coefficients become complex functions of radial and angle
coordinates. We study the qualitative behavior of the pressure, expansion, and
energy/momentum densities for two specific black holes: the slowly rotating
black hole, with the angular momentum of one percent of the black hole mass
squared, and the extremal Kerr black hole. For the Kerr solution in the Boyer-
Lindquist coordinates, these transport coefficients generally have poles at
different values of the radial coordinate in the range between the horizon and
the Schwarzschild radius of the black hole, in dependence on the fixed angle
direction. We briefly discuss our findings in the context of a relation
between the Membrane Paradigm and the AdS/CFT correspondence, the KSS bound
violation, the coordinate choice, and a non-stationary extension of the Kerr
solution.

##  1 Introduction

This work is aimed at the application of the Membrane Paradigm within the
Parikh-Wilczek approach to rotating black holes. Developing pioneering
researches on Electrodynamics and Gravitodynamics near black hole horizons [1,
2], the Membrane Paradigm [3, 4, 5, 6, 7] appeals to the effective dynamics of
a viscous fluid, simulated characteristics of a stretched membrane located at
the horizon. Later on, the Duality between the hydrodynamic limit of a
strongly coupled Gauge Theory and weakly coupled Gravity with the matter was
used for computing transport coefficients of the Gauge Theory on the Gravity
side [8]. Surprisingly, some of the results of the “old” Membrane Paradigm and
of the “new” Gauge/Gravity Duality matched in the early development of the
latter (see, e.g., [10, 9, 11]). Notwithstanding, the Membrane Paradigm does
not refer anyway to conformal symmetry, which plays a significant role as in
the original AdS/CFT correspondence, as well as in its “light” version, known
as the Gauge/Gravity Duality.

To date, the strong differentiation between the results of computation of
shear viscosity on both – the Membrane Paradigm and the Gauge/Gravity Duality
– sides was not determined. Considering this rationale as a guideline, before
doing non-trivial computations in the Gauge/Gravity Duality on the Kerr
spacetime, we applied technically more simple Membrane Paradigm to compute
this and other transport coefficients of the dual fluid. In the course of our
studies, we recovered old results for the shear and bulk viscosities, which
are constants in the Kerr geometry, and got new upshots on the functional
dependence on the radial and angle coordinates for other transport
coefficients. The shear viscosity value coincides with that of the
Schwarzschild geometry, which, in particular, holds up the “shear
viscosity/entropy density” bound $\eta/s=\hbar/4\pi k_{B}$, known as the KSS
bound [8]. The dual fluid pressure, expansion, energy, and momentum densities,
being functions of radial and polar angle coordinates, show different
properties in dependence on the considered case. For a slowly rotating black
hole, these quantities may be regular in the range from the location of
circular photon orbits to spatial infinity, as it happens in the case of the
black hole whose angular momentum value is one percent of the mass squared.
For the extremal Kerr black hole solution in the Boyer-Lindquist coordinates,
the mentioned quantities display complex behavior. In particular, all of them
have poles, which in specific cases cause the flipping of the sign in the
vicinity of the singular point.

To set up the stage for such conclusions, in Section 2, we consider the 1+1+2
split of the Kerr metric similar to that of [6, 7], early developed for non-
rotating black holes. Note that the standard 1+3 separation of coordinates has
been employed to rotating black holes as in the framework of the Membrane
Paradigm and in the fluid/gravity correspondence as well (see, e.g., [12, 13,
14, 15, 16, 17, 18, 19]). The explicit 1+1+2 decomposition of metric for
rotating black holes a là Parikh-Wilczek has not been considered in the
literature, to the extent we know.

Computing the external curvature of a 3D stretched membrane hypersurface in
the vicinity of the black hole horizon, we form the membrane stress-energy
tensor. Next, comparing this tensor to that of a viscous fluid, we derive the
correspondence between gravitational geometric quantities and transport
coefficients of the dual fluid. We recover in this way the energy density,
pressure, expansion, and momentum density of the dual medium in terms of the
geometric characteristics of the Kerr solution. Specifically, we find that the
effects of rotation do not impact on the shear and bulk viscosity, the values
of which for the Kerr black hole coincide, within the Membrane Paradigm, with
that of a static neutral black hole of the same mass. What concerns the other
transport coefficients is that they, as we have mentioned, become functions of
radial and angular coordinates that we explicitly obtain.

In Section 3, we analyze the functional dependence of the corresponding
quantities on the radial coordinate by fixing the angle direction in the
equatorial/pole planes of either extremal (fast rotating) or non-extremal
slowly rotating Kerr black hole. Here, we establish the previously announced
properties of the pressure, expansion, and energy/momentum densities of the
dual fluid: they are smooth functions in the range of the radial coordinate
from the circular photon orbits in the equatorial plane (as the reference
point) to spatial infinity for the slowly rotating black hole of the mass $M$
and of the angular momentum $J=M^{2}/100$. For other relations between $J$ and
$M^{2}$ and for other polar angle planes, these quantities generally have
poles. This feature becomes manifest in the extremal Kerr black hole case,
where all the mentioned characteristics of the dual fluid are divergent at the
Schwarzschild radius in the equatorial plane.

The summary of the results, their discussion, and our conclusions are
collected in the last section. We also include, for the sake of completeness,
two technical Appendices. There, we reproduce the relevant to the
Schwarzschild black hole computations of characteristics of the dual to the
stretched membrane viscous fluid, as well as the dimensional analysis of
quantities we deal with.

Notation and conventions. Here we use the “mostly plus” signature of the
metric and $c=1$ units. The black hole’s effective mass is scaled with the
gravitational constant: $M=mG$, where $m$ is the usual mass of the object.
Indices of small letters of the Latin alphabet – $a,b,c,\dots$ – run from $0$
to $3$; indices of capital letters – $A,B,C,\dots$ – are two-dimensional ones
(of the angular subspace of the Kerr spacetime). BH is used as the shortening
of a “black hole”.

##  2 Viscous fluid description of the Kerr stretched horizon

We begin with the standard form of the Kerr metric [20] in the Boyer-Lindquist
coordinates [21]:

$ds^{2}=-F^{2}_{t}dt^{2}-2\omega
F_{t}\,dtd\varphi+F^{2}_{r}dr^{2}+\rho^{2}(r,\theta)d\theta^{2}+F^{2}_{\varphi}d\varphi^{2},$
(1)

where we have introduced the following functions of $(r,\theta)$:

$F_{t}=\left(1-\frac{2Mr}{\rho^{2}(r,\theta)}\right)^{1/2},\qquad\rho^{2}(r,\theta)=r^{2}+a^{2}\cos^{2}\theta,$
(2) $\omega=-\frac{2Mr}{\rho^{2}(r,\theta)}\,a\sin^{2}\theta\cdot
F^{-1}_{t},\qquad
F_{r}=\frac{\rho(r,\theta)}{\Delta^{1/2}},\qquad\Delta=r^{2}-2Mr+a^{2},$ (3)
$F_{\varphi}=\left((r^{2}+a^{2})\sin^{2}\theta+\frac{2Mr}{\rho^{2}(r,\theta)}\,a^{2}\sin^{4}\theta\right)^{1/2}.$
(4)

To employ the 1+1+2 Parikh-Wilczek [6] decomposition of the metric, we present
the line element (1) in the following equivalent form:

$ds^{2}=-\left(F_{t}dt+\omega
d\varphi\right)^{2}+F_{r}^{2}dr^{2}+\rho^{2}(r,\theta)d\theta^{2}+\left(F^{2}_{\varphi}+\omega^{2}\right)d\varphi^{2},$
(5)

so that $ds^{2}=\left(-u_{a}u_{b}+n_{a}n_{b}+\gamma_{ab}\right)dx^{a}dx^{b}$
is realized by 1-forms

$u_{a}dx^{a}=F_{t}dt+\omega d\varphi,\qquad n_{a}dx^{a}=F_{r}dr,$ (6)

and by the metric of a 2D subspace of angular coordinates

$\gamma_{ab}\,dx^{a}dx^{b}=\rho^{2}(r,\theta)d\theta^{2}+\left(F^{2}_{\varphi}+\omega^{2}\right)d\varphi^{2},\quad\gamma_{ab}=g_{ab}-n_{a}n_{b}+u_{a}u_{b}.$
(7)

In dual basis, vectors $u^{a}$ and $n^{a}$ have the components

$u^{a}=-F^{-1}_{t}\partial_{t},\qquad n^{a}=F^{-1}_{r}\partial_{r},$ (8)

and the norm of $u^{a}$ and $n^{a}$ is $-1$ and $+1$, respectively.

The $n_{a}$ field acceleration $n^{b}{}_{b}n_{a}$ becomes non-trivial in the
case. (Indirectly, it means the presence of a non-trivial Hájiček field [22],
see below.) However, one can still use the standard expression of [6] for a
stretched membrane stress-tensor

$8\pi Gt_{ab}=Kh_{ab}-K_{ab},\qquad h_{ab}=g_{ab}-n_{a}n_{b},$ (9)

even in this case (see Appendix A for details). The 3D extrinsic curvature
tensor of the orthogonal to $n^{a}$ hypersurface is

$K_{ab}={h_{a}}^{c}{}_{c}n_{b}=\frac{1}{2F_{r}}\left(\begin{array}[]{cccc}-\partial_{r}F^{2}_{t}&0&0&-\partial_{r}(\omega
F_{t})\\\ 0&0&0&0\\\ 0&0&\partial_{r}\rho^{2}&0\\\ -\partial_{r}(\omega
F_{t})&0&0&\partial_{r}F^{2}_{\varphi}\end{array}\right).$ (10)

The trace of this tensor comes to

$K\equiv{\rm
Tr}\,K_{ab}=\frac{1}{2F_{r}}\partial_{r}\ln\left[\rho^{2}(r,\theta)F^{2}_{t}\left(\omega^{2}+F^{2}_{\varphi}\right)\right],$
(11)

or, in terms of the standard for the Kerr solution variables,

$K=\frac{1}{\Delta^{1/2}\rho^{3}(r,\theta)}\left(\rho^{2}(r,\theta)(r-M)+r\Delta\right).$
(12)

Now, following [6], we have to decouple the time-like direction and to present
the 3D extrinsic curvature $K_{ab}$ in terms of 2D extrinsic curvature
$k_{AB}$ of the normal to $u^{a}$ hypersurface. Clearly, the metric on this
hypersurface is a $4\times 4$ matrix $\gamma_{ab}$, the only non-trivial part
of which is the $2\times 2$ block $\gamma_{AB}$ determined by eq. (7). Details
on the relation between 3D and 2D extrinsic curvatures can be found in
Appendix A. Here we just recall that $K_{ab}=\alpha^{-1}\left(k_{ab}-g_{{\cal
H}}u_{a}u_{b}\right)+\Omega_{a}u_{b}+\Omega_{b}u_{a}$, where $\alpha$ is the
renormalization factor making divergent quantities finite on the event
horizon; $\Omega_{a}$ is the Hájiček field and $g_{{\cal H}}$ is the surface
gravity. (Actually, we will not use this relation for $K_{ab}$ in what
follows, so the explicit form of the Hájiček field does not matter.)

In our case, we choose the renormalization factor $\alpha=F^{-1}_{r}$ with
$F_{r}(r,\theta)$ from eq. (3), which makes it possible to form the null-
vector $l^{a}=F^{-1}_{r}n^{a}$. (The norm of this vector is
$l^{a}l_{a}=F^{-2}_{r}=\Delta/\rho^{2}(r,\theta)$, and it is equal to zero on
the BH horizon $r_{h}$, defined by $\Delta(r_{h})=0$.) Then, we can compute
the 2D extrinsic curvature tensor as the Lie derivative of the 2D metric
$\gamma_{AB}$ along the null-vector $l^{a}$. We get

$k_{AB}=\frac{1}{2F^{2}_{r}}\left(\begin{array}[]{cc}\partial_{r}\rho^{2}(r,\theta)&0\\\
0&\partial_{r}\left(\omega^{2}+F^{2}_{\varphi}\right)\end{array}\right).$ (13)

Separating the traceless part of $k_{AB}$, we write the 2D extrinsic curvature
as the combination of the 2D shear tensor $\sigma_{AB}$ and the expansion
$\Theta$: $k_{AB}=\sigma_{AB}+1/2\,\Theta\gamma_{AB}$, ${\rm
Tr}\,\sigma_{AB}=0$.

On account of the diagonal structure of the 2D metric (cf. eq. (7)), we have 3
equations to identify $\sigma_{AB}$ and $\Theta$:

$\left.\begin{aligned}
k_{\theta\theta}=\sigma_{\theta\theta}+\frac{1}{2}\Theta\gamma_{\theta\theta},\\\
k_{\varphi\varphi}=\sigma_{\varphi\varphi}+\frac{1}{2}\Theta\gamma_{\varphi\varphi}\end{aligned}\right\\}$
(14)

and
$\gamma^{\theta\theta}\sigma_{\theta\theta}+\gamma^{\varphi\varphi}\sigma_{\varphi\varphi}=0$.
Therefore, the expansion is defined by

$\Theta=\gamma^{AB}k_{AB}=\frac{1}{2F^{2}_{r}}\partial_{r}\ln\left(\rho^{2}(r,\theta)\left(\omega^{2}+F^{2}_{\varphi}\right)\right),$
(15)

or, in terms of the standard for the Kerr spacetime variables,

$\Theta=\frac{\Delta}{\rho^{4}(r,\theta)}\,r+\frac{4a^{2}M^{2}r\sin^{2}\theta}{\rho^{4}(r,\theta)(\rho^{2}(r,\theta)-2Mr)}\left(1-\frac{Mr}{\rho^{2}(r,\theta)}\right)\left(1-\frac{2r^{2}}{\rho^{2}(r,\theta)}\right)$
$+\frac{1}{\rho^{2}(r,\theta)}\left(1-\frac{2Mr}{\rho^{2}(r,\theta)}\right)\left(r+\frac{a^{2}M\sin^{2}\theta}{\rho^{2}(r,\theta)}\left(1-\frac{2r^{2}}{\rho^{2}(r,\theta)}\right)\right).$
(16)

Next, taking eqs. (13)-(15) into account, we obtain the following non-trivial
components of the shear tensor $\sigma_{AB}$:

$\sigma_{\theta\theta}=\frac{\rho^{2}(r,\theta)}{4F^{2}_{r}}\,\partial_{r}\ln\left(\frac{\rho^{2}(r,\theta)}{\omega^{2}+F^{2}_{\varphi}}\right)=-\frac{a^{2}\sin^{2}\theta(M-r)}{2(\rho^{2}(r,\theta)-2Mr)},$
(17)
$\sigma_{\varphi\varphi}=\frac{\omega^{2}+F^{2}_{\varphi}}{4F^{2}_{r}}\,\partial_{r}\ln\left(\frac{\omega^{2}+F^{2}_{\varphi}}{\rho^{2}(r,\theta)}\right)=\frac{a^{2}\sin^{4}\theta\Delta(M-r)}{2(\rho^{2}(r,\theta)-2Mr)^{2}}.$
(18)

Now, we are ready to move on to solving the main task: to derive transport
coefficients of the dual fluid in terms of the Kerr geometry variables. Let us
turn to the stretched membrane stress-tensor $8\pi Gt_{ab}=Kh_{ab}-K_{ab}$,
which is (cf. eqs. (10), (11))

$8\pi
Gt_{ab}=\left(\begin{array}[]{cccc}-KF^{2}_{t}+\frac{1}{2F_{r}}\partial_{r}F^{2}_{t}&0&0&-K\omega
F_{t}+\frac{1}{2F_{r}}\partial_{r}(\omega F_{t})\\\ 0&0&0&0\\\
0&0&K\rho^{2}(r,\theta)-\frac{1}{2F_{r}}\partial_{r}\rho^{2}(r,\theta)&0\\\
-K\omega F_{t}+\frac{1}{2F_{r}}\partial_{r}(\omega
F_{t})&0&0&KF^{2}_{\varphi}-\frac{1}{2F_{r}}\partial_{r}F^{2}_{\varphi}\end{array}\right).$
(19)

To derive the transport coefficients, we have to compare the stress-tensor
(19) to the stress-tensor of a viscous fluid. The latter has the following
general form (see, e.g., [23], [24]):

$t_{ab}=\frac{1}{\alpha}\rho
u_{a}u_{b}+\frac{1}{\alpha}\gamma_{aA}\gamma_{bB}\left(p\gamma^{AB}-2\eta\sigma^{AB}-\zeta\Theta\gamma^{AB}\right)+\pi^{A}\left(\gamma_{aA}u_{b}+\gamma_{bA}u_{a}\right).$
(20)

Recall that indices $a,b$ are indices of the 4-dimensional spacetime; indices
$A,B$ are two-dimensional indices of a surface parameterized by angles
$\\{\theta,\varphi\\}$. $\rho$ is the energy density, $p$ is the fluid
pressure, and $\sigma_{AB}$ and $\Theta$ are the shear tensor and the
expansion of the null geodesics near the event horizon. Also, $\eta$ is the
shear viscosity, $\zeta$ is the bulk viscosity, and $\pi^{A}$ is the momentum
density. As we have noticed, the regularization factor is chosen to be
$\alpha=1/F_{r}$.

Comparing two stress-tensors from eqs. (19), (20), we derive the corresponding
dynamical characteristics of the fluid in terms of the Kerr geometry. For $tt$
components of both stress tensors, we have

$t_{tt}=\frac{1}{\alpha}\rho
u_{t}u_{t}\qquad\leadsto\qquad\Big{|}\alpha=F^{-1}_{r},\,u_{t}=F_{t}\Big{|}\qquad\leadsto$
$\rho=\frac{1}{8\pi
G}\left(-\frac{K}{F_{r}}+\frac{1}{2F^{2}_{r}}\,\partial_{r}\ln
F^{2}_{t}\right).$ (21)

Equivalently,

$\rho=\frac{1}{8\pi
G\rho^{4}(r,\theta)}\left[\frac{M\Delta\left(r^{2}-a^{2}\cos^{2}\theta\right)}{\rho^{2}(r,\theta)-2Mr}-\left(\rho^{2}(r,\theta)(r-M)+r\Delta\right)\right].$
(22)

Next, we have to recover the momentum density $\pi^{A}$ in the dual viscous
fluid from the non-diagonal elements of the stretched membrane stress-energy
tensor. Clearly, comparing (19) and (20), we get $\pi^{\theta}=0$, and

$\pi^{\varphi}=\frac{1}{8\pi
G}\frac{1}{F_{t}\left(\omega^{2}+F_{\varphi}^{2}\right)}\left(-K\omega
F_{t}+\frac{1}{2F_{r}}\partial_{r}\left(\omega
F_{t}\right)\right)-\frac{F_{r}\omega}{\omega^{2}+F_{\varphi}^{2}}\,\rho.$
(23)

With the energy density (21), it becomes

$\pi^{\varphi}=\frac{1}{16\pi
G}\frac{\omega}{F_{r}(\omega^{2}+F_{\varphi}^{2})}\,\partial_{r}\ln\frac{\omega}{F_{t}}.$
(24)

Therefore, the 2D momentum density vector is
$\pi^{A}=(\pi^{\theta},\pi^{\varphi})=(0,\pi^{\varphi})$ with

$\pi^{\varphi}=\frac{1}{8\pi
G}\frac{aM(r^{2}-a^{2}\cos^{2}\theta)}{\rho^{4}\Delta^{1/2}(\rho^{2}(r,\theta)-2Mr)^{1/2}}.$
(25)

The next components of (19) we focus on are:

$t_{\theta\theta}=\frac{1}{8\pi
G}\left(K\rho^{2}(r,\theta)-\frac{1}{2F_{r}}\partial_{r}\rho^{2}(r,\theta)\right),$
(26) $t_{\varphi\varphi}=\frac{1}{8\pi
G}\left(KF^{2}_{\varphi}-\frac{1}{2F_{r}}\partial_{r}F^{2}_{\varphi}\right).$
(27)

On the other hand, see eq. (20),

$t_{\theta\theta}=\frac{1}{\alpha}\gamma_{\theta\theta}\left(p-\zeta\Theta\right)-\frac{1}{\alpha}2\eta\sigma_{\theta\theta},$
(28)
$t_{\varphi\varphi}=\frac{1}{\alpha}\gamma_{\varphi\varphi}\left(p-\zeta\Theta\right)-\frac{1}{\alpha}2\eta\sigma_{\varphi\varphi}+\frac{1}{\alpha}\rho
u_{\varphi}u_{\varphi}+2\pi^{\varphi}\gamma_{\varphi\varphi}u_{\varphi}.$ (29)

Extracting the $(p-\zeta\Theta)$ combination from eqs. (28), (29) and equating
the results we get

$2\eta=\frac{1}{F_{r}(\sigma_{\theta\theta}\gamma_{\varphi\varphi}-\sigma_{\varphi\varphi}\gamma_{\theta\theta})}\left(t_{\varphi\varphi}\gamma_{\theta\theta}-t_{\theta\theta}\gamma_{\varphi\varphi}-\rho
F_{r}u^{2}_{\varphi}\gamma_{\theta\theta}-2\pi^{\varphi}\gamma_{\varphi\varphi}\gamma_{\theta\theta}u_{\varphi}\right).$
(30)

Or, by use of eqs. (6), (7), (17), (18), (21), (24), (26), (27),

$\eta=\frac{1}{16\pi
G}\left[1+\frac{\partial_{r}\omega^{2}-2K\omega^{2}F_{r}-2\omega^{2}\partial_{r}\ln\frac{\omega}{F_{t}}}{(\omega^{2}+F^{2}_{\varphi})\,\partial_{r}\ln\left(\rho^{2}(r,\theta)/(\omega^{2}+F^{2}_{\varphi})\right)}\right]-\frac{\rho\omega^{2}F^{2}_{r}}{(\omega^{2}+F^{2}_{\varphi})\,\partial_{r}\ln\left(\rho^{2}(r,\theta)/(\omega^{2}+F^{2}_{\varphi})\right)}.$
(31)

With eq. (11), one simplifies the shear viscosity to

$\eta=\frac{1}{16\pi G}.$ (32)

Hence, we recover the standard classical result [2, 3, 4]: the shear viscosity
of the dual to a Kerr BH effective fluid is constant, and it is the same as
for the Schwarzschild BH. (See Appendix A.)

Next, multiplying both sides of (28) and (29) with $\gamma^{\theta\theta}$ and
$\gamma^{\varphi\varphi}$, respectively, and using the traceless of the shear
tensor $\sigma_{AB}$, we get

$p-\zeta\Theta=-\frac{1}{2}\rho\gamma^{\varphi\varphi}u_{\varphi}^{2}-\frac{1}{F_{r}}\pi^{\varphi}u_{\varphi}+\frac{1}{2F_{r}}\left(\gamma^{\theta\theta}t_{\theta\theta}+\gamma^{\varphi\varphi}t_{\varphi\varphi}\right).$
(33)

As a matter of fact, the last term on the r.h.s. of (33) turns into

$\frac{1}{2F_{r}}\left(\gamma^{\theta\theta}t_{\theta\theta}+\gamma^{\varphi\varphi}t_{\varphi\varphi}\right)=-\frac{1}{16\pi
G}\Theta+\frac{1}{16\pi
GF_{r}}\left(2K-\frac{K\omega^{2}}{\omega^{2}+F^{2}_{\varphi}}+\frac{\partial_{r}\omega^{2}}{2F_{r}(\omega^{2}+F^{2}_{\varphi})}\right)$
(34)

with the expansion $\Theta$ of (15), so that

$p=\frac{1}{8\pi G}\frac{K}{F_{r}}+\left(\zeta-\frac{1}{16\pi
G}\right)\Theta.$ (35)

To fix the value of the bulk viscosity, we will require the equality of the
fluid pressure (35), in the limit $a\rightarrow 0$, to the pressure of the
dual to the corresponding Schwarzschild BH fluid (cf. eqs. (A.16), (A.18) in
Appendix A). Taking this limit on account of eqs. (11), (15), one should get

$\lim_{a\rightarrow 0}p=\frac{1}{8\pi
G}\left(\frac{1}{2}\partial_{r}f+\frac{2f}{r}\right)+\left(\zeta-\frac{1}{16\pi
G}\right)\frac{2f}{r}=\frac{1}{16\pi G}\,\partial_{r}f,\quad f=1-2M/r.$ (36)

It becomes possible for

$\zeta=-\frac{1}{16\pi G}.$ (37)

Therefore, the bulk viscosity of the dual to a Kerr BH fluid is constant, and
it coincides with the corresponding quantity in the Schwarzschild geometry [2,
3, 4]. (See also Appendix A.)

With this choice of the bulk viscosity, the pressure, eq. (35), is reduced to

$p=\frac{1}{8\pi G}\left(\frac{K}{F_{r}}-\Theta\right)=\frac{1}{8\pi
G}\frac{1}{F_{r}^{2}}\partial_{r}\ln F_{t},$ (38)

or

$p=\frac{1}{8\pi
G}\frac{M\Delta\left(r^{2}-a^{2}\cos^{2}\theta\right)}{\rho^{4}(r,\theta)\left(\rho^{2}(r,\theta)-2Mr\right)}.$
(39)

This is the last transport coefficient in a viscous fluid stress-tensor (20).

To summarize the intermediate results of the work, we have established the
correspondence between the Kerr geometry and transport coefficients of the
dual to the event horizon stretched membrane viscous fluid. We have recovered
the standard for the Membrane Paradigm results for the shear and bulk
viscosities, and have explicitly obtained the functional dependence of the
other transport coefficients on the radial and polar angle coordinates. In the
next section, we make an analysis of these functions for two cases: of fast
and slowly rotating black holes.

##  3 Comments on pressure, expansion, and energy/momentum densities of the
dual fluid

Now let us review the behavior of the transport coefficients which are
functions of coordinates. We begin with the fluid pressure determined by eq.
(39). This expression is convenient to analyze in terms of the Schwarzschild
radius $r_{s}=2M$:

$p(r,\theta)=\frac{1}{16\pi
G}\frac{r_{s}\left(r^{2}-r_{s}r+a^{2}\right)\left(r^{2}-a^{2}\cos^{2}\theta\right)}{\left(r^{2}-r_{s}r+a^{2}\cos^{2}\theta\right)\left(r^{2}+a^{2}\cos^{2}\theta\right)^{2}}.$
(40)

Next, we recall that the horizons of a Kerr BH are located at

$r^{+}_{{\cal H}}=\frac{1}{2}\left(r_{s}+\sqrt{r^{2}_{s}-4a^{2}}\right),\qquad
r^{-}_{{\cal H}}=\frac{1}{2}\left(r_{s}-\sqrt{r^{2}_{s}-4a^{2}}\right).$ (41)

Apparently, the parameter $a$, related to the BH angular momentum, falls into
the range

$0\leq\frac{a}{M}\leq 1.$ (42)

Introducing $C\equiv a/M$, the BH horizons are determined by

$r_{{\cal H}}^{\pm}=\frac{r_{s}}{2}\left(1\pm\sqrt{1-C^{2}}\right).$ (43)

From now on, we have to specify the type of black hole from the point of view
of the intensity of its rotation. We start from the high end of the $C$ range,
corresponding to the so-called extremal Kerr black hole, when the two horizons
coincide: $C=1$ and $r^{extr.}_{{\cal H}}=r_{s}/2=M$. Also, we have two
different circular photon orbits in the equatorial plane of the Kerr black
hole, which are located at [25, 26]

$r_{1}=r_{s}\left[1+\cos\left(\frac{2}{3}\arccos\left(-\frac{|a|}{M}\right)\right)\right],$
(44)
$r_{2}=r_{s}\left[1+\cos\left(\frac{2}{3}\arccos\left(\frac{|a|}{M}\right)\right)\right].$
(45)

These radii coincide at $C=|a|/M=0$ and obey $r_{1}\leq r_{2}$. According to
the direction of their rotation, these two photon rings are divided into
prograde (in the same rotation direction as the black hole; for $r_{1}$) and
retrograde (in the opposite direction; for $r_{2}$). Fixing $C=1$ (which
corresponds to the case of a fast rotating BH), we find $r_{1}=M$ and
$r_{2}=4M$. We attract the reader’s attention to the fact that eqs. (44) and
(45) describe stable photon orbits in the equatorial plane; in any other
plane, the location of photon orbits is not determined by (44), (45), and
these orbits are generally unstable (see [26]). We will use $r_{1,2}$ values
as the reference points in our discussion of quantities in non-equatorial
planes, in particular, in the (north) pole plane.

Fixing the polar angle $\theta=\pi/2$ and the Schwarzschild radius $r_{s}=1$,
we get the following dependence of the rescaled by $16\pi G$ pressure on the
radial coordinate for the extremal Kerr BH, Fig.1. In this case, the radial
location of the prograde photon orbits coincides with the BH horizon,
$r_{1}=r_{s}/2=r^{extr.}_{{\cal H}}$. We note that near the double extremal
horizon location, equal to the Schwarzschild radius $r_{s}$, the pressure
diverges and undergoes the sign flip. On both sides of the singular point
$16\pi Gp(r,\pi/2)$ is a decaying function. For the retrograde photon orbits,
located far from the point of singularity, the pressure is a smooth decaying
function: from $9/32\times 1/16\pi G$ at $r_{2}=4r^{extr.}_{{\cal H}}$ to
$1/r^{2}$ at the spatial infinity.

Figure 1: The functional dependence of the pressure rescaled by $16\pi G$:
from the prograde circular photon orbits in the equatorial plane to the radial
infinity (the extremal Kerr BH). The orbits are located on the BH horizon
$r^{extr.}_{{\cal H}}=r_{s}/2=1/2$, where the pressure is equal to zero. The
pressure diverges near the double extremal horizon $r_{s}$ and undergoes the
sign flip.

Let us take a look at the pressure as the radial coordinate function at
$\theta=0$ (the north pole). Here, we fix the “prograde” and “retrograde”
photon orbits (44), (45) as the reference points. For $r\in[r_{1},\infty[$, we
get Fig.2, so that $16\pi Gp(r,0)$ is a smooth function of the radial
direction: from zero on the horizon, having the maximum at $r_{s}$, and
further falling as $1/r^{2}$. The same qualitative behavior $1/r^{2}$ is
observed for the pressure from the “retrograde” photon orbits reference point
$r_{2}=4r^{extr.}_{{\cal H}}$ ($p=60/289\times 1/16\pi G$ there) to the
spatial infinity.

Figure 2: The rescaled by $16\pi G$ pressure in the north pole plane (the
extremal Kerr BH). The reference point is the BH horizon $r^{extr.}_{{\cal
H}}=r_{s}/2=1/2$, where the pressure is equal to zero.

For a slowly rotating black hole, when, say, $C=0.01$, the circular photon
orbit radii in the equatorial plane are almost the same. For $r_{s}=1$, we get
$r_{1}=1.49422$ and $r_{2}=1.50576$. These values are around $r=3/2r_{s}$
which corresponds to the circular photon orbit of the Schwarzschild black hole
of the same mass. Plotting the dependence of $16\pi Gp$ on the radial
coordinate in this case, we observe that for the equatorial/pole planes $16\pi
Gp$ come to be almost the same: from $16\pi Gp\approx 0.45$ at the reference
point, falling as $1/r^{2}$ to the spatial infinity (see Fig.3).

Figure 3: The plot of $16\pi Gp$ from the prograde circular photon orbits in
the equatorial/pole planes of a slowly rotating black hole ($C\equiv
a/M=0.01$). In both cases the reference point is $r_{1}=1.49422\approx
3r_{s}/2$.

Now, we turn to the analysis of the expansion (16), which is useful to present
as a function of the Schwarzschild radius $r_{s}=2M$, $C\equiv a/M$ and
radial/angle coordinates. It becomes

$\Theta=\frac{r\Delta}{\rho^{4}(r,\theta)}+\frac{C^{2}r^{4}_{s}r\sin^{2}\theta}{4\rho^{4}(r,\theta)\left(\rho^{2}(r,\theta)-r_{s}r\right)}\left(1-\frac{r_{s}r}{2\rho^{2}(r,\theta)}\right)\left(1-\frac{2r^{2}}{\rho^{2}(r,\theta)}\right)$
$+\frac{1}{\rho^{2}(r,\theta)}\left(1-\frac{r_{s}r}{\rho^{2}(r,\theta)}\right)\left(r+\frac{C^{2}r_{s}^{3}\sin^{2}\theta}{8\rho^{2}(r,\theta)}\left(1-\frac{2r^{2}}{\rho^{2}(r,\theta)}\right)\right),$
(46)

where

$\Delta=r^{2}-r_{s}r+\frac{1}{4}C^{2}r^{2}_{s},\qquad\rho^{2}(r,\theta)=r^{2}+\frac{1}{4}C^{2}r^{2}_{s}\cos^{2}\theta.$
(47)

The expansion in the equatorial plane of the extremal Kerr BH diverges at the
Schwarzschild radius $r_{s}$ (cf. Fig.4) and undergoes the sign flip at this
point. Its behavior is even more interesting at $r>r_{s}$ since the maximum of
the expansion in this domain is reached at $r=2r_{s}$; after this point the
expansion falls as $1/r$ (cf. Fig.5). On the north pole, the expansion becomes
a smooth function, reaching the same maximum $\Theta=1/2$ as in the previously
considered case, and in the same point, $r=2r_{s}$. It also falls to the
spatial infinity as $1/r$ (see Fig.6).

Figure 4: The expansion $\Theta$ in the equatorial plane of the extremal Kerr
BH. Figure 5: The expansion in the equatorial plane of the extremal Kerr BH in
an extended domain of the radial coordinate. Figure 6: The expansion in the
north pole plane of the extremal Kerr BH. The reference point is the BH
horizon.

In the case of the non-extremal Kerr BH with $C=0.01$, we get almost the same
functional dependence of the expansion in both – equatorial and north pole –
planes, when the maximum of $\Theta$ is at $r=2r_{s}$, and it falls to the
spatial infinity as $1/r$ (see Fig.7).

Figure 7: The expansion in the equatorial plane of the non-extremal slowly
rotating Kerr BH with $C\equiv a/M=0.01$ from the prograde photon orbits
location to infinity.

For the momentum density $\pi^{\varphi}$, eq. (25), written in terms of
$r_{s}$ and $C=a/M$,

$\pi^{\varphi}(r,\theta)=\frac{1}{8\pi
G}\frac{Cr^{2}_{s}\left(r^{2}-\frac{1}{4}C^{2}r^{2}_{s}\cos^{2}\theta\right)}{4\Delta^{1/2}\rho^{4}(r,\theta)\left(\rho^{2}(r,\theta)-r_{s}r\right)^{1/2}},$
(48)

we obtain the following behavior in both equatorial/pole planes of the
extremal Kerr BH, Fig.8. The momentum density diverges at the BH horizon in
the pole plane and at the double BH horizon in the equatorial plane. In both
cases, $\pi^{\varphi}$ decays as $1/r^{4}$. For a slowly rotating BH,
$\pi^{\varphi}$ is a regular function, falling in both planes to the spatial
infinity also as $1/r^{4}$ (see Fig.9).

Figure 8: The momentum density in the equatorial/pole planes of the extremal
Kerr BH. Figure 9: The momentum density in the equatorial/pole planes of a
non-extremal Kerr BH. ($C\equiv a/M=0.01$; $r$ starts from the “prograde”
photon orbits.)

Finally, the energy density (22) has the following equivalent representation:

$\rho=\frac{1}{8\pi
G\rho^{4}(r,\theta)}\left[\frac{r_{s}\Delta\left(r^{2}-\frac{1}{4}C^{2}r^{2}_{s}\cos^{2}\theta\right)}{2\left(\rho^{2}(r,\theta)-r_{s}r\right)}-\left(\rho^{2}(r,\theta)\left(r-\frac{1}{2}r_{s}\right)+r\Delta\right)\right].$
(49)

In the case of the extremal Kerr BH, the functional dependence of the energy
density in the equatorial/pole planes is plotted in Fig.10, with the singular
point at the Schwarzschild radius in the equatorial plane. For the considered
here non-extremal BH (with $C=0.01$), we get almost coincided regular curves
in both planes; see Fig.11.

Figure 10: The energy density in the equatorial/pole planes of the extremal
Kerr BH. Figure 11: The energy density in the equatorial/pole planes of the
non-extremal Kerr BH with $C\equiv a/M=0.01$. $r$ starts from the “prograde”
photon orbits.

To summarize, the transport coefficients of the dual fluid $p$, $\Theta$,
$\rho$, and $\pi^{\varphi}$ are complex functions of radial and angle
variables. In accordance with the structure of all these functions (cf. eqs.
(40), (46), (48), and (49)), they possess the poles defined by
$r^{2}+C^{2}r^{2}_{s}\cos^{2}\theta/4-r_{s}r=0$. These poles may be hidden in
specific requirements, an example of which is the transfer of a reference
point beyond a singular point. Then, one may find a set of parameters of the
Kerr solution (defining the relevant reference point) at which the mentioned
coefficients are regular. Generally, we have to take into account the
irregularity of these functions.

##  4 Summary, discussion, and final remarks

In this paper, we have applied the Membrane Paradigm to neutral rotating black
holes. Following the Parikh-Wilczek approach, well-developed for non-rotating
black holes [6, 7], we have constructed the stress-tensor of a stretched
membrane, located near the black hole horizon, and to compare it to the
stress-tensor of a relativistic viscous fluid. In the nomenclature of the
Gauge/Gravity Duality, this effective fluid is dual to the stretched membrane.
So Physics on (or in the vicinity of) the black hole horizon is equivalently
determined by the corresponding hydrodynamic transport coefficients and
susceptibilities of the dual fluid.

In the course of our studies, we have made the following major findings.

First, all but two main characteristics of the dual viscous fluid, expressed
in terms of geometric variables of the Kerr spacetime, become functions of two
variables: the radial coordinate and the polar angle. We have obtained the
qualitative behavior of the pressure, expansion, and energy/momentum densities
for two specific black holes: the slowly rotating Kerr BH, whose angular
momentum is one percent of the black hole mass squared, and the extremal (fast
rotating) Kerr BH. Generally, these transport coefficients have poles at
different values of the radial coordinate – in the range between the horizon
and the Schwarzschild radius of the black hole – in dependence on the fixed
angle direction. However, if the point of observation is located far from the
Schwarzschild radius (this, in particular, happens in the considered here case
of the slowly rotating BH with $J=M^{2}/100$, when the reference point has
been chosen to be the photon orbits in the equatorial plane), these functions
become smooth in all polar angle planes. It would be interesting to trace back
a correspondence of regularity/non-regularity of the transport coefficients on
the dual side of the Kerr black hole to the stability of this object, recently
proved in [27] and further developed in [28].

And second, the transport coefficients – the shear and bulk viscosities –
remain constants, and we have recovered their values as before (see, e.g., [2,
3, 4]). That, in particular, means the famous $\eta/s$ ratio, computed for the
most black holes and black branes of String Theory in AdS spacetimes to be
$\eta/s\leq\hbar/4\pi k_{B}$ [29], holds in the case; no violation of the KSS
bound [8] occurs. (Recall that the issue of violation of this bound was under
the focus since establishing this result (see, e.g., [30, 31, 32, 33, 34] for
details).) Nevertheless, the presence of a negative bulk viscosity still
becomes a problem: as it has been noted even in the early works on the
Membrane Paradigm (cf. [3]), it requires non-causal teleological boundary
conditions (see [35] for a review) which means that the horizon fluid is anti-
causal in a response [36]. It does not concern isolated and dynamical horizons
[37, 38], where the response is causal; hence, the bulk viscosity is positive.
However, dynamical horizons exist rather for time-dependent solutions than for
static/stationary ones. Time-dependent extensions of the Kerr solution
developed and studied in [39, 40] look perspective to this end.

As we have noted at the beginning of the paper, our motivation was to compute
main characteristics of the dual viscous fluid within the Membrane Paradigm
before starting similar computations on the Gauge/Gravity Duality side.
Mainly, our focus was on the computation of the shear viscosity due to
importance of this quantity for the Gauge/Gravity Duality. Though we have
recovered the classical result of the old Membrane Paradigm, which coincides
with the KSS bound coming from the AdS/CFT correspondence, it still looks
enigmatic how these different approaches, strictly speaking, unrelated to each
other (see [41, 42] for details), may lead to the same answer. For example,
results for 4D bulk viscosity do not look so unambiguous anymore. The AdS/CFT
leads to [43, 44]:

$\frac{\zeta}{\eta}\geq 2\left(\frac{1}{3}-c^{2}_{s}\right).$

Since the physical intuition requires a positive value of the speed of sound
square $c^{2}_{s}$, possible values of the bulk viscosity are restricted by

$\frac{\zeta}{\eta}\leq\frac{2}{3}\qquad\leadsto\qquad\zeta\leq\frac{1}{24\pi
G}\,\,\,\,\,\,\left(\text{for}\,\,\eta=\frac{1}{16\pi G}\right),$

which does not exclude positive values of the bulk viscosity in a holographic
fluid.

We have reported that the transport coefficients of the extremal Kerr black
hole are divergent functions in the radial direction. This result for
"internal" degrees of freedom of the BH horizon seems to be in contradiction
with the early obtained claims on the regularity of external, w.r.t. the
horizon, matter fields for static/stationary solutions of the Einstein
equations (see [39, 40] and early papers [45, 46, 47]). However, we have to
note that the obtained here results are referred to the Boyer-Lindquist
parametrization of the Kerr metric. To avoid truly unphysical singularities,
one needs to use the Eddington-Finkelstein parametrization, which is beyond
the scope of the paper and the study of which we postpone for future
investigations.

To sum up, the obtained results, which have shown differences from the case of
non-rotating black holes where outcomes of both approaches partially coincide,
call for extra theoretical work in figuring out the power of the Membrane
Paradigm and its actual correspondence to and differences with the AdS/CFT.
Answering to this and other questions will be handy in more deep understanding
of microscopic degrees of freedom of a black hole, and what they really are: a
condensate of gravitons [48, 49, 50, 51], strings/D-branes [52, 53, 54], or
other “molecules”.

Acknowledgements. We are grateful to all our colleagues around the world for
their support during these turbulent times for us. We would also like to
express our sincere gratitude to everyone who kindly help Ukrainian refugees
forced to leave their homes. AJN personally thanks Leonid and Tetyana Dudnik
for their warm hospitality during the leave of Kharkiv.

##  Appendix A Details of computations: stress tensors of stretched membrane
in Schwarzschild geometry and its dual viscous fluid

According to the Membrane Paradigm [3, 4, 5], the membrane stress tensor in
1+1+2 coordinate split comes as follows [6]:

$t_{ab}=\frac{1}{8\pi G}\left(Kh_{ab}-K_{ab}\right),\qquad
h_{ab}=\gamma_{ab}-u_{a}u_{b},$ (A.1)

where $h_{ab}$ is the metric tensor of a 3D hypersurface orthogonal to the
space-like unit vector $n^{a}$ (radial direction for the Schwarzschild);
$\gamma_{ab}$ is the metric of a 2D hypersurface orthogonal as to $n^{a}$, as
well as to the time-like vector $u^{a}$. $K_{ab}$ is the extrinsic curvature
of the 3D hypersurface; $K$ is the trace of $K_{ab}$.

The stretched membrane stress tensor (A.1) can be equivalently represented by
the stress-tensor of a viscous fluid [24]:

$t_{ab}=\frac{1}{\alpha}\rho
u_{a}u_{b}+\frac{1}{\alpha}\gamma_{aA}\gamma_{bB}\left(p\gamma^{AB}-2\eta\sigma^{AB}-\zeta\Theta\gamma^{AB}\right)+\pi^{A}\left(\gamma_{aA}u_{b}+\gamma_{bA}u_{a}\right).$
(A.2)

Recall that indices $a,b$ are 4-dimensional indices; indices $A,B$ are two-
dimensional indices of a surface parameterized by angles
$\\{\theta,\varphi\\}$. In (A.2), $\rho$ is the energy density, $p$ is the
fluid pressure, $\sigma_{ab}$, and $\Theta$ are the shear tensor and the
expansion of the null geodesics near the event horizon. Also, $\eta$ is the
shear viscosity, $\zeta$ is the bulk viscosity, and $\pi^{A}$ is the momentum
density. In particular, note the presence of the renormalization parameter
$\alpha$, the role of which is to make the divergent on the true horizon
quantities finite (see [6] and further discussion for details).

Let us review how eq. (A.1) can be used to read off various characteristics of
an effective fluid dynamics from eq. (A.2) for the Schwarzschild solution.

The Schwarzschild geometry is determined by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),\qquad f(r)=1-\frac{2M}{r}.$ (A.3)

We are aimed at splitting this metric into $1+1+2$ form by the use of time-
like and space-like vectors $u^{a}$ and $n^{a}$ near the horizon, turned into
the same null-vector $l^{a}$ on the horizon. This split takes the form

$ds^{2}=-u_{a}u_{b}\,dx^{a}dx^{b}+n_{a}n_{b}\,dx^{a}dx^{b}+\gamma_{ab}\,dx^{a}dx^{b},$
(A.4)

where

$\begin{split}&u_{a}dx^{a}=f^{1/2}dt\rightarrow u_{a}=f^{1/2}\delta_{a}^{t}\\\
&n_{a}dx^{a}=f^{-1/2}dr\rightarrow n_{a}=f^{-1/2}\delta_{a}^{r}\\\
\gamma_{ab}dx^{a}dx^{b}=r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}&\rightarrow\gamma_{AB}=\begin{pmatrix}r^{2}&&0\\\
0&&r^{2}\sin^{2}\theta\end{pmatrix}\;\;A,B=\\{\theta,\varphi\\}.\end{split}$
(A.5)

Therefore, in the dual to 1-forms basis,

$u^{a}=-f^{-1/2}\partial_{t},\qquad n^{a}=f^{1/2}\partial_{r},$ (A.6)

so that $u^{a}u_{a}=-1$, $n^{a}n_{a}=1$, $u^{a}n_{a}=0$, and
$n^{b}{}_{b}n_{a}=0$. The latter relation justifies the form of the stress
tensor (A.1) (see details in [6]).

To recover (A.1) in the background (A.3), we have to compute the extrinsic
curvature of a stretched horizon (a 3-dimensional surface orthogonal to
$n_{a}$) and its trace. For the extrinsic curvature tensor, we have

$K_{ab}={}_{a}n_{b}=-f^{1/2}\begin{pmatrix}\frac{1}{2}\partial_{r}f&&0&&0&&0\\\
0&&0&&0&&0\\\ 0&&0&&-r&&0\\\
0&&0&&0&&-r\sin^{2}\theta\end{pmatrix}=-\frac{1}{2}f^{-1/2}\partial_{r}fu_{a}u_{b}+\frac{f^{1/2}}{r}\gamma_{ab},$
(A.7)

and the trace becomes

$K=K^{a}_{a}=\frac{1}{2}f^{-1/2}\partial_{r}f+\frac{2f^{1/2}}{r}.$ (A.8)

Within the Parikh-Wilczek aproach [6], the extrinsic curvature of the
stretched horizon is replaced accordingly with

$K_{ab}\rightarrow\alpha^{-1}k_{ab}-\alpha^{-1}g_{{\cal H}}u_{a}u_{b}.$ (A.9)

Technically, in (A.9), we further split $K_{ab}$ onto a two-dimensional
surface (orthogonal to $u_{a}$ of (A.5)) extrinsic curvature $k_{AB}$ (then
$k_{ab}=\gamma_{aA}\gamma_{bB}k^{AB}$) and the surface gravity $g_{{\cal H}}$.
At this point, one may wonder how the extrinsic curvature of the orthogonal to
$u^{a}$ 2D hypersurface, lying completely inside the 3D hypersurface, is
related to the extrinsic curvature tensor $K_{ab}$, defining the geometry of
the hypersurface normal to a vector ($n^{a}$ in the case) directed outside.
This match happens due to a coincidence of $u^{a}$ and $n^{a}$ vectors in the
null limit when $\alpha u^{a}\rightarrow l^{a}$ and $\alpha n^{a}\rightarrow
l^{a}$ at $\alpha\rightarrow 0$. Here, $\alpha$ is the renormalization factor,
needed to keep finiteness of $K_{ab}$ components on the horizon, and $l^{a}$
is the null vector normal and tangential to the horizon. The renormalization
factor $\alpha$ is chosen as $\alpha=f^{1/2}$ in the case.333The requirement
for $\alpha$ comes as follows [6, 7]. In the $\alpha\rightarrow 0$ limit,
$\alpha u^{a}\rightarrow l^{a}$ and $\alpha n^{a}\rightarrow l^{a}$, where
$l^{a}$ is a null vector ($l^{a}l_{a}=0$) which is tangential and normal to
the horizon simultaneously. Clearly, the specific representation of such a
null vector depends on the choice. The norm of $\alpha u^{a}$ and $\alpha
n^{a}$ of (A.6) with $\alpha=f^{1/2}$ is equal to $f$ which becomes zero on
the horizon.

Indeed, in the limit $\alpha\rightarrow 0$, when the stretched membrane turns
into the true event horizon surface (null hypersurface), the trace of $K_{ab}$
(cf. (A.8)) diverges due to the simple pole from $f(r)$ at $r=r_{{\cal H}}$.
In this limit,

$\lim_{\alpha\rightarrow
0}K=\frac{1}{2}\,f^{-1/2}\partial_{r}f\Big{|}_{r=r_{{\cal H}}}\sim{\rm
Tr}\left(\alpha^{-1}k_{ab}-\alpha^{-1}g_{{\cal
H}}u_{a}u_{b}\right)\Big{|}_{r=r_{{\cal H}}},$ (A.10)

hence choosing $\alpha\sim f^{1/2}$ makes geometric quantities regular on the
horizon. The same concerns components of $K_{ab}$ (see (A.7)).
$K_{tt}=K_{ab}u^{a}u^{b}$ clearly diverges in the null limit:

$\lim_{\alpha\rightarrow
0}K_{tt}=-\frac{1}{2}f^{-1/2}\partial_{r}f\Big{|}_{r=r_{{\cal
H}}}\longrightarrow-\alpha^{-1}g_{{\cal H}}\Big{|}_{r=r_{{\cal H}}},$ (A.11)

but fixing the renormalization parameter $\alpha\sim f^{1/2}$ results in the
correct expression for the surface gravity on the horizon (see eq. (A.18) in
below).

Next, we can divide the 2-dimensional extrinsic curvature $k_{AB}=k_{BA}$ onto
the traceless part and the corresponding trace $\Theta=\gamma^{AB}k_{AB}$:

$k_{AB}=\sigma_{AB}+\frac{1}{2}\Theta\gamma_{AB},\qquad{\rm
Tr}\,\sigma_{AB}=0.$ (A.12)

Hence, inserting (A.9) and (A.12) into the stress tensor (A.1), we arrive at

$t_{ab}=\frac{1}{8\pi G}\left(\left(\alpha^{-1}\Theta+\alpha^{-1}g_{{\cal
H}}\right)\left(\gamma_{ab}-u_{a}u_{b}\right)+\alpha^{-1}g_{{\cal
H}}u_{a}u_{b}-\alpha^{-1}\left(\sigma_{ab}+\frac{1}{2}\Theta\gamma_{ab}\right)\right).$
(A.13)

And after some algebra we get

$t_{ab}=\frac{1}{8\pi
G\alpha}\left[-\Theta\,u_{a}u_{b}-\sigma_{ab}+\left(\frac{1}{2}\Theta+g_{{\cal
H}}\right)\gamma_{ab}\right].$ (A.14)

Now, we can compare stress tensors (A.14) and (A.2) with the following
outcome:

$\rho=-\frac{1}{8\pi G}\Theta,\qquad 2\eta=\frac{1}{8\pi G},$
$p-\zeta\Theta=\frac{1}{8\pi G}\left(\frac{1}{2}\Theta+g_{{\cal H}}\right).$

So we get the following correspondence between characteristics of the dual
fluid and properties of the stretched membrane in the specific geometric
background:

$\rho=-\frac{1}{8\pi G}\Theta,\qquad\eta=\frac{1}{16\pi G},$ (A.15)
$p=\frac{g_{{\cal H}}}{8\pi G},\qquad\zeta=-\frac{1}{16\pi G}.$ (A.16)

Also, the momentum density in angular directions is $\pi^{A}=0$.

The other task is to relate $\Theta$, $g_{{\cal H}}$, and $\sigma_{AB}$ to the
geometric configuration of the space-time in hands. We have to compare the
stress tensor (A.14) to $t_{ab}=(Kh_{ab}-K_{ab})/8\pi G$, which is (cf. (A.7),
(A.8))

$t_{ab}=\frac{1}{8\pi
G}\left[\left(\frac{1}{2f^{1/2}}\partial_{r}f+\frac{f^{1/2}}{r}\right)\gamma_{ab}-\frac{2f^{1/2}}{r}u_{a}u_{b}\right].$
(A.17)

We obtain

$\Theta=\frac{2f(r)}{r},\qquad\sigma_{AB}=0,\qquad g_{{\cal
H}}=\frac{\partial_{r}f(r)}{2}.$ (A.18)

Hence, the shear tensor and the expansion are trivial on the black hole
horizon, and the surface gravity is finite there and is equal to $g_{{\cal
H}}=(4M)^{-1}$ as it should be.444The fact of finiteness of the stress-tesor
components near the horizon of static and stationary black holes even on
account of quantum corrections to gravity action was pointed out in [45, 46,
47, 39, 40].

Additionally, one may check the relevance of (A.18) computing the
2-dimensional extrinsic curvature $k_{AB}$. By definition,

$k_{AB}=\frac{1}{2}\mathcal{L}_{l}\gamma_{AB},$ (A.19)

where $l^{a}$ is the null-vector on the horizon, determined by $l^{a}=\alpha
n^{a}$ (recall, $\alpha=f^{1/2}$), and $\mathcal{L}_{l}$ is the Lie derivative
along the null-vector. Then, the $2\times 2$ angular part of the full $K_{ab}$
tensor becomes

$k_{AB}=\left(\begin{array}[]{cc}rf&0\\\
0&rf\sin^{2}\theta\end{array}\right)=\frac{1}{2}\left(\frac{2f}{r}\right)\gamma_{AB}.$
(A.20)

Comparing this result to the general structure of $k_{AB}$, eq. (A.12), we
arrive at

$\sigma_{AB}=0,\qquad\Theta=\frac{2f}{r}$ (A.21)

again.

For the Schwarzschild geometry, we get the trivial (everywhere in the
acceptable domain of coordinates) value of the shear tensor. One may wonder
how we have got the value of the shear viscosity (A.15) if $\sigma_{ab}$ is
presented in (A.14) just formally. The value of the shear viscosity can not be
fixed from $\eta\sigma_{ab}=0$ equation in this case, which requires taking
additional arguments. For instance, it could be (projected) Einstein equations
which can be written in the form of Navier-Stokes-like equations (Damour-
Navier-Stokes (DNS) equations) [2, 55, 56]. Then, the effective shear
viscosity can be extracted from the structure of the DNS equations.

Now, we briefly describe the reason for using the standard stress-tensor (A.1)
(or eq. (9) in the main text) in the case of non-trivial acceleration of
$n_{a}$ defined by $a^{c}=n^{b}{}_{b}n^{c}$. Here, we follow [57] and slightly
correct the result mentioned therein. The relevant part of the gravitational
action variation that includes the $n_{a}$ field acceleration is (cf. eq. (A1)
of [6]):

$\delta S_{\text{out}}\Big{|}_{a^{c}\neq 0}=\int
d^{3}x\,\sqrt{-h}\left[\left(a_{a}n^{a}h^{bc}+n^{c}a^{b}+n^{b}a^{c}\right)\delta
h_{bc}-h^{bc}a_{c}n^{a}\delta h_{ab}-n^{a}a^{b}\delta h_{ab}\right].$ (A.22)

The difference in (A.22) and eq. (A1) of [57] is in the term
$a_{a}n^{a}h^{bc}\delta h_{bc}$ which is equal to zero, due to $n^{a}a_{a}=0$.
Then, following directly [57], we can use the symmetry of $\delta h_{ab}$ and
further write (A.22) as

$\delta S_{\text{out}}\Big{|}_{a^{c}\neq 0}=\int
d^{3}x\,\sqrt{-h}\left(n^{a}a^{b}-h^{bc}a_{c}n^{a}\right)\delta h_{ab}=\int
d^{3}x\,\sqrt{-h}\left(g^{bc}-h^{bc}\right)n^{a}a_{c}\delta h_{ab}$ $=\int
d^{3}x\,\sqrt{-h}\,n^{a}n^{b}n^{c}a_{c}\delta h_{ab}.$ (A.23)

The variation is equal to zero due to $n^{c}a_{c}=0$ again.

##  Appendix B Dimension analysis

To control the correctness of different quantities, it is useful to make the
dimension analysis of variables. Since the metric tensor is chosen to be
dimensionless, different quantities entering line elements have different
dimensions in length. The line element by itself has the dimension $l^{2}$.
Then, $[dt^{2}]=l^{2}=[dr^{2}]$, spherical angles are dimensionless –
$[d\theta^{2}]=l^{0}=[d\varphi^{2}]$ – so that the entities in the Kerr metric
(1) get received

$[F_{t}]=l^{0},\qquad[F_{r}]=l^{0},\qquad[f]=l,\qquad[\rho^{2}(r,\theta)]=l^{2},\qquad[F_{\varphi}]=l.$
(B.1)

As a result, the BH mass has the dimension of length: $[M]=l$. And different
components of the 3D extrinsic curvature tensor $K_{ab}$ have different
dimensions:

$[K_{tt}]=l^{-1},\qquad[K_{t\varphi}]=l^{0},\qquad[K_{\theta\theta}]=l,\qquad[K_{\varphi\varphi}]=l.$
(B.2)

The latter in particulr means that

$[\sigma_{\theta\theta}]=[\sigma_{\varphi\varphi}]=l,\qquad[\Theta]=l^{-1},\qquad[\gamma_{AB}]=l^{2}.$
(B.3)

From the exact form of $\rho^{2}(r,\theta)$, it is clear that $[a]=l$; hence,
the dimension of $J=Ma$ is $[J]=l^{2}$.

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