KEK-TH-1867

Black rings at large $D$

Kentaro Tanabe

Theory Center, Institute of Particles and Nuclear Studies, KEK,

Tsukuba, Ibaraki, 305-0801, Japan

<EMAIL_ADDRESS>

Abstract

We study the effective theory of slowly rotating black holes at the infinite
limit of the spacetime dimension $D$. This large $D$ effective theory is
obtained by integrating the Einstein equation with respect to the radial
direction. The effective theory gives equations for non-linear dynamical
deformations of a slowly rotating black hole by effective equations. The
effective equations contain the slowly rotating Myers-Perry black hole, slowly
boosted black string, non-uniform black string and black ring as stationary
solutions. We obtain the analytic solution of the black ring by solving
effective equations. Furthermore, by perturbation analysis of effective
equations, we find a quasinormal mode condition of the black ring in analytic
way. As a result we confirm that thin black ring is unstable against non-
axisymmetric perturbations. We also include $1/D$ corrections to the effective
equations and discuss the effects by $1/D$ corrections.

###### Contents

  1. 1 Introduction
  2. 2 Effective equations
    1. 2.1 Setup
    2. 2.2 Effective equations
    3. 2.3 Effective energy-momentum tensor
    4. 2.4 $1/D$ corrections
  3. 3 Black ring and its physical properties
    1. 3.1 Ring coordinate embedding
    2. 3.2 Black ring solution
    3. 3.3 Quasinormal modes
    4. 3.4 Black string limit
    5. 3.5 $1/D$ corrections
    6. 3.6 Phase diagram
  4. 4 Summary
  5. A Other solutions
    1. A.1 Slowly rotating Myers-Perry black hole
    2. A.2 Slowly boosted black string
  6. B Trivial perturbations in ring coordinate
  7. C $1/D$ corrections of effective equations

##  1 Introduction

Stationary asymptotically flat black holes have only spherical horizon
topology in four dimensions by the Hawking’s topology theorem [1]. This
situation changes drastically in higher dimensions, and higher dimensional
black hole has various horizon topology. As one concrete example of this fact,
Emparan and Reall found the five dimensional black hole solution, black ring,
whose horizon topology is $S^{1}\times S^{2}$ [2]. This is the first
asymptotically flat black hole solution which has a non-spherical horizon
topology. After this discovery many new five dimensional black hole solutions
with multiple horizons have been constructed by the solution generating
technique such as the inverse scattering method ( e.g., see [3] for a review).
In these solutions the topology of each horizon is $S^{3}$ or $S^{1}\times
S^{2}$. We expect that black hole solutions can have much more various and
non-trivial horizon topology in higher dimensions than five. To study this
variety it would be the first step to extend the Emparan-Reall’s discovery to
higher dimensions, that is, the construction of the black ring in higher
dimensions. However the solution generating technique cannot be applied to the
asymptotically flat solution in higher dimensions than five. Then there are
some efforts to construct the black ring in higher dimensions by numerical
methods [4, 5] and by analytical approximation methods such as the blackfold
method [6, 7]111 The blackfold method has achieved a great success along this
direction. In fact black holes with various horizon topology, including
expected and unexpected ones, has been found [8, 9]. .

In this paper we consider the construction of the $D$ dimensional black ring
solution by using another analytical approximation method, the large $D$
expansion method [10, 11]. Recently this large $D$ expansion method was
extended to construct the stationary [12, 13] and time-dependent black hole
solutions [14, 15]. So it is interesting to see if we can obtain the black
ring solution by the large $D$ expansion method. The paper [13] discussed the
effective theory and gave the effective equation for the large $D$ stationary
rotating black holes. This stationary solution was assumed to have the $O(1)$
horizon angular velocity at the large $D$ limit. On the other hand the
analysis by the blackfold method [6] found that the horizon angular velocity
of the $D$ dimensional thin black ring becomes

$\displaystyle\Omega_{\text{H}}=\frac{1}{\sqrt{D-3}}\frac{1}{R},$ (1.1)

where $R$ is a ring radius. This means that the horizon angular velocity of
the thin black ring is $O(1/\sqrt{D})$, not $O(1)$ at the large $D$ limit.
Thus the effective equation in [13] cannot be straightforwardly applied to the
(thin) black ring. Hence, to construct the black ring solution by the large
$D$ expansion method, we should study the effective theory of the black hole
with $O(1/\sqrt{D})$ horizon angular velocity anew. The purposes of this paper
are to give the effective theory of black holes with $O(1/\sqrt{D})$ horizon
angular velocity by including time-dependence and to construct the large $D$
black ring solution by solving the effective equations. We call a black hole
with $O(1/\sqrt{D})$ horizon angular velocity slowly rotating large $D$ black
hole in this paper. We expect that there is no black ring with $O(1)$ horizon
angular velocity. The horizon angular velocity of the black ring is determined
by the balance condition between the string tension and centrifugal force by
the boost, which is the horizon angular velocity. At large $D$ the tension of
a black string becomes small compared to its mass by $O(D^{-1})$ [11]. As a
result the horizon angular velocity should be also small at large $D$. Hence,
generically, there is no black ring with $O(1)$ horizon angular velocity at
large $D$. This is why we do not use the word “slowly rotating black ring” in
our analysis.

The large $D$ effective theory for dynamical black holes has been considered
in [14, 15], where it was shown that the large $D$ expansion method gives
simple effective equations for the dynamical black holes. The equations for
dynamical black branes obtained in [15] can be solved very easily, and the
numerical solution of the equations capture the non-linear evolution and
endpoint of the Gregory-Laflamme instability of the black brane [16] in higher
dimensions than the critical dimension [17]. We will study the large $D$
effective equation for slowly rotating large $D$ black holes in the similar
manner with [15]. Then we will see that the large $D$ black ring is found as
an analytic stationary solution of the effective equations. Furthermore, by
perturbing effective equations, we can obtain a quasinormal mode condition of
the black ring. As found numerically in [18], the quasinormal mode says that
the thin black ring is unstable against non-axisymmetric perturbations, and
relatively fat black ring is stable. Our large $D$ effective equations
describe the non-linear evolution of such non-axisymmetric instabilities of
the black ring. We can also include the $1/D$ corrections to the effective
equations. We find that the $1/D$ corrections give striking features to the
instability modes of the black ring.

This paper is organized as following. In section 2 we give the effective
theory of slowly rotating large $D$ black holes by the effective equations. We
also discuss some general properties of the stationary solution of the
equations and give formula for thermodynamic quantities such as the mass and
angular momentum of the solution with $1/D$ corrections. In section 3 we
construct the black ring solution analytically by solving the effective
equations. Some physical properties such as quasinormal modes and phase
diagram of the black ring will be also discussed there. We close this paper by
giving discussion and outlook of this work in section 4. The appendices
contain technical details and some useful byproducts of the main results in
this paper. Especially the slowly rotating Myers-Perry black hole and slowly
boosted black string will be rediscovered with their quasinormal mode
frequencies analytically in Appendix A.

Note that we perform the large $D$ expansion of the Einstein equation to find
$D$ dimensional black hole solutions. In the expansion, defining $n$ by

$\displaystyle D=n+4,$ (1.2)

we use $1/n$ as the expansion parameter instead of $1/D$ in this paper. This
definition of $n$ is same with one of [6, 7] for black rings.

##  2 Effective equations

We consider the large $D$ effective theory for slowly rotating large $D$ black
holes. The effective equations will be given as equations for the energy and
momentum density of a dynamical black hole. In this section we study general
properties of solutions of the effective equations without specifying an
embedding of solutions.

###  2.1 Setup

The effective equations describe non-linear dynamical deformations of black
holes. To obtain them, it is natural to use the ingoing Eddington-Finkelstein
coordinate for the metric ansatz as

$\displaystyle ds^{2}=-Adv^{2}-2(u_{v}dv+u_{a}dX^{a})dr$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}G_{ab}dX^{a}dX^{b}-2C_{a}dvdX^{a}+r^{2}H^{2}d\Omega^{2}_{n},$
(2.1)

where $X^{a}=(z,\Phi)$. $A$, $u_{v}$, $G_{ab}$, $C_{a}$ are functions of
$(v,r,X^{a})$. $u_{a}$ and $H$ are functions of $(v,X^{a})$. In this paper we
use $v$ as a time coordinate in this system and the effective theory obtained
below, while $t$ was used for it in [15]. This metric ansatz partially fixes
the gauge of the radial coordinate $r$. The residual gauge can be fixed in
solving the Einstein equations. $X^{a}=(z,\Phi)$ are inhomogeneous spatial
directions of black holes, and we take $\Phi$ as the rotational direction. We
give large $D$ scalings and boundary conditions of the metric functions.

#### Large $D$ scalings

To solve the Einstein equation by $1/n$ expansion, we should specify behaviors
of each metric functions at large $D$. In the analysis of the blackfold [6]
they found that the horizon angular velocity of the $D=n+4$ dimensional thin
black ring is given by

$\displaystyle\Omega_{H}=\frac{1}{\sqrt{n+1}}\frac{1}{R}+O(R^{-2}),$ (2.2)

where $R$ is a ring radius. This result implies that the metric function
$C_{\Phi}$ in eq. (2.1) becomes $O(1/\sqrt{n})$ at large $D$. From this
observation we define the slowly rotating large $D$ black hole by
$C_{\Phi}=O(1/\sqrt{n})$ 222 The stationary solution considered in [13] was
assumed to have $C_{\Phi}=O(1)$. . One may think that this assumption is
restrictive for a black ring and describes the dynamics only of a thin black
ring solution. However this is not correct. As we will see, solutions obtained
in this paper cover wider range than one by the blackfold because we do not
require the much larger ring radius than the ring thickness. In our analysis
ring radius and thickness can be comparable, and our large $D$ solution can
describe the dynamics also of not-thin black ring.

In our setting $\partial_{\Phi}$ is not a Killing vector. Thus the metric
functions have $\Phi$-dependences in general. Then, to have consistent $1/n$
expansions of the Einstein equations, we should assume
$\partial_{\Phi}=O(\sqrt{n})$ 333 We consider the decoupled mode excitation
[19]. So we assume $\partial_{v}=O(1)$ and $\partial_{r}=O(D)$. . If we take
$\partial_{\Phi}=O(1)$, the Einstein equations have not only $1/n$ series but
also $1/\sqrt{n}$ series, and the analysis becomes a bit involved. To treat
the scaling $\partial_{\Phi}=O(\sqrt{n})$ it is useful to introduce new
coordinate $\phi$ defined by

$\displaystyle\Phi=\frac{\phi}{\sqrt{n}}.$ (2.3)

Then $\partial_{\Phi}=O(\sqrt{n})$ is equivalent to $\partial_{\phi}=O(1)$.
Summarizing, our metric ansatz for the slowly rotating large $D$ black holes
is

$\displaystyle ds^{2}=-Adv^{2}-2(u_{v}dv+u_{a}dx^{a})dr$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}G_{ab}dx^{a}dx^{b}-2C_{a}dvdx^{a}+r^{2}H^{2}d\Omega^{2}_{n},$
(2.4)

where $x^{a}=(z,\phi)$. Each metric functions have following large $D$
expansions

$\displaystyle A=\sum_{k\geq
0}\frac{A^{(k)}(v,r,x^{a})}{n^{k}},~{}~{}C_{z}=\sum_{k\geq
0}\frac{C_{z}^{(k)}(v,r,x^{a})}{n^{k+1}},~{}~{}C_{\phi}=\sum_{k\geq
0}\frac{C_{\phi}^{(k)}(v,r,x^{a})}{n^{k+1}},$ (2.5) $\displaystyle
u_{v}=\sum_{k\geq
0}\frac{u_{v}^{(k)}(v,r,x^{a})}{n^{k}},~{}~{}u_{z}=\sum_{k\geq
0}\frac{u_{z}^{(k)}(v,x^{a})}{n^{k+1}},~{}~{}u_{\phi}=\sum_{k\geq
0}\frac{u_{\phi}^{(k)}(v,x^{a})}{n^{k+1}},$ (2.6) $\displaystyle
G_{zz}=1+\sum_{k\geq
0}\frac{G_{zz}^{(k)}(v,r,x^{a})}{n^{k+1}},~{}~{}G_{z\phi}=\sum_{k\geq
0}\frac{G_{z\phi}^{(k)}(v,r,x^{a})}{n^{k+2}},~{}~{}$ (2.7)

and

$\displaystyle G_{\phi\phi}=\frac{G(z)^{2}}{n}\left(1+\sum_{k\geq
0}\frac{G_{\phi\phi}^{(k)}(v,r,x^{a})}{n^{k+1}}\right),~{}~{}H=H(z)$ (2.8)

Under these scaling assumptions, the Einstein equation can be consistently
expanded in $1/n$. $\phi$ is the rotational direction of the black hole, and
we assume $\partial_{v}$ and $\partial_{\phi}$ become the Killing vectors in a
background geometry. This assumption implies that the asymptotic metric at far
from the horizon does not have $v$ and $\phi$ dependences. So the metric
function $H(z)$ and the leading order of $G_{\phi\phi}$ has only $z$
dependences as seen in eq. (2.8). Finally we introduce new radial coordinate
$\mathsf{R}$ to realize $\partial_{r}=O(n)$ defined by

$\displaystyle\mathsf{R}=\left(\frac{r}{r_{0}}\right)^{n},$ (2.9)

where a constant $r_{0}$ is a fiducial horizon size, and we set to $r_{0}=1$.

#### Boundary conditions

We give boundary conditions for metric functions. In this paper we consider
non-linear dynamics of the decoupled mode excitations [19]. The decoupled mode
condition demands following boundary conditions in the asymptotic region
$\mathsf{R}\gg 1$

$\displaystyle
A=1+O(\mathsf{R}^{-1}),~{}~{}C_{a}=O(\mathsf{R}^{-1})~{}~{},G_{z\phi}=O(\mathsf{R}^{-1}).$
(2.10)

The dynamical solutions have the horizon-like surface where $A$ vanishes at
the leading order in $1/n$ expansion 444 In this paper we consider the slowly
rotating black holes. Thus the horizon position is determined mainly by
$g_{uu}=-A$. . We regard the surface as a horizon of the dynamical black hole.
The boundary condition on the horizon is the regularity condition of the each
metric functions.

###  2.2 Effective equations

We solve the Einstein equation by the large $D$ expansion. The leading order
equations of the Einstein equations contain only $\mathsf{R}$-derivatives, so
we can integrate them easily. Then, the leading order solutions after imposing
the boundary conditions are obtained as

$\displaystyle
A^{(0)}=1-\frac{p_{v}(v,x^{a})}{\mathsf{R}},~{}~{}C_{a}^{(0)}=\frac{p_{a}(v,x^{a})}{\mathsf{R}},~{}~{}u_{v}^{(0)}=-\frac{H(z)}{\sqrt{1-H^{\prime}(z)^{2}}},$
(2.11) $\displaystyle
u_{a}^{(0)}=u_{a}^{(0)}(z),~{}~{}G_{zz}^{(0)}=0,~{}~{}G_{z\phi}^{(0)}=\frac{p_{z}p_{\phi}}{p_{v}}\frac{1}{\mathsf{R}},$
(2.12)

and

$\displaystyle
G_{\phi\phi}^{(0)}=-\left(2+\frac{2H(z)G^{\prime}(z)H^{\prime}(z)}{G(z)(1-H^{\prime}(z)^{2})}\right)\log{\mathsf{R}}+\frac{p_{\phi}^{2}}{G(z)^{2}p_{v}}\frac{1}{\mathsf{R}}.$
(2.13)

$p_{v}(v,x^{a})$ and $p_{a}(v,x^{a})$ are mass and momentum density of the
solution. They are introduced as integration functions of
$\mathsf{R}$-integrations of the Einstein equations. We can see that the
horizon position of this dynamical solution is at $\mathsf{R}=p_{v}(v,x^{a})$.
Furthermore we found that the function $H(z)$ should satisfy the following
condition

$\displaystyle 1-H^{\prime}(z)^{2}+H(z)H^{\prime\prime}(z)=0,$ (2.14)

from the boundary condition of $G_{z\phi}^{(0)}$ at asymptotic region
$\mathsf{R}\gg 1$. In eq. (2.12) we already imposed the condition (2.14). This
condition is equivalent to the constant condition of the mean curvature of
$r=\text{const.}$ surface in the asymptotic region for static solutions
[12]555 If we perform $(D-1)+1$ decomposition of the Einstein equation on
$\mathsf{R}=\text{const.}$ surface, this condition is obtained from the
momentum constraint as shown in [12]. . In this paper we consider small
rotation $\Omega_{\text{H}}=O(1/\sqrt{n})$, so the leading order solution has
no effects of rotations and gives same equations with static solutions. The
condition (2.14) is integrated as

$\displaystyle\frac{\sqrt{1-H^{\prime}(z)^{2}}}{H(z)}=2\hat{\kappa},$ (2.15)

where $\hat{\kappa}$ is a constant related with the surface gravity of the
horizon as we will see later. So we call $\hat{\kappa}$ a reduced surface
gravity. At the leading order the integration functions $p_{v}$ and $p_{a}$
are arbitrary functions of $(v,x^{a})$, and there is no constraint among them.
$u_{a}^{(0)}$ is also arbitrary function of $z$ at the leading order. At
higher order in $1/n$, constraints for $u_{a}^{(0)}$ appear, and we can
determine their explicit forms.

Solving the next-to leading order Einstein equations we obtain non-trivial
conditions which $p_{v}(v,x^{a})$ and $p_{a}(v,x^{a})$ should satisfy. The
non-trivial conditions can be obtained also from the momentum constraints on
$\mathsf{R}=\text{const}.$ surface. These conditions give effective equations
for the slowly rotating large $D$ black holes. Their forms are

$\displaystyle\partial_{v}p_{v}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{v}-\frac{\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}G(z)^{2}}+\frac{\partial_{\phi}p_{\phi}}{G(z)^{2}}+\frac{H^{\prime}(z)}{H(z)}p_{z}=0,$
(2.16)
$\displaystyle\partial_{v}p_{\phi}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{\phi}-\frac{\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}G(z)^{2}}+\frac{1}{G(z)^{2}}\partial_{\phi}\Biggl{[}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{4\hat{\kappa}^{2}G(z)H(z)^{2}+2G^{\prime}(z)H(z)H^{\prime}(z)}{4\hat{\kappa}^{2}G(z)H(z)^{2}}\partial_{\phi}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{H^{\prime}(z)}{H(z)}\frac{p_{z}p_{\phi}}{p_{v}}+\frac{G^{\prime}(z)H^{\prime}(z)}{\hat{\kappa}G(z)H(z)}p_{\phi}=0,$
(2.17)

and

$\displaystyle\partial_{v}p_{z}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{z}-\frac{\partial_{\phi}^{2}p_{z}}{2\hat{\kappa}G(z)^{2}}+\partial_{z}p_{v}+\frac{1}{G(z)^{2}}\partial_{\phi}\Biggl{[}\frac{p_{\phi}p_{z}}{p_{v}}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{H^{\prime}(z)}{H(z)}\frac{p^{2}_{z}}{p_{v}}-\frac{G^{\prime}(z)}{G(z)^{3}}\frac{p^{2}_{\phi}}{p_{v}}+\frac{G^{\prime}(z)}{\hat{\kappa}G(z)^{3}}\partial_{\phi}p_{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{H(z)G^{\prime}(z)^{2}H^{\prime}(z)+G(z)(G^{\prime}(z)-H(z)H^{\prime}(z)G^{\prime\prime}(z))}{4\hat{\kappa}^{2}G(z)^{2}H(z)^{2}}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\frac{1-2H^{\prime}(z)^{2}}{2\hat{\kappa}H(z)^{2}}p_{z}=0.$
(2.18)

These equations describe dynamical non-linear deformations of mass and
momentum density of the dynamical black hole. To solve these equations we
should specify the embedding functions $G(z)$ and $H(z)$. Before doing that,
we study some general properties of solutions of these effective equations.
Note that the effective equations for dynamical black strings obtained in [15]
can be reproduced from eqs. (2.16), (2.17) and (2.18) as one example.

As one simple solution of the effective equations, we consider a stationary
solution

$\displaystyle p_{v}=p_{v}(z),~{}~{}p_{a}=p_{a}(z).$ (2.19)

This solution has two Killing vectors $\partial_{v}$ and $\partial_{\phi}$.
The effective equations can be solved under this ansatz. Eqs. (2.16) and
(2.17) give

$\displaystyle
p_{z}=\frac{p_{v}^{\prime}(z)}{2\hat{\kappa}},~{}~{}p_{\phi}(z)=\hat{\Omega}_{H}G(z)^{2}p_{v}(z).$
(2.20)

$\hat{\Omega}_{H}$ is an integration constant of $z$-integration of eq.
(2.17). To see the physical meaning of $\hat{\Omega}_{H}$, we rewrite the
$(v,\phi)$ part of the leading order metric of this stationary solution as

$\displaystyle
ds^{2}_{(v,\phi)}=-\left(1-\frac{p_{v}(z)}{\mathsf{R}}\right)dv^{2}+\frac{G(z)^{2}}{n}\left(d\phi-\hat{\Omega}_{H}\frac{p_{v}(z)}{\mathsf{R}}dv\right)^{2}+O(1/n).$
(2.21)

From this expression we can see that
$\Omega_{\text{H}}=\hat{\Omega}_{H}/\sqrt{n}$ gives the horizon angular
velocity. Actually horizon generating Killing vector $\xi$ is

$\displaystyle\xi=\frac{\partial}{\partial
v}+\hat{\Omega}_{H}\frac{\partial}{\partial\phi}=\frac{\partial}{\partial
v}+\Omega_{H}\frac{\partial}{\partial\Phi}.$ (2.22)

The Killing vector $\xi$ becomes null at the horizon $\mathsf{R}=p_{v}(z)$.
Furthermore we can calculate the surface gravity $\kappa$ of the black hole by
using $\xi$ as

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle-\frac{\partial_{r}(\xi_{\mu}\xi^{\mu})}{2\xi_{r}}\Bigl{|}_{\mathsf{R}=p_{v}(z)}$
(2.23) $\displaystyle=$ $\displaystyle
n\frac{\sqrt{1-H^{\prime}(z)^{2}}}{2H(z)}$ $\displaystyle=$ $\displaystyle
n\hat{\kappa},$

where $x^{\mu}=(v,x^{a})$. Here we omit $O(1/n)$ terms for simplification.
Thus the integration constant $\hat{\kappa}$ represents the surface gravity of
the black hole at the leading order in $1/n$ expansions. Finally, substituting
the solutions (2.20) into eq. (2.18), we obtain an equation for
$p_{v}(z)=e^{P(z)}$ as

$\displaystyle P^{\prime\prime}(z)-\frac{H^{\prime}(z)}{H(z)}P(z)$
$\displaystyle-\Bigl{[}\frac{G^{\prime}(z)^{2}}{G(z)^{2}}-\frac{G^{\prime\prime}(z)}{G(z)}+\frac{G^{\prime}(z)}{G(z)H(z)H^{\prime}(z)}-\hat{\Omega}^{2}_{\text{H}}\frac{G(z)G^{\prime}(z)(1-H^{\prime}(z)^{2})}{H(z)H^{\prime}(z)}\Bigr{]}=0.$
(2.24)

To solve this equation for $P(z)$ we should specify the functions $G(z)$ and
$H(z)$. The functions can be determined by embedding the solution into a
background geometry. In next section we will embed the leading order solution
into a flat background in the ring coordinate. Such solution describes
dynamical black rings. In Appendix A we show other solutions such as the
Myers-Perry black hole and boosted black strings by considering the embeddings
into other backgrounds.

###  2.3 Effective energy-momentum tensor

Next we study physical quantities of the solution. The mass and angular
momentum of dynamical black holes can be evaluated by the effective energy-
momentum tensor. The effective energy momentum tensor, $T_{\mu\nu}$, is
defined by 666 The effective energy momentum tensor is originally defined on
$r=\text{const}.$ surface. Thus its component runs over $(v,x^{a},x^{I})$
where $x^{I}$ is a coordinate on $S^{n}$ in eq. (2.4). However the metric
components of $g_{IJ}$ have been taken as non-dynamical ones by the gauge
choice. So we consider only $x^{\mu}=(v,x^{a})$ components of the effective
energy momentum tensor.

$\displaystyle
T_{\mu\nu}=-\frac{1}{8\pi\mathsf{G}_{D}}\left(\Bigl{[}K_{\mu\nu}\Bigr{]}_{\Sigma}-\bar{g}_{\mu\nu}\Bigl{[}K\Bigr{]}_{\Sigma}\right),$
(2.25)

where $x^{\mu}=(v,x^{a})$ in eq. (2.4). $\Sigma$ is a $r=\text{const.}$
surface in the asymptotic region $\mathsf{R}\gg 1$. $\mathsf{G}_{D}$ is the
gravitational constant in $D$ dimensions. $K_{\mu\nu}$ is the extrinsic
curvature of $r=\text{const.}$ surface. The square bracket represents the
background subtraction at $\Sigma$. $\bar{g}_{\mu\nu}$ is the background
induced metric on $\Sigma$. The background metric $\bar{g}_{\mu\nu}$ in the
asymptotic region $\mathsf{R}\gg 1$ is obtained from the leading order
solution as

$\displaystyle\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}$ $\displaystyle=$
$\displaystyle-dv^{2}+r^{2}dz^{2}$ (2.26)
$\displaystyle+\frac{G(z)^{2}}{n}\left(1-\frac{2H(z)H^{\prime}(z)G^{\prime}(z)\log{\mathsf{R}}}{nG(z)(1-H^{\prime}(z)^{2})}+O(n^{-2})\right)d\phi^{2}.$

The background extrinsic curvature, say $\bar{K}_{\mu\nu}$, is calculated as

$\displaystyle\bar{K}_{vv}=O(1/n),~{}~{}\bar{K}_{zz}=2\hat{\kappa}+O(1/n),~{}~{}\bar{K}_{\phi\phi}=-\frac{G(z)G^{\prime}(z)H^{\prime}(z)}{2\hat{\kappa}H(z)}+O(1/n),$
(2.27)

in our gauge choice for the radial coordinate $r$. The trace part of the
background extrinsic curvature, $\bar{K}$, contains contributions from
$\bar{K}_{IJ}$ where $x^{I}$ is a coordinate on $S^{n}$ in eq. (2.4). It is
obtained as

$\displaystyle\bar{K}=n\Biggl{[}2\hat{\kappa}+\frac{1}{n}\left(2\hat{\kappa}-\frac{G^{\prime}(z)H^{\prime}(z)}{2\hat{\kappa}G(z)H(z)}-2\hat{\kappa}\log{\mathsf{R}}\right)+O(n^{-2})\Biggr{]}.$
(2.28)

By definition, the background extrinsic curvature $\bar{K}_{\mu\nu}$ satisfies

$\displaystyle
K_{\mu\nu}-\bar{K}_{\mu\nu}=O(1/\mathsf{R}),~{}~{}K-\bar{K}=O(1/\mathsf{R}).$
(2.29)

Then we can compute the effective energy-momentum tensor $T_{\mu\nu}$. The
results are

$\displaystyle
T_{vv}=\frac{n\hat{\kappa}p_{v}}{8\pi\mathsf{G}_{D}\mathsf{R}}\left(1+O(n^{-1},\mathsf{R}^{-1})\right),~{}~{}T_{va}=\frac{2\hat{\kappa}p_{a}-\partial_{a}p_{v}}{16\pi\mathsf{G}_{D}\mathsf{R}}\left(1+O(n^{-1},\mathsf{R}^{-1})\right),$
(2.30)

and

$\displaystyle
T_{zz}=-\frac{G(z)^{2}(H^{\prime}(z)p_{z}+H(z)\partial_{z}p_{v})+H(z)\partial_{\phi}p_{\phi}}{16\pi\mathsf{G}_{D}G(z)^{2}H(z)\mathsf{R}}\left(1+O(n^{-1},\mathsf{R}^{-1})\right),$
$\displaystyle
T_{z\phi}=-\frac{1}{n}\frac{G(z)(2\hat{\kappa}p_{z}p_{\phi}-p_{v}(\partial_{z}p_{\phi}+\partial_{\phi}p_{z}))+2G^{\prime}(z)p_{\phi}p_{v}}{16\pi\mathsf{G}_{D}G(z)p_{v}\mathsf{R}}\left(1+O(n^{-1},\mathsf{R}^{-1})\right),$
$\displaystyle
T_{\phi\phi}=\frac{1}{n}\frac{1}{32\pi\mathsf{G}_{D}\hat{\kappa}H(z)p_{v}\mathsf{R}}\Bigl{[}2\hat{\kappa}H(z)(2\hat{\kappa}p_{\phi}^{2}+p_{v}(\partial_{\phi}p_{\phi}+G(z)^{2}(\partial_{v}p_{v}-2\hat{\kappa}p_{v})))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-G(z)H^{\prime}(z)p_{v}(G^{\prime}(z)p_{v}-2\hat{\kappa}G(z)p_{z})\Bigr{]}\left(1+O(n^{-1},\mathsf{R}^{-1})\right).$
(2.31)

Then we can define the mass $\mathcal{M}$ and angular momentum
$\mathcal{J}_{\Phi}$ of a dynamical black hole from this $T_{\mu\nu}$ as

$\displaystyle\mathcal{M}=\frac{n\hat{\kappa}\Omega_{n}}{8\pi\mathsf{G}_{D}}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}p_{v}(v,x^{a}),$ (2.32)
$\displaystyle\mathcal{J}_{\Phi}=\sqrt{n}\frac{\hat{\kappa}\Omega_{n}}{8\pi\mathsf{G}_{D}}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}p_{\phi}(v,x^{a}),$ (2.33)

where $H(z)^{n}$ term comes from the volume factor on $S^{n}$. $\Omega_{n}$ is
the volume of unit $S^{n}$. The $\sqrt{n}$ factor in $\mathcal{J}_{\Phi}$
comes from the relation between $\phi$ and $\Phi$ in eq. (2.3). For stationary
solutions we can define the mass and angular momentum of the black hole by the
Komar integral of each Killing vector. It can be shown easily that their
definitions are equivalent.

The horizon area of the solution, $\mathcal{A}_{\text{H}}$, becomes

$\displaystyle\mathcal{A}_{\text{H}}=\Omega_{n}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}p_{v}(v,x^{a}),$ (2.34)

where we used the fact that the horizon is at $\mathsf{R}=p_{v}$. Then we find
that the stationary solution satisfies the Smarr formula given by

$\displaystyle\frac{n+1}{n+2}\mathcal{M}=\frac{\kappa}{8\pi\mathsf{G}_{D}}\mathcal{A}+\Omega_{\text{H}}\mathcal{J}_{\Phi},$
(2.35)

at the leading order in $1/n$ expansions. Note that since
$\Omega_{\text{H}}=O(1/\sqrt{n})$ and $\kappa=O(n)$,
$\Omega_{\text{H}}\mathcal{J}_{\Phi}$ term does not contribute to the Smarr
formula at the leading order. The mass and area of the solution can be
calculated also for non-stationary solutions $p_{v}=p_{v}(v,x^{a})$. For this
non-stationary solution, we can see that the Smarr formula (2.35) holds at the
leading order in $1/n$ expansion. But this formula does not hold at the higher
order for non-stationary solutions.

###  2.4 $1/D$ corrections

Solving the next-to-next-to leading order of the Einstein equations, we can
obtain $1/n$ corrections to the effective equations (2.16), (2.17) and (2.18).
The explicit forms of the $1/n$ corrections to the effective equations are not
simple, so we show them in Appendix C. Here we show only physical effects by
$1/n$ corrections to the solution.

At first we define the mass and momentum density, $p_{v}$ and $p_{a}$, up to
$1/n$ corrections. The mass and momentum density are introduced as integration
functions of $\mathsf{R}-$integrations of the Einstein equations. Thus we
should specify the normalization of the integration functions. Expanding $A$
and $C_{a}$ up to $1/n$ corrections we define the mass and momentum density,
$p_{v}$ and $p_{a}$, by coefficients of $1/\mathsf{R}$ as

$\displaystyle
A=1-\frac{p_{v}(v,x^{a})}{\mathsf{R}}+O(n^{-2},\mathsf{R}^{-2}),~{}~{}C_{a}=\frac{p_{a}(v,x^{a})}{\mathsf{R}}+O(n^{-2},\mathsf{R}^{-2}).$
(2.36)

This definition normalizes the integration functions up to $O(1/n)$ by the
asymptotic behavior of the metric functions. Using this definition and the
next-to-next-to leading order stationary solutions, the surface gravity up to
$1/n$ correction is obtained as

$\displaystyle\frac{\kappa}{n}=\hat{\kappa}-\frac{1}{n}\Biggl{[}\frac{G^{\prime}(z)H^{\prime}(z)}{4\hat{\kappa}G(z)H(z)}+\hat{\kappa}\log{p_{v}}+\frac{H^{\prime}(z)}{2H(z)}\frac{p_{z}}{p_{v}}+\frac{\hat{\kappa}}{2G(z)^{2}}\frac{p_{\phi}^{2}}{p_{v}^{2}}\Biggr{]}+O(n^{-2}).$
(2.37)

One may think that this expression is strange since the surface gravity
$\kappa$ seems not to be constant when the solution is stationary where
$p_{v}=p_{v}(z)$ and $p_{a}=p_{a}(z)$. However we can show that

$\displaystyle\frac{d}{dz}\Biggl{[}\frac{G^{\prime}(z)H^{\prime}(z)}{4\hat{\kappa}G(z)H(z)}+\hat{\kappa}\log{p_{v}}+\frac{H^{\prime}(z)}{2H(z)}\frac{p_{z}}{p_{v}}+\frac{\hat{\kappa}}{2G(z)^{2}}\frac{p_{\phi}^{2}}{p_{v}^{2}}\Biggr{]}=0,$
(2.38)

by using the effective equations (2.16), (2.17) and (2.18) for the stationary
solution. Hence the surface gravity becomes constant for the stationary
solution as expected. The horizon angular velocity
$\Omega_{\text{H}}=\hat{\Omega}_{\text{H}}/\sqrt{n}$ is also obtained as

$\displaystyle\hat{\Omega}_{\text{H}}=\frac{p_{\phi}}{G(z)^{2}p_{v}}+\frac{1}{n}\frac{G^{\prime}(z)H^{\prime}(z)}{2\hat{\kappa}^{2}G(z)^{3}H(z)}\frac{p_{\phi}\log{p_{v}}}{p_{v}}+O(n^{-2}).$
(2.39)

The first term in r.h.s. is equivalent to $\hat{\Omega}_{\text{H}}$ at the
leading order as seen in eq. (2.20). But, at higher order in $1/n$ expansion,
they are different. We can show that, when the solution is stationary, the
horizon angular velocity is constant up to $1/n$ correction by using the
effective equations with $1/n$ corrections given in Appendix C.

We can calculate the mass, angular momentum and area of the solution up to
$O(1/n)$. The results are complicate, so we show the results only for the
stationary solutions $p_{v}=p_{v}(z)$ and $p_{a}=p_{a}(z)$. The mass and
angular momentum formula up to $O(1/n)$ become

$\displaystyle\mathcal{M}=\frac{n\hat{\kappa}\Omega_{n}}{8\pi\mathsf{G}_{D}}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}M(z),$ (2.40)
$\displaystyle\mathcal{J}_{\Phi}=\sqrt{n}\frac{\hat{\kappa}\Omega_{n}}{8\pi\mathsf{G}_{D}}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}J_{\phi}(z),$ (2.41)

where

$\displaystyle
M(z)=p_{v}+\frac{1}{n}\Biggl{[}p_{v}-\frac{p_{z}}{2\hat{\kappa}}\frac{H^{\prime}(z)}{H(z)}-\frac{p_{v}}{4\hat{\kappa}^{2}}\frac{G^{\prime}(z)H^{\prime}(z)}{G(z)H(z)}\Biggr{]},$
(2.42) $\displaystyle
J_{\phi}(z)=p_{\phi}-\frac{1}{n}\Biggl{[}\frac{p_{z}p_{\phi}}{2\hat{\kappa}p_{v}}\frac{H^{\prime}(z)}{H(z)}+\frac{p_{\phi}}{4\hat{\kappa}^{2}}\frac{G^{\prime}(z)H^{\prime}(z)}{G(z)H(z)}\Biggr{]}.$
(2.43)

The mass and angular momentum can be obtained from the Komar integral for the
stationary solution or from the effective energy momentum tensor for general
time-dependent solution. These definitions coincide for the stationary
solution. The formula of the horizon area also has the $1/n$ correction as

$\displaystyle\mathcal{A}_{\text{H}}=\Omega_{n}\int
dz\frac{d\phi}{\sqrt{n}}~{}G(z)H(z)^{n}A_{\text{H}}(v,x^{a})$ (2.44)

where

$\displaystyle
A_{\text{H}}=p_{v}+\frac{1}{n}\Biggl{[}p_{v}\log{p_{v}}-\frac{1}{2G(z)^{2}}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}.$
(2.45)

Then we can see that the Smarr formula,

$\displaystyle\frac{n+1}{n+2}\mathcal{M}=\frac{\kappa}{8\pi\mathsf{G}_{D}}\mathcal{A}_{\text{H}}+\Omega_{\text{H}}\mathcal{J}_{\Phi},$
(2.46)

can be satisfied up to $1/n$ corrections.

##  3 Black ring and its physical properties

Next we solve the effective equations in an explicit embedding. Especially we
consider an embedding into a flat background in the ring coordinate and find a
stationary solution, that is, the black ring solution analytically.

###  3.1 Ring coordinate embedding

The leading order metric has the following asymptotic form at $\mathsf{R}\gg
1$

$\displaystyle ds^{2}|_{\mathsf{R}\gg
1}=-dv^{2}+2\left(\frac{H(z)}{\sqrt{1-H^{\prime}(z)^{2}}}dv-u^{(0)}_{a}(z)\frac{dx^{a}}{n}\right)dr$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}dz^{2}+r^{2}G(z)^{2}d\Phi^{2}+r^{2}H(z)^{2}d\Omega^{2}_{n}.$
(3.1)

We embed this leading order metric into a flat background in the ring
coordinate. The $D=n+4$ dimensional flat metric in the ring coordinate is [20]
777 In this note we use the ring coordinate by $(r,\theta)$ in [20]. Of course
the following analysis can be done also by using $(x,y)$ coordinate in [20].

$\displaystyle
ds^{2}=-dt^{2}+\frac{R^{2}}{(R+r\cos{\theta})^{2}}\Biggl{[}\frac{R^{2}dr^{2}}{R^{2}-r^{2}}+(R^{2}-r^{2})d\Phi^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\Omega^{2}_{n})\Biggr{]},$
(3.2)

where $0\leq r\leq R$, $0\leq\theta\leq\pi$ and $0\leq\Phi\leq 2\pi$. $R$ is a
ring radius. $r=0$ is the origin of the ring coordinate. The asymptotic
infinity and the axis of $\Phi$-rotation are at $r=R$. In this ring coordinate
$r=\text{const.}$ surface has a topology of $S^{1}\times S^{n+1}$. $\theta=0$
is an inner equatorial plane, and $\theta=\pi$ is an outer equatorial plane.
Remembering the definition of $\mathsf{R}=(r/r_{0})^{n}$, we can embed the
leading order induced metric on $\mathsf{R}=\text{const.}$ surface into the
flat background in this ring coordinate by $r=r_{0}$ where $r_{0}$ is a
constant. As done in the previous section we set to $r_{0}=1$. Then, comparing
eqs. (3.1) and (3.2) on $r=1$ surface, we find that the embedding gives
following identifications

$\displaystyle
H(z)=\frac{R\sin{\theta}}{R+\cos{\theta}},~{}~{}G(z)=\frac{R\sqrt{R^{2}-1}}{R+\cos{\theta}},~{}~{}\frac{d\theta}{dz}=\frac{R+\cos{\theta}}{R}.$
(3.3)

It is easy to confirm that this identification actually satisfies eq. (2.14).
We can calculate the surface gravity, $\kappa=n\hat{\kappa}$, by these
identifications as

$\displaystyle\hat{\kappa}=\frac{\sqrt{1-H^{\prime}(z)^{2}}}{2H(z)}=\frac{\sqrt{R^{2}-1}}{2R}.$
(3.4)

The embedded solution is the black ring solution because the horizon topology
is now $S^{1}\times S^{n+1}$. From $R\geq r$, the ring radius should be larger
than unity $R\geq 1$. $R\simeq 1$ corresponds to the fat black ring, and thin
black ring is described by $R\gg 1$.

Using identifications (3.3) and the surface gravity (3.4), we obtain the
leading order effective equations for the black ring from eqs. (2.16), (2.17)
and (2.18) as

$\displaystyle\partial_{v}p_{v}+\frac{(R+y)(1+Ry)}{R\sqrt{R^{2}-1}}\partial_{y}p_{v}-\frac{(R+y)^{2}}{R(R^{2}-1)^{3/2}}\partial_{\phi}^{2}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(R+y)^{2}}{R^{2}(R^{2}-1)}\partial_{\phi}p_{\phi}+\frac{1+Ry}{R\sqrt{1-y^{2}}}p_{z}=0,$
(3.5)
$\displaystyle\partial_{v}p_{\phi}+\frac{(R+y)(1+Ry)}{R\sqrt{R^{2}-1}}\partial_{y}p_{\phi}-\frac{(R+y)^{2}}{R(R^{2}-1)^{3/2}}\partial_{\phi}^{2}p_{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(R+y)^{2}}{R^{2}(R^{2}-1)}\partial_{\phi}\Biggl{[}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}-\frac{1+2Ry+R^{2}}{R^{2}-1}\partial_{\phi}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1+Ry}{R\sqrt{1-y^{2}}}\frac{p_{z}p_{\phi}}{p_{v}}+\frac{2(1+Ry)}{R\sqrt{R^{2}-1}}p_{\phi}=0$
(3.6)

and

$\displaystyle\partial_{v}p_{z}+\frac{(R+y)(1+Ry)}{R\sqrt{R^{2}-1}}\partial_{y}p_{z}-\frac{(R+y)^{2}}{R(R^{2}-1)^{3/2}}\partial_{\phi}^{2}p_{z}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{(R+y)\sqrt{1-y^{2}}}{R}\partial_{y}p_{v}+\frac{(R+y)^{2}}{R^{2}(R^{2}-1)}\partial_{\phi}\Biggl{[}\frac{p_{z}p_{\phi}}{p_{v}}\Biggr{]}+\frac{1+Ry}{R\sqrt{1-y^{2}}}\frac{p_{z}^{2}}{p_{v}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{(R+y)^{2}\sqrt{1-y^{2}}}{R^{3}(R^{2}-1)}\frac{p_{\phi}^{2}}{p_{v}}+\frac{2(R+y)^{2}\sqrt{1-y^{2}}}{R^{2}(R^{2}-1)^{3/2}}\partial_{\phi}p_{\phi}+\frac{(R+y)\sqrt{1-y^{2}}}{R^{2}-1}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{2+2Ry-y^{2}+R^{2}(2y^{2}-1)}{R\sqrt{R^{2}-1}(1-y^{2})}p_{z}=0.$
(3.7)

Here we introduced a coordinate $y$ defined by

$\displaystyle y=\cos{\theta}.$ (3.8)

Our effective equations (3.5), (3.6) and (3.7) describe non-linear dynamical
deformations of the black ring from thin $R\gg 1$ to not-thin $R>1$ region.

###  3.2 Black ring solution

The black ring solution is obtained as the stationary solution of the
effective equations (3.5), (3.6) and (3.7). The stationary solution is given
by

$\displaystyle p_{v}=e^{P(y)},~{}~{}p_{a}=p_{a}(y).$ (3.9)

As done in eq. (2.20), we find from eqs. (3.5) and (3.6)

$\displaystyle
p_{z}(y)=-\frac{(R+y)\sqrt{1-y^{2}}}{\sqrt{R^{2}-1}}p_{v}^{\prime}(y),~{}~{}p_{\phi}=\hat{\Omega}_{H}\frac{R^{2}(R^{2}-1)}{(R+y)^{2}}p_{v}(y).$
(3.10)

Furthermore eq. (3.7) gives an equation for $p_{v}=e^{P(y)}$ as eq. (2.24)

$\displaystyle
P^{\prime\prime}(y)+\frac{2}{R+y}P^{\prime}(y)-\frac{R}{(R+y)(1+Ry)}+\hat{\Omega}^{2}_{\text{H}}\frac{R^{2}(R^{2}-1)^{2}}{(R+y)^{4}(1+Ry)}=0.$
(3.11)

This equation contains a pole at $y=-1/R$ in the source term. The solution can
have a singular behavior by the source term at $y=-1/R$, too. To obtain a
regular solution at $y=-1/R$, $\hat{\Omega}_{H}$ should be

$\displaystyle\hat{\Omega}_{H}=\frac{\sqrt{R^{2}-1}}{R^{2}}.$ (3.12)

If $\hat{\Omega}_{\text{H}}$ does not take this value, the function $P(y)$ has
a logarithmic divergence at $y=-1/R$. Thus, as seen in five dimensional black
ring [2] and higher dimensional black ring by the blackfold [6, 7], the
regularity condition determines the horizon angular velocity of the black
ring. The solution of eq. (3.11) under the condition (3.12) is obtained
analytically as

$\displaystyle
P(y)=p_{0}+\frac{d_{0}}{R+y}+\frac{(1+Ry)(1+Ry+2R(R+y)\log{(R+y)})}{2R^{2}(R+y)^{2}},$
(3.13)

where $p_{0}$ and $d_{0}$ are integration constants. They are degree of
freedom associated with trivial deformations and do not affect physical
properties of black rings. $p_{0}$ is the $1/n$ redefinition of $r_{0}$, and
$d_{0}$ comes from the redefinition of $\phi$ coordinate of the ring
coordinate as discussed in Appendix B. Summarizing above results, we found the
black ring solution at the leading order of large $D$ expansion by the metric

$\displaystyle
ds^{2}=-\left(1-\frac{p_{v}(z)}{\mathsf{R}}\right)dv^{2}+2\left(\frac{dv}{2\hat{\kappa}}-u_{a}^{(0)}(z)\frac{dx^{a}}{n}\right)dr$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2\left(\frac{R+y}{2\hat{\kappa}R}\frac{p_{v}^{\prime}(z)}{\mathsf{R}}\frac{dz}{n}+\hat{\Omega}_{\text{H}}\frac{R^{2}(R^{2}-1)}{(R+\cos{\theta)^{2}}}\frac{p_{v}(z)}{\mathsf{R}}\frac{d\phi}{n}\right)dv$
$\displaystyle~{}~{}~{}~{}+r^{2}dz^{2}+\hat{\Omega}_{\text{H}}\frac{8\hat{\kappa}^{2}R}{R+\cos{\theta}}\frac{p_{v}^{\prime}(z)}{\mathsf{R}}\frac{dzd\phi}{n^{2}}$
$\displaystyle~{}~{}~{}~{}+r^{2}\frac{R^{2}(R^{2}-1)}{(R+\cos{\theta})^{2}}\left(1-\frac{2R(R+\cos{\theta})}{R^{2}-1}\frac{\log{\mathsf{R}}}{n}+\hat{\Omega}^{2}_{\text{H}}\frac{R^{2}(R^{2}-1)}{(R+\cos{\theta})^{2}}\frac{p_{v}(z)}{n\mathsf{R}}\right)\frac{d\phi^{2}}{n}$
$\displaystyle~{}~{}~{}~{}+\frac{r^{2}R^{2}\sin^{2}{\theta}}{(R+\cos{\theta})^{2}}d\Omega^{2}_{n},$
(3.14)

where $\mathsf{R}=(r/r_{0})^{n}$ with $r_{0}=1$. The coordinates $z$ and
$\theta$ are related by eq. (3.3). $\hat{\kappa}$ and
$\hat{\Omega}_{\text{H}}$ are given in eqs. (3.4) and (3.12). The function
$p_{v}=e^{P(y)}$ is the solution of the equation (3.11), and it is obtained in
eq. (3.13) with $y=\cos{\theta}$. Our black ring solution breaks down at
$R=1$, and the very fat black ring cannot be captured by our large $D$
solution 888 A very fat black ring means the solution with $R=1+O(n^{-1})$. .
However, the black ring with not so large ring radius can be described by eq.
(3.14). So we can study the properties of not only the thin black ring but
also not-thin black ring $R>1$ by using our solution (3.14).

###  3.3 Quasinormal modes

We investigate quasinormal modes of the black ring solution. The quasinormal
modes are obtained by perturbation analysis of the effective equations around
the black ring solution. The perturbation ansatz is

$\displaystyle p_{v}(v,y,\phi)=e^{P(y)}\left(1+\epsilon e^{-i\omega
v}e^{im\phi}F_{v}(y)\right),$ (3.15) $\displaystyle
p_{z}(v,y,\phi)=-\frac{(R+y)\sqrt{1-y^{2}}}{2\hat{\kappa}R}p_{v}^{\prime}(y)\left(1+\epsilon
e^{-i\omega v}e^{im\phi}F_{z}(y)\right),$ (3.16) $\displaystyle
p_{\phi}(v,y,\phi)=\frac{R(R^{2}-1)^{3/2}}{(R+y)^{2}}p_{v}(y)\left(1+\epsilon
e^{-i\omega v}e^{im\phi}F_{\phi}(y)\right),$ (3.17)

where we used eq. (3.12). There is one remark on the quantum number $m$
associated with $\partial_{\phi}$. In the ring coordinate (3.2) the coordinate
$\Phi$ has the range of $0\leq\Phi\leq 2\pi$. Thus the quantum number
$m_{\Phi}$ associated with $\partial_{\Phi}$ is quantized as $m_{\Phi}=0,\pm
1,\pm 2,...$. From the relation (2.3), these quantum numbers are related by

$\displaystyle m=\frac{m_{\Phi}}{\sqrt{n}}.$ (3.18)

So $m$ in the perturbations can take non-integer values in general.

Perturbing the effective equations (3.5), (3.6) and (3.7) with respect to
$\epsilon$, we obtain perturbation equations for $F_{v}(y)$, $F_{z}(y)$ and
$F_{\phi}(y)$. The perturbation equations have a pole at $y=-1/R$ again. To
solve the perturbation equations we should impose regularity conditions. If we
specify the behavior of the perturbation fields at the pole $y=-1/R$ as the
regularity condition by

$\displaystyle F_{v}(y)\propto(1+Ry)^{\ell}\left(1+O(1+Ry)\right),$ (3.19)

where $\ell$ is a non-negative integer, we get one non-trivial condition for
the frequency $\omega$ as

$\displaystyle\frac{1}{\sqrt{R^{2}-1}(m^{2}+imR+\ell
R^{2})-iR^{3}\omega}\Bigl{[}R^{9}\omega^{3}+iR^{6}\sqrt{R^{2}-1}\Bigl{(}3m^{2}+3imR$
$\displaystyle~{}~{}~{}+(3\ell-2)R^{2}\Bigr{)}\omega^{2}-R^{3}(R^{2}-1)\Bigl{(}3m^{4}+6im^{3}R+2(3\ell-4)m^{2}R^{2}$
$\displaystyle~{}~{}~{}+2i(3\ell-2)mR^{3}+3(\ell-1)\ell
R^{4}\Bigr{)}\omega-i(R^{2}-1)^{3/2}\Bigl{(}m^{6}+3im^{5}R+3(\ell-2)m^{4}R^{2}$
$\displaystyle~{}~{}~{}+6i(\ell-1)m^{3}R^{3}-(4-7\ell+3\ell^{2})m^{2}R^{4}+3i\ell(\ell-1)mR^{5}+\ell^{2}(\ell-1)R^{6}\Bigr{)}\Bigr{]}$
$\displaystyle~{}~{}~{}=0.$ (3.20)

This is the quasinormal mode condition for the black ring. One may feel
strange about this derivation since we derive the quasinormal mode condition
from the local condition (3.19) at $y=-1/R$. If the condition (3.20) is
satisfied, the perturbation actually becomes regular at $y=-1/R$, but its
global structure, e.g., of $F_{v}(y)$, is still unknown. Furthermore the
quantum number $\ell$ associated with the harmonics in the ring coordinate
should be given by the global solution of $F_{v}(y)$. But the quantum number
is introduced by the local condition (3.19) in our derivation at the large $D$
limit. This localization of the quantum number associated with the harmonics
at the large $D$ limit has been observed in the spherical harmonics and
spheroidal harmonics in [13]. So we expect that the same feature would appear
in the harmonics in the ring coordinate. It is interesting to investigate this
property in detail, although we do not pursue this structure in this paper. In
the following we give results derived from eq. (3.20) for non-axisymmetric and
axisymmetric modes separately.

#### Non-axisymmetric modes $(m\neq 0)$

The quasinormal mode condition (3.20) can be solved in a simple form for
$\ell=0$ by

$\displaystyle\omega^{(\ell=0)}_{\pm}=\frac{\sqrt{R^{2}-1}}{R}\Bigl{[}\hat{m}\pm
i\hat{m}(1\mp\hat{m})\Bigr{]},~{}~{}\omega^{(\ell=0)}_{0}=\frac{\sqrt{R^{2}-1}}{R}\Bigl{[}\hat{m}-i(\hat{m}^{2}-2)\Bigr{]},$
(3.21)

Figure 1:  Plots of the leading order quasinormal modes with $\ell=0$ and
$m=2$, $\omega^{(\ell=0)}_{\pm}$, of the black ring. The left panel shows the
real part of the frequency normalized by the reduced surface gravity
$\hat{\kappa}$. The right panel shows the imaginary part. The black and gray
line represent $\omega^{(\ell=0)}_{+}$ and $\omega^{(\ell=0)}_{-}$
respectively. The real part of $\omega^{(\ell=0)}_{+}$ and
$\omega^{(\ell=0)}_{-}$ are same. $\omega^{(\ell=0)}_{+}$ is unstable for
$R>2$.

where $\hat{m}=m/R$. $\omega^{(\ell=0)}_{+}$ is an instability mode when
$R>m$. $\omega_{\pm}^{(\ell=0)}$ can be understood as the quasinormal mode of
the boosted black string as we will see below. We regard
$\omega_{0}^{(\ell=0)}$ in eq. (3.21) as a gauge mode in this paper 999 We
have not still studied residual gauge of $(v,z,\phi)$ coordinate in detail.
The gauge of $r$-coordinate was fixed in eq. (2.1). So we should check the
gauge transformation of perturbations to check which modes are physical.
Actually there are residual gauge. In Appendix B we study stationary one of
them. It is a bit involved task to eliminate whole gauge modes. So instead we
use another simple method to identify physical modes here. . At large radius
limit $R\gg 1$ the quasinormal modes should reproduce the quasinormal modes of
the boosted black string. Although $\omega^{(\ell=0)}_{\pm}$ has corresponding
modes, $\omega^{(\ell=0)}_{0}$ does not have. So it might be natural to
consider that $\omega^{(\ell=0)}_{0}$ is a gauge mode. In Figure 1 we show
plots of the quasinormal mode $\omega^{(\ell=0)}_{\pm}$ with $m=2$.
$\omega^{(\ell=0)}_{+}$ becomes stable in relatively fat region $R<m$. This
plot reproduces the behavior of the numerical results in [18] 101010 The
perturbation considered in [18] is the mode with $m_{\Phi}=2$, not $m=2$ as
seen in eq. (3.18). Thus our result for $m=2$ does not exactly correspond to
the mode in [18]. However, in $D=5~{}(n=1)$, we can expect that $m_{\Phi}=2$
and $m=2$ are not different so much. . We could not find any instabilities for
$\ell\neq 0$ and $m\neq 0$ mode perturbations.

Note that the instability mode $\omega^{(\ell=0)}_{+}$ always saturates the
superradiant condition 111111 The quantum numbers $m$ and $m_{\Phi}$ has the
relation (3.18). Then the superradiant factor is
$\displaystyle\text{Re}\Bigl{[}\omega-
m_{\Phi}\Omega_{\text{H}}\Bigr{]}=\text{Re}\Bigl{[}\omega-m\hat{\Omega}_{\text{H}}\Bigr{]}.$

$\displaystyle\text{Re}\Bigl{[}\omega^{(\ell=0)}_{+}-m\hat{\Omega}_{\text{H}}\Bigr{]}=0.$
(3.22)

The nature of this saturation at the large $D$ limit is unclear. The
dynamically unstable mode of the Myers-Perry black hole also shows the
peculiar relation with the superradiant condition at the large $D$ limit [13,
21]. For equally spin Myers-Perry black hole, the coincidence of the onset of
the superradiant and dynamically unstable regime was confirmed up to $1/D$
corrections [21], and it is consistent with the numerical result [22]. For
singly rotating Myers-Perry black hole the coincidence was also found at the
large $D$ limit [13]. However it is known numerically that their onsets are at
different rotation parameters for the singly rotating Myers-Perry black hole
in finite dimensions [23, 24]. Thus the coincidence of the onsets of
superradiant and dynamically unstable regime is only the feature at the large
$D$ limit for the singly rotating Myers-Perry black hole. As we will see
below, the saturation of the superradiant condition of the black ring does not
always hold at higher order in $1/n$ expansion, and we find that onsets of the
superradiant and dynamically unstable regime are same also for the black ring
at the large $D$ limit.

##### Endpoints of instability?

We found non-axisymmetric instabilities of the black ring solution. One may
ask what its endpoint is. In [25] they discussed that the instability of the
five dimensional thin black ring leads to the fragmentation into black holes
by following two reasons. One is that the dynamical timescale to release the
inhomogeneity by gravitational wave radiations is much longer than the
timescale of the black ring instability. Thus the inhomogeneity by the
instability of the thin black ring will grow in time. Another reason is that
the fragmenting solution can be more entropic than the black ring. This can be
understood by the fact that the non-uniform black string is less entropic than
localized black holes in five dimensions. Then we expect that going to the
fragmenting solution is preferable as the endpoint of the instability than
going back to the (fatter) stable black ring. So the growth of the
inhomogeneity of the black ring would not stop, and the fragmentation occurs.
In our case this second reason of the discussion cannot hold. The non-uniform
black string can be the endpoint of the Gregory-Laflamme instability as seen
in [15] since the non-uniform black string becomes more entropic solution in
higher dimensions than the critical dimension [17]. Then, also for the black
ring instability, we can say same thing. In enough higher dimensions the non-
uniform solution is more entropic than the fragmenting solutions. So we expect
that the black ring would not go to the fragmentation by the instability. On
the other hand the first reason can be applied to our case. Actually the
inhomogeneity by the black ring instability and its rotation give time-
dependent quadrupole moments to gravitational fields. Then the timescale of
gravitational wave radiations by time-dependent quadrupole moments,
$t_{\text{GW}}$, is estimated as done in [25] by

$\displaystyle
t_{\text{GW}}\sim\frac{1}{\Omega_{\text{H}}}\frac{R^{n+1}}{\mathsf{G}_{D}\mathcal{M}}\sim\frac{R^{n+2}}{\mathsf{G}_{D}\mathcal{M}}\sim
e^{n/2}R\left(\frac{R}{r_{0}}\right)^{n},$ (3.23)

where the exponential factor $e^{n/2}$ comes from $\Omega_{n}$ in
$\mathsf{G}_{D}\mathcal{M}$ as seen in eq. (2.32). We restored the black ring
thickness $r_{0}$ for usefulness. The timescale of the black ring instability,
$t_{\text{BR}}$, is estimated by the quasinormal mode frequency as

$\displaystyle
t_{\text{BR}}=\left(\text{Im}\left[\omega^{(\ell=0)}_{+}\right]\right)^{-1}\sim
r_{0}.$ (3.24)

So $t_{\text{BR}}$ is $O(1)$ quantity in $1/n$ expansion. Then, remembering
that our large $D$ black ring solution satisfies $R>r_{0}$, the ratio of
timescales is exponentially small in $n$ at large $D$

$\displaystyle\frac{t_{\text{BR}}}{t_{\text{GW}}}\sim
e^{-n/2}\left(\frac{R}{r_{0}}\right)^{-n-1}\ll O(1).$ (3.25)

This implies that inhomogeneity by the black ring instability cannot be
dissipated by the gravitational wave radiation to the infinity in $1/n$
expansion. But the inhomogeneous black ring solution would not fragment into
small black holes unlike in five dimensions as discussed above. Thus the
inhomogeneity would stop at some point like the non-uniform black string
observed in [15]. Then we reach the conclusion for the endpoint of the black
ring instability that the black ring evolves to a non-uniform black ring,
NUBR, as a metastable solution at large $D$ 121212 The author thanks Roberto
Emparan for the suggestion of this possibility. . The metastable solution at
large $D$ means that the solution is stationary in $1/n$ expansion, but it is
not stationary in $O(e^{-n/2})$ by the gravitational wave emissions. The
stationary solution with inhomogeneities along rotating direction is
prohibited by the rigidity theorem [1, 26]. Our statement for the endpoint of
the black ring instability is consistent with the rigidity theorem since NUBR
is not stationary in $O(e^{-n/2})$. Such exponentially suppressed evolutions
cannot be described by our effective equations (3.5), (3.6) and (3.7) by $1/n$
expansions. It is interesting to investigate this possibility by solving the
effective equations for the black ring (3.5), (3.6) and (3.7) directly as the
investigation of the endpoint of the Gregory-Laflamme instability in [15]. If
there is a metastable solution such as NUBR, we can find a stationary and non-
axisymmetric solution of the effective equations.

This statement for the endpoint of the black ring instability is only for the
leading order result in $1/n$ expansion. If we include $1/n$ corrections to
the effective equations, the results show the dimensional dependences, and we
can observe the critical dimension in which the stability of NUBR would
change. Actually we have investigated the critical dimension of the non-
uniform black string by considering $1/n$ corrections [29]. So our conjecture
that the endpoint of the black ring instability is a stable NUBR is valid only
at the leading order in $1/n$ expansion.

#### Axisymmetric modes $(m=0)$

For $m=0$ modes, we can also solve eq. (3.20) in a simple form as

$\displaystyle\omega^{(m=0)}_{\pm}=\pm{\sqrt{\ell-1}}-i(\ell-1).$ (3.26)

Note that the numerator of eq. (3.20) is a cubic algebraic equation of
$\omega$, but one solution of the numerator equation is canceled with the
denominator of eq. (3.20) for $m=0$ modes. So there are only two modes as eq.
(3.26). The quasinormal mode frequencies (3.26) are same with one of the
Schwarzschild black hole at large $D$ [19]. This is because we consider the
decoupled mode excitations. In the decoupled mode excitation, the dynamics of
perturbations is determined almost locally on the horizon. For $m=0$ modes,
the perturbation does not have interactions along $\phi$ direction. Thus the
perturbation feels the horizon as one of the $D-1$ dimensional Schwarzschild
black hole, and its quasinormal mode becomes one of the Schwarzschild black
hole.

The axisymmetric perturbation does not have any instabilities at the leading
order of $1/n$ expansion. Thus the instability found in [25, 27] for the fat
black ring against axisymmetric perturbations is not contained in our setup.
Such instability occurs due to interactions between horizons of the black
ring, so it may be non-decoupled mode instability. However, by including $1/n$
corrections, we find a suggestion for the axisymmetric instabilities of the
black ring as we will see later.

###  3.4 Black string limit

As one useful observation let us see the large ring radius limit $R\gg 1$ of
the black ring solution found above. This limit gives the expression of the
black ring as a boosted black string [6]. At the large radius limit we can set
the behavior of $P(y)$ in eq. (3.13) to

$\displaystyle P(y)=O(1/R),$ (3.27)

by $p_{0}=0$ and $d_{0}=0$. Then we get the large radius limit of the black
ring solution as

$\displaystyle
ds^{2}=-\left(1-\frac{1}{\mathsf{R}}\right)dv^{2}+2dvdr-\frac{1}{\sqrt{n}\mathsf{R}}dvdx+\left(1-\frac{1}{n\mathsf{R}}\right)dx^{2}+r^{2}d\Omega^{2}_{n},$
(3.28)

where we defined the black string direction $dx$ by
$dx=Rd\phi/\sqrt{n}=Rd\Phi$. This large radius limit solution is actually
regarded as the large $D$ limit metric of the boosted string with the boost
velocity

$\displaystyle\sinh{\alpha}=\frac{1}{\sqrt{n}},$ (3.29)

as found in [6]. Using this boost relation we can reproduce the large radius
limit of the quasinormal modes, $\omega^{(\ell=0)}_{\pm}$, of the black ring
from the quasinormal modes of the black string. The quasinormal modes of the
black string, $\omega^{\text{BS}}_{\pm}$ was obtained in [11] by large $D$
expansion. For the perturbation $\sim e^{-i\omega v}e^{ikx}$, the leading
order result is

$\displaystyle\omega^{\text{BS}}_{\pm}=\pm i\hat{k}(1\mp\hat{k}),$ (3.30)

where $\hat{k}=k/\sqrt{n}$. Now the boost transformation on the black string
is acting by

$\displaystyle
dv\rightarrow\cosh{\alpha}dv-\sinh{\alpha}dx,~{}~{}dx\rightarrow\cosh{\alpha}dx-\sinh{\alpha}dv.$
(3.31)

This transformation on $\omega^{\text{BS}}_{\pm}$ gives quasinormal modes of
the boosted black string, $\omega^{\text{bBS}}_{\pm}$, as

$\displaystyle\omega^{\text{BS}}_{\pm}\rightarrow\omega^{\text{bBS}}_{\pm}=k\sinh{\alpha}+\omega^{\text{BS}}_{\pm}\cosh{\alpha}.$
(3.32)

Using eq. (3.29) and $\hat{k}=k/\sqrt{n}$, we obtain the quasinormal modes of
the boosted black string with the boost velocity (3.29) as

$\displaystyle\omega^{\text{bBS}}_{\pm}=\hat{k}\pm i\hat{k}(1\mp\hat{k}).$
(3.33)

This quasinormal modes precisely reproduce the large radius limit of
quasinormal modes of the black ring by identifying $\hat{k}=\hat{m}$ 131313
The boost transformation is one by the exact symmetry of the black string.
Thus we can obtain the quasinormal mode of the boosted black string from the
black string. As for the singly rotting Myers-Perry black hole, the boost
transformation comes from an approximation symmetry appearing only at the
leading order of large $D$ limit. Thus quasinormal modes of the Myers-Perry
black hole do not follow boost transformation rules. . In Appendix A we give
the direct derivation of the quasinormal modes of the boosted black string
from the effective equations.

###  3.5 $1/D$ corrections

We can obtain $1/n$ corrections to results obtained above by solving the
effective equations up to $O(1/n)$. In the following we set to $p_{0}=0$ and
$d_{0}=0$ for $P(y)$ in eq. (3.13). $p_{v}(z)$ and $p_{\phi}(z)$ for the
stationary solution are

$\displaystyle p_{v}(y)=e^{P(y)}\Biggl{[}1+\frac{P^{(1)}(y)}{n}\Biggr{]},$
(3.34) $\displaystyle p_{\phi}(y)=\frac{R(R^{2}-1)^{3/2}}{(R+y)^{2}}e^{P(y)}$
$\displaystyle~{}~{}~{}\times\Biggl{[}1+\frac{1}{n}\Bigl{(}P^{(1)}(y)-\frac{2(1+Ry)}{R^{2}-1}P(y)+\frac{\log{(R-R^{-1})}}{R^{2}}\Bigr{)}\Biggr{]},$
(3.35)

where $P^{(1)}(y)$ is given by

$\displaystyle
P^{(1)}(y)=p_{1}+\frac{d_{1}}{R+y}+\frac{(R^{2}-1)^{2}}{R^{4}(R+y)^{2}}\log{(R-R^{-1})}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{2(1+Ry)}{R^{3}(R+y)}\text{Li}_{2}\left(\frac{1+Ry}{R^{2}-1}\right)+\frac{(R^{2}-1)(y-R(1-2y^{2}))}{2R^{3}(R+y)^{2}}(\log{(R+y)})^{2}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{1}{4R^{4}(R^{2}-1)(R+y)^{4}}\Bigl{[}3+12Ry+2y^{2}+2R^{8}(4-3y^{2})+2R^{7}y(5-4y^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-R^{2}(5-6y^{2})-12R^{5}y(1+y^{2})-R^{6}(12-8y^{2}+5y^{4})-2R^{3}y(10+y^{4})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+R^{4}(4-30y^{2}-5y^{4})\Bigr{]}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{\log{(R+y)}}{R^{3}(R^{2}-1)(R+y)^{3}}\Bigl{[}-2+R^{7}y-y^{2}+3R^{2}(1-2y^{2})+R^{6}(2+y^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+3R^{3}y(2+y^{2})-2Ry(3+y^{2})+3R^{5}y(1+y^{2})-R^{4}(2-12y^{2}+y^{4})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2(1+Ry)(R^{2}-1)(R+y)^{2}\log{(R-R^{-1})}\Bigr{]}.$
(3.36)

$p_{1}$ and $d_{1}$ are integration constants of $1/n$ corrections.
$\text{Li}_{2}(x)$ is the polylogarithm function. This solution has the
surface gravity

$\displaystyle\kappa=n\frac{\sqrt{R^{2}-1}}{2R}\Biggl{[}1-\frac{1}{2n}+O(n^{-2})\Biggr{]},$
(3.37)

and the horizon angular velocity

$\displaystyle\Omega_{H}=\frac{1}{\sqrt{n}}\frac{\sqrt{R^{2}-1}}{R^{2}}\Biggl{[}1-\frac{1}{n}\frac{R^{2}-2\log{(R-R^{-1})}}{2R^{2}}+O(n^{-2})\Biggr{]}.$
(3.38)

The horizon angular velocity up to $O(1/n)$ can be obtained by imposing the
regularity of $p_{v}(z)$ at the pole $y=-1/R$ up to $O(1/n)$. We can see that
this horizon angular velocity reproduces eq. (2.2) by the blackfold method at
the large radius limit up to $O(1/n)$. The surface gravity has the following
large radius limit

$\displaystyle\lim_{R\rightarrow\infty}\kappa$ $\displaystyle=$
$\displaystyle\frac{n}{2}\Bigl{[}1-\frac{1}{2n}+O(n^{-2})\Bigr{]}$ (3.39)
$\displaystyle=$
$\displaystyle\frac{n}{2\cosh{\alpha}}\left(1+O(n^{-2})\right),$

Then this expression implies that the surface gravity is reproduced by the
boosted black string with the boost velocity (3.29) up to $1/n$ corrections.
We can also compare the $1/n$ corrections of our results with the $1/R^{2}$
corrections by the blackfold method [7] to the horizon angular velocity.
However, in this comparison, there are some subtle things such as a definition
of $R$, so the comparison is not clear. The fact that $1/n$ corrections do not
contain $1/R$ contribution to $\Omega_{H}$ is consistent with the result in
[6].

Finally we give $1/n$ corrections to quasinormal modes of the black ring.
$\omega^{(\ell=0)}_{\pm}$ up to $1/n$ corrections is

$\displaystyle\omega_{\pm}^{(\ell=0)}=\frac{\sqrt{R^{2}-1}}{R}\Biggl{[}\hat{m}\pm
i\hat{m}(1\mp\hat{m})+\frac{\delta\hat{\omega}^{(\ell=0)}_{\pm}}{n}\Biggr{]},$
(3.40)

where

$\displaystyle\delta\hat{\omega}^{(\ell=0)}_{\pm}=\frac{1}{2\hat{m}R^{2}}\Bigl{[}\hat{m}^{2}(R^{2}-4+4R^{2}\hat{m}^{2})+i\hat{m}(2+(R^{2}+16)\hat{m}^{2}+8\hat{m}^{4})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\hat{m}^{2}(1-2i\hat{m})\log{(R-R^{-1})}\mp\Bigl{(}2\hat{m}(2+(3R^{2}-4)\hat{m}^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i(2+(3R^{2}+1)\hat{m}^{2}-2(R^{2}-10)\hat{m}^{4}-2\hat{m}^{2}\log{(R-R^{-1})})\Bigr{)}\Bigr{]}.$
(3.41)

At the large radius limit $R\rightarrow\infty$ with fixed $\hat{m}$, this
quasinormal mode reduces to one of the boosted black string with the boost
parameter given in eq. (3.29) as seen in Appendix A. Thus we could confirm
that the blackfold analysis by [6] is correct up to $1/n$ corrections.

From this $1/n$ correction of $\omega^{(\ell=0)}_{+}$, we find that the black
ring becomes unstable against non-axisymmetric perturbations for $R>R_{D}$
where

$\displaystyle
R_{D}=m\Biggl{[}1-\frac{1}{n}\frac{3-2\log{(m-m^{-1})}}{2m^{2}}+O(n^{-2})\Biggr{]}.$
(3.42)

Figure 2:  Plots of the quasinormal mode $\omega^{(\ell=0)}_{+}$ with $\ell=0$
and $m=2$ in $n=10$ of the black ring up to $O(1/n)$ corrections. The left
panel shows the real part of the frequency normalized by the reduced surface
gravity. The right panel shows the imaginary part. The dashed line represents
the leading order result. The thick line is the result with $O(1/n)$
corrections.

At the large radius limit this threshold ring radius $R_{D}$ gives the
marginally stable ”wave number $\hat{m}_{D}$” as

$\displaystyle\hat{m}_{D}\equiv\frac{m}{R_{D}}=1+O(n^{-2},R^{-2}).$ (3.43)

This corresponds to the Gregory-Laflamme mode of the boosted black string with
the boost velocity (3.29) as seen in Appendix A. In Figure 2 we show the plots
of the quasinormal mode frequency $\omega^{(\ell=0)}_{+}$ with $m=2$ in $n=10$
up to the $1/n$ corrections. The plots in Figure 2 show that the results
become much closer to numerical results [18] by $1/n$ corrections. At $R=1$
our quasinormal mode formula breaks down due to the term $\log{(R-R^{-1})}$.
Thus the quasinormal mode frequency behavior around $R=1$ is not reliable.

One interesting observation on $1/n$ corrections to quasinormal modes is the
relation of the superradiant and dynamically unstable regime. At the leading
order the real part of the quasinormal mode frequency $\omega^{(\ell=0)}_{+}$
is marginal in the superradiant condition

$\displaystyle\text{Re}\left[\omega_{+}^{(\ell=0)}-m\hat{\Omega}_{\text{H}}\right]=O(n^{-1}).$
(3.44)

By including the $1/n$ corrections, we see that the real part of the
quasinormal mode frequency $\omega_{+}^{(\ell=0)}$ deviates from the
superradiant condition. Actually the real part of the frequency does not
always satisfy the superradiant condition

$\displaystyle\text{Re}\left[\omega^{(\ell=0)}_{+}-m\hat{\Omega}_{\text{H}}\right]=\frac{2\hat{\kappa}}{n}\frac{(\hat{m}-1)(2-(R^{2}-4)\hat{m}+2R^{2}\hat{m}^{2})}{R^{2}}+O(n^{-2}).$
(3.45)

Figure 3:  The plot of the superradiant factor is shown for the dynamically
unstable mode $\omega^{(\ell=0)}_{+}$ with $m=2$ and $n=10$. The superradiant
factor can be negative and positive in dynamically unstable regime $R>m$,
while it is always positive in stable regime.

In Figure 3 we show the plot of eq. (3.45) for $n=10$ and $m=2$. The
superradiant regime can be seen in the dynamically unstable regime $R>m$. But
there is also the dynamically unstable mode which does not satisfy the
superradiant condition at $R>R_{S}>m$,

$\displaystyle R_{S}=\frac{m^{2}+1+\sqrt{1+6m^{2}+m^{4}}}{m}.$ (3.46)

Thus the black ring has dynamically unstable modes which satisfy the
superradiant condition for $R_{S}>R>m$, and the superradiant condition becomes
not to be satisfied by unstable modes at $R>R_{S}$. This behavior of the
dynamically unstable modes is the striking property of the black ring also
seen in numerical results [18]. In fact the all known dynamically unstable
modes of black holes such as one of the Myers-Perry black holes always satisfy
the superradiant condition. The physics behind this feature is not still
clear.

We can also obtain $1/n$ corrections to the quasinormal modes with $m=0$ in a
simple form. The result is

$\displaystyle\omega^{(m=0)}_{\pm}=\frac{\sqrt{R^{2}-1}}{R}\Biggl{[}\pm\sqrt{\ell-1}-i(\ell-1)+\frac{\delta\hat{\omega}^{(m=0)}_{\pm}}{n}\Biggr{]},$
(3.47)

where

$\displaystyle\delta\hat{\omega}^{(m=0)}_{\pm}=\pm\frac{R^{2}\sqrt{\ell-1}}{2(R^{2}-1)}\Biggl{[}3\ell-5-\frac{7\ell^{2}-\ell-10}{2R^{2}(\ell-1)}+\frac{2(4\ell-5)}{R^{4}(\ell-1)}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-i\frac{R^{2}(\ell-1)}{2(R^{2}-1)}\Biggl{[}2\ell-5-\frac{4\ell^{2}-7\ell-10}{2R^{2}(\ell-1)}+\frac{2(2\ell-5)}{R^{4}(\ell-1)}\Biggr{]}.$
(3.48)

At the large radius limit $R\rightarrow\infty$ the quasinormal mode frequency
$\omega^{(m=0)}_{\pm}$ deviates from one of Schwarzschild black hole [19] in
$1/n$ corrections. This deviation can be understood as a boost effect of the
black string as discussed in Appendix A. Around $R=1$ the imaginary part of
the $1/n$ correction $\delta\hat{\omega}^{(m=0)}_{\pm}$ becomes positive. This
might mean that the axisymmetric perturbation becomes also unstable around
$R=1$. Actually we find that there is a marginally stable mode at $R=R_{f}$ as

$\displaystyle\omega^{(m=0)}_{\pm}\Bigl{|}_{R=R_{f}}=-\ell\sqrt{\ell-1}+O(n^{-1}),$
(3.49)

where $R_{f}$ is the marginal radius for the apparent axisymmetric instability
of the black ring given by

$\displaystyle R_{f}=1-\frac{1}{n}\frac{\ell}{8(\ell-1)}+O(n^{-2}).$ (3.50)

The imaginary part becomes positive at $1>R>R_{f}$, and the axisymmetric
perturbation may be unstable there. As mentioned above our formula of the
quasinormal mode breaks down around $R=1$. Thus we cannot say immediately that
the axisymmetric instability mode of the black ring is found from our
quasinormal mode formula. However $R_{f}<1$ might suggest that the unstable
region for the axisymmetric perturbations exists around $R=1$. This may be
related with the instability of the fat black ring [25, 27]

###  3.6 Phase diagram

Let us draw the phase diagram of the black ring solution obtained by the large
$D$ expansion method above. To do it we collect the formula for thermodynamic
quantities of the black ring. The mass $\mathcal{M}$ and the angular momentum
$\mathcal{J}_{\Phi}$ formula are given in eqs. (2.32) and (2.33). For the
black ring, using the ring coordinate embedding by eq. (3.3) and the leading
order stationary solution (3.14), these formula become

$\displaystyle\mathcal{M}=\frac{n\Omega_{n}}{8\mathsf{G}_{D}}\hat{\mathcal{M}},~{}~{}\mathcal{J}_{\Phi}=\sqrt{n}\frac{\Omega_{n}}{8\mathsf{G}_{D}}\hat{\mathcal{J}}_{\Phi},$
(3.51)

where

$\displaystyle\hat{\mathcal{M}}=\int
dy\frac{(R^{2}-1)e^{P(y)}}{R\sqrt{1-y^{2}}}\left(\frac{R\sqrt{1-y^{2}}}{R+y}\right)^{n},$
(3.52)

and

$\displaystyle\hat{\mathcal{J}}_{\Phi}=\int
dy\frac{(R^{2}-1)^{5/2}e^{P(y)}}{R(R+y)^{2}\sqrt{1-y^{2}}}\left(\frac{R\sqrt{1-y^{2}}}{R+y}\right)^{n}.$
(3.53)

Note that $y=\cos{\theta}$ in the ring coordinate (3.2). The formula (2.34)
gives the area of the black ring as

$\displaystyle\mathcal{A}_{\text{H}}=2\pi\Omega_{n}\hat{\mathcal{A}}_{\text{H}},$
(3.54)

where

$\displaystyle\hat{\mathcal{A}}_{\text{H}}=\int
dy\frac{\sqrt{R^{2}-1}e^{P(y)}}{\sqrt{1-y^{2}}}\left(\frac{R\sqrt{1-y^{2}}}{R+y}\right)^{n}.$
(3.55)

$P(y)$ is given in eq. (3.13). We set to $p_{0}=d_{0}=0$ in eq. (3.13) 141414
These values of $p_{0}$ and $d_{0}$ are chosen so that the solution becomes
$P(y)=O(1/R)$ at the large radius limit $R\gg 1$. These choices do not affect
the essential feature of the phase diagram. . In these formula the black ring
thickness $r_{0}$ is set to unity. Here we consider only the leading order
contributions to the mass, angular momentum and area. This is because it seems
to be difficult to track the effect of $1/n$ corrections precisely in
$\mathcal{M}$, $\mathcal{J}_{\Phi}$ and $\mathcal{A}_{\text{H}}$ due to a term
in integrands with a power of $n$. At large $n$ the integrations of such terms
can be very large, and it is not clear how to control the size of such
integrations in $1/n$ expansion. In this paper, we take into account only the
leading order contributions to observe general feature, not detail numerical
values, of the phase diagram of the large $D$ black ring. Hence the phase
diagrams for the large $D$ black ring shown below have $O(1/n)$ errors. On the
other hand the temperature and horizon angular velocity do not have such
troublesome terms in their definitions, so we include $1/n$ corrections. The
temperature $T_{\text{H}}$ of the black ring is

$\displaystyle
T_{\text{H}}\equiv\frac{\kappa}{2\pi}=\frac{n}{4\pi}\frac{\sqrt{R^{2}-1}}{R}\Biggl{[}1-\frac{1}{2n}\Biggr{]}.$
(3.56)

The horizon angular velocity of the black ring is

$\displaystyle\Omega_{H}=\frac{\hat{\Omega}_{\text{H}}}{\sqrt{n}}=\frac{1}{\sqrt{n}}\frac{\sqrt{R^{2}-1}}{R^{2}}\Biggl{[}1-\frac{1}{n}\frac{R^{2}-2\log{(R-R^{-1})}}{2R^{2}}\Biggr{]}.$
(3.57)

To draw the phase diagram we define reduced quantities by the mass as
following

$\displaystyle
j_{\Phi}^{n+1}=c_{j}\frac{\mathcal{J}_{\Phi}^{n+1}}{\mathsf{G}_{D}\mathcal{M}^{n+2}},~{}~{}a_{\text{H}}^{n+1}=c_{a}\frac{\mathcal{A}^{n+1}_{\text{H}}}{(\mathsf{G}_{D}\mathcal{M})^{n+2}},$
(3.58)

and

$\displaystyle
t_{\text{H}}=c_{T}T_{\text{H}}(\mathsf{G}_{D}\mathcal{M})^{1/(n+1)},~{}~{}\omega_{\text{H}}=c_{\omega}\Omega_{\text{H}}(\mathsf{G}_{D}\mathcal{M})^{1/(n+1)}.$
(3.59)

The normalization numerical coefficients are taken from [6] as

$\displaystyle
c_{j}=\frac{\Omega_{n+1}}{2^{n+5}}\frac{(n+2)^{n+2}}{(n+1)^{(n+1)/2}},~{}~{}c_{a}=\frac{\Omega_{n+1}}{2(16\pi)^{n+1}}\frac{n^{(n+1)/2}(n+2)^{n+2}}{(n+1)^{(n+1)/2}},$
(3.60)

and

$\displaystyle
c_{t}=\frac{4\pi\sqrt{n+1}}{\sqrt{n}}\left(\frac{(n+2)\Omega_{n+1}}{2}\right)^{-1/(n+1)},c_{\omega}=\sqrt{n+1}\left(\frac{(n+2)\Omega_{n+1}}{16}\right)^{-1/(n+1)}.$
(3.61)

Then we obtain

$\displaystyle
j_{\Phi}^{n+1}=\frac{1}{2^{n+5}}\frac{\Omega_{n+1}}{\Omega_{n}}\left(1+\frac{2}{n}\right)^{n+2}\left(1+\frac{1}{n}\right)^{-(n+1)/2}\frac{\hat{\mathcal{J}}_{\Phi}^{n+1}}{\hat{\mathcal{M}}^{n+2}},$
(3.62) $\displaystyle
a_{\text{H}}^{n+1}=\frac{1}{2^{n}}\frac{\Omega_{n+1}}{\Omega_{n}}\left(1+\frac{2}{n}\right)^{n+2}\left(1+\frac{1}{n}\right)^{-(n+1)/2}\frac{\hat{\mathcal{A}}_{\text{H}}^{n+1}}{\hat{\mathcal{M}}^{n+2}},$
(3.63) $\displaystyle
t_{\text{H}}=2^{n/(n+1)}\left(1+\frac{1}{n}\right)^{1/2}\left(\frac{\Omega_{n}}{\Omega_{n+1}}\right)^{1/(n+1)}\left(1+\frac{2}{n}\right)^{-1/(n+1)}\kappa\hat{\mathcal{M}}^{1/(n+1)},$
(3.64)

and

$\displaystyle\omega_{\text{H}}=2^{2/(n+1)}\left(1+\frac{1}{n}\right)^{1/2}\left(\frac{\Omega_{n}}{\Omega_{n+1}}\right)^{1/(n+1)}\left(1+\frac{2}{n}\right)^{-1/(n+1)}\hat{\Omega}_{\text{H}}\hat{\mathcal{M}}^{1/(n+1)}.$
(3.65)

We evaluate $j_{\Phi}$, $a_{\text{H}}$, $t_{\text{H}}$ and $\omega_{\text{H}}$
numerically for the large $D$ black ring solution by using eqs. (3.52) and
(3.53) with $P(y)$ given in eq. (3.13). As mentioned above, we set to
$p_{0}=d_{0}=0$. Figure 4 shows the phase diagram of $(j_{\Phi},a_{\text{H}})$
for the black ring solution by the blackfold and large $D$ expansion method
from $R/r_{0}=1.1$ to $R/{r_{0}}=20$ in $n=10$. $R$ and $r_{0}$ are a ring
radius and ring thickness of the black ring.

Figure 4:  The $(j_{\Phi},a_{\text{H}})$ phase diagram of the black ring in
$n=10$ is shown. The curve by black dots is the numerical evaluation of the
black ring solution obtained by the large $D$ expansion method. The gray
dashed line is the leading order result by the blackfold method. The thick
gray line is the result with $O(1/R^{2})$ corrections by [7]. The left panel
shows the phase diagram in not-thin region, and right panel is for the thin
region $R\gg r_{0}$. At the large radius $R\gg r_{0}$ these results show
similar behavior as expected. The difference of values are within $O(1/n)$
errors in thin region.

The curve by black dots is the result by numerical evaluations of eqs. (3.62)
and (3.63) for our large $D$ black ring solution. The gray dashed line is the
leading order result by the blackfold method [6]. The thick gray line is
$O(1/R^{2})$ correction of the blackfold [7]. As expected we can see that
these results show similar behavior at the large radius region $R\gg 1$. Two
results by the blackfold and large $D$ expansion method seems to have a
difference by the constant offset at $R\gg 1$. This difference can be
understood as $O(1/n)$ errors in the large $D$ expansion method. By taking the
large radius limit $R\gg 1$, from $P(y)=O(1/R)$, we find that

$\displaystyle\mathcal{M}=\frac{n\Omega_{n+1}}{8\mathsf{G}_{D}}Rr_{0}^{n}\left(1+O(n^{-1},R^{-1})\right),~{}\mathcal{J}_{\Phi}=\frac{\sqrt{n}\Omega_{n+1}}{8\mathsf{G}_{D}}R^{2}r_{0}^{n}\left(1+O(n^{-1},R^{-1})\right),$
(3.66)

and

$\displaystyle\mathcal{A}_{\text{H}}=2\pi\Omega_{n+1}Rr_{0}^{n+1}\left(1+O(n^{-1},R^{-1})\right),$
(3.67)

where we restored the black ring thickness $r_{0}$. On the other hand the
leading order results by the blackfold [6] gives

$\displaystyle\mathcal{M}=\frac{(n+2)\Omega_{n+1}}{8\mathsf{G}_{D}}Rr_{0}^{n}\left(1+O(R^{-1})\right),~{}\mathcal{J}_{\Phi}=\frac{\sqrt{n+1}\Omega_{n+1}}{8\mathsf{G}_{D}}R^{2}r_{0}^{n}\left(1+O(R^{-1})\right),$
(3.68)

and

$\displaystyle\mathcal{A}_{\text{H}}=2\pi\sqrt{\frac{n+1}{n}}\Omega_{n+1}Rr_{0}^{n+1}\left(1+O(R^{-1})\right).$
(3.69)

These results coincide at the leading order in $1/n$ expansion. $1/n$
corrections in eqs. (3.68) and (3.69) gives smaller $a_{\text{H}}$ and
$j_{\Phi}$ than the leading order results in $1/n$ expansion. So $1/n$
correction would reduce the difference seen in Figure 4. This $O(1/n)$ error
would give the difference of the onset at $R\gg 1$ seen in Figure 4. It might
be difficult to observe $O(1/R^{2})$ corrections in the results by the large
$D$ expansion method. This is because the definitions of thermodynamic
quantities contain the $y$-integrations of $R^{n}/(R+y)^{n}$. The blackfold by
the $1/R$ expansion and the large $D$ expansion method by $1/n$ expansion do
not give same results for such integrations in higher order corrections of
$O(n^{-1},R^{-2})$.

Figure 5:  The $(j_{\Phi},t_{\text{H}})$ (left) and
$(j_{\Phi},\omega_{\text{H}})$ (right) phase diagram in $n=10$ are shown. The
gray dashed and thick line corresponds to the leading order and $O(1/R^{2})$
corrected results of the blackfold method. The curve by black dots is the
numerical evaluations for the large $D$ black ring solution.

Figure 5 are plots of the phase diagram of $(j_{\Phi},t_{\text{H}})$ and
$(j_{\Phi},\omega_{\text{H}})$ for the black ring by the blackfold (gray
dashed and thick lines) and the large $D$ expansion method (curve by black
dots). In these diagrams we see that results by the blackfold and the large
$D$ expansion method show similar values and behaviors at the large radius
region $R\gg 1$. These results show different behavior in not-thin region
$j_{\Phi}\sim 1$. This might be because our results contain $O(1/n)$ errors
which become larger at fat ring region, and the blackfold method also has not-
small errors there 151515 In not so much higher dimension, it was observed
that the blackfold method has a good accuracy even in not thin region
$j_{\Phi}\sim 1$ if we include higher order corrections [7]. .

##  4 Summary

We constructed the effective theory for the slowly rotating large $D$ black
holes. The slow rotation is defined by $O(1/\sqrt{D})$ horizon angular
velocity at large $D$. This solution class of large $D$ black holes contains
the slowly rotating Myers-Perry black hole, slowly boosted black string and
black ring solution. The black ring should be slowly rotating at large $D$
since the tension effect which determines the horizon angular velocity is
small. So our analysis of the black ring in this paper by the large $D$
effective theory is not restricted one. We solved the effective equations and
found the black ring solution analytically. Furthermore, by perturbation
analysis, the quasinormal mode condition of the black ring was obtained. The
quasinormal mode says that the thin black ring has the Gregory-Laflamme type
instability in the non-axisymmetric perturbations as found numerically. The
black ring solution obtained in this paper describes not only thin black ring,
but also not-thin black ring. For not-thin black ring, we found that the black
ring becomes stable against non-axisymmetric perturbations. We gave some
discussions on the endpoint of the non-axisymmetric instabilities of the black
ring. At large $D$ the gravitational wave emission becomes much less efficient
by $O(e^{-D/2})$ than the dynamical instability, and the instability would not
lead to the fragmentation of the horizon in enough higher dimensions. This
fact suggests the existence of a metastable solution, non-uniform black ring,
as the endpoint of the black ring instability at large $D$. We also studied
$1/D$ corrections to the quasinormal modes and phase diagram.

In this paper we considered the construction of the black ring solution by the
large $D$ expansion method as the first step to reveal the variety of higher
dimensional black holes. So we have some natural extensions of our current
work. One is to consider the $O(1)$ horizon angular velocity solution. In [13]
the effective theory for the stationary solution with $O(1)$ horizon angular
velocity was constructed, but they did not include the time dependent
deformations. It is possible to include the time dependent deformations in the
work [13] by the same way in this paper. The non-liner dynamical deformations
of the singly rotating Myers-Perry black hole can be described by such
framework. The singly rotating Myers-Perry black hole is known to have the
unstable modes, and it is interesting to study the instability mode and
stationary deformed solution associated with the zero mode, so called bumpy
black hole, as the endpoint of the instability by the large $D$ effective
theory. The second possible extension is to investigate the non-linear
evolution of the instability of the black ring. The effective equations
obtained in this paper describe such evolutions. It is interesting to solve
the effective equations numerically and study the endpoint of the black ring
instability. Our effective equations have much simpler form than original
Einstein equations, and they are expected to be much more tractable
numerically. In that analysis our conjecture for the existence of the non-
uniform black ring can be also studied. The third is to include further
angular momentum to the effective equations. The effective equations in this
paper describe the dynamical black hole with essentially only one angular
momentum. By adding further number of angular momenta to the effective
equations, we can describe much richer dynamics of the black holes, and it
will be found that the variety of the black hole horizon becomes much richer
than one of the singly rotating black hole. These work would give new
interesting insights to the higher dimensional black hole physics.

## Acknowledgments

The author was grateful to Roberto Emparan for useful discussions and comments
on the draft. This work was supported by JSPS Grant-in-Aid for Scientific
Research No.26-3387.

##  Appendix A Other solutions

In this appendix we study other embeddings of the solution. In the asymptotic
region $\mathsf{R}\gg 1$ the leading order metric on
$\mathsf{R}=\text{const.}$ surface has the following form

$\displaystyle
ds^{2}|_{\mathsf{R}=\text{const.}}=-dv^{2}+r^{2}\left(dz^{2}+G(z)^{2}d\Phi^{2}+H(z)^{2}d\Omega^{2}_{n}\right)+O(\mathsf{R}^{-1}).$
(A.1)

$H(z)$ should satisfy the equation given by

$\displaystyle 1-H^{\prime}(z)^{2}+H(z)H^{\prime\prime}(z)=0.$ (A.2)

In the following this metric is embedded into a flat background in spherical
coordinates. Such solutions describes the slowly rotating Myers-Perry black
hole and slowly boosted black string solution.

###  A.1 Slowly rotating Myers-Perry black hole

The first spherical coordinate is the spherical coordinate of $D=n+4$
dimensional flat spacetime. The metric in the spherical coordinate is

$\displaystyle
ds^{2}=-dt^{2}+dr^{2}+r^{2}(dz^{2}+\sin^{2}{z}~{}d\Phi^{2}+\cos^{2}{z}~{}d\Omega^{2}_{n}).$
(A.3)

The embedding of the leading order metric is given by $r=r_{0}$ in this flat
background metric. Then the embedding says that $H(z)$ and $G(z)$ are
identified by

$\displaystyle H(z)=\cos{z},~{}~{}G(z)=\sin{z}.$ (A.4)

Note that this $H(z)$ actually satisfies eq. (A.2). The surface gravity of
this spherical coordinate embedding is

$\displaystyle\kappa=n\hat{\kappa}=\frac{n}{2}.$ (A.5)

$r=\text{const.}$ surfaces in the metric (A.3) have the topology of $S^{n+2}$.
So the topology of the black hole horizon is $S^{n+2}$ in this embedding.

#### Effective equations

The effective equations (2.16), (2.17) and (2.18) in the spherical coordinate
embedding are

$\displaystyle\partial_{v}p_{v}+\partial_{z}p_{v}\tan{z}-\frac{\partial_{\phi}^{2}p_{v}}{\sin^{2}{z}}+\frac{\partial_{\phi}p_{\phi}}{\sin^{2}{z}}-p_{z}\tan{z}=0,$
(A.6)
$\displaystyle\partial_{v}p_{\phi}+\partial_{z}p_{\phi}\tan{z}-\frac{\partial_{\phi}^{2}p_{\phi}}{\sin^{2}{z}}+\partial_{\phi}p_{v}+\frac{1}{\sin^{2}{z}}\partial_{\phi}\Biggl{[}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}-2p_{\phi}-\frac{p_{\phi}p_{z}}{p_{v}}\tan{z}=0,$
(A.7)

and

$\displaystyle~{}~{}\partial_{v}p_{z}+\partial_{z}p_{z}\tan{z}-\frac{\partial_{\phi}^{2}p_{z}}{\sin^{2}{z}}+\partial_{z}p_{v}+\frac{1}{\sin^{2}{z}}\partial_{z}\Biggl{[}\frac{p_{z}p_{\phi}}{p_{v}}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{p_{\phi}^{2}\cot{z}+p_{z}^{2}\tan{z}\sin^{2}{z}}{p_{v}\sin^{2}{z}}+\frac{2\cot{z}}{\sin^{2}{z}}\partial_{\phi}p_{\phi}-\frac{\cos{2z}}{\cos^{2}{z}}p_{z}=0.$
(A.8)

#### Stationary solution

We solve eqs. (A.6), (A.7) and (A.8) for the stationary solution. The
stationary solution ansatz assumes that $\partial_{v}$ and $\partial_{\phi}$
are the Killing vectors. Thus we assume

$\displaystyle
p_{v}=e^{P(z)},~{}~{}p_{\phi}=p_{\phi}(z),~{}~{}p_{z}=p_{z}(z).$ (A.9)

$p_{a}(z)$ are given by $p_{v}(z)$ as derived in eq. (2.20) with the
integration constant $\hat{\Omega}_{\text{H}}$. The equation for $P(z)$ given
in eq. (2.24) in the spherical coordinate embedding is

$\displaystyle
P^{\prime\prime}(z)+P^{\prime}(z)\tan{z}-\hat{a}^{2}\cos^{2}{z}=0.$ (A.10)

Here we rename $\hat{\Omega}_{\text{H}}$ in eq. (2.24) by $\hat{a}$ just for
usefulness. The solution of this equation is

$\displaystyle P(z)=p_{0}+d_{0}\sin{z}-\frac{1}{2}\hat{a}^{2}\cos^{2}{z}.$
(A.11)

Integration constants, $p_{0}$ and $d_{0}$, describe trivial deformations of
the solution, so we set to $p_{0}=d_{0}=0$. Then, using eq. (2.20), we find
that the stationary solutions of eqs. (A.6), (A.7) and (A.8) are

$\displaystyle
p_{v}(z)=e^{-\hat{a}^{2}\cos^{2}{z}/2},~{}~{}p_{\phi}(z)=\hat{a}p_{v}(z)\sin^{2}{z},~{}~{}p_{z}(z)=p_{v}^{\prime}(z).$
(A.12)

This stationary solution describes the $D=n+4$ dimensional slowly rotating
Myers-Perry black hole [28] with $a=\hat{a}/\sqrt{n}$. This can be confirmed
directly by performing a coordinate transformation from the Myers-Perry black
hole. As we can see below, quasinormal modes of the stationary solution also
coincide to one of the Myers-Perry black hole obtained found in [13].

#### Quasinormal modes

By perturbing the equations (A.6), (A.7) and (A.8) around the stationary
solution, we obtain quasinormal mode frequencies. The perturbations are

$\displaystyle p_{v}(v,z,\phi)=e^{-a^{2}\cos^{2}{z}/2}\left(1+\epsilon
F_{v}(z)e^{-i\omega v}e^{im\phi}\right),$ $\displaystyle
p_{\phi}(v,z,\phi)=a\sin^{2}{z}e^{-a^{2}\cos^{2}{z}/2}\left(1+\epsilon
F_{\phi}(z)e^{-i\omega v}e^{im\phi}\right),$ $\displaystyle
p_{z}(v,z,\phi)=a^{2}\cos{z}\sin{z}e^{-a^{2}\cos^{2}{z}/2}\left(1+\epsilon
F_{z}(z)e^{-i\omega v}e^{im\phi}\right).$ (A.13)

We impose the boundary condition on the perturbations at $z=0$ as 161616 This
boundary condition was also used in [13].

$\displaystyle F_{v}(z)\propto z^{\ell}\left(1+O(z)\right).$ (A.14)

Then the frequency is discretized by quantum numbers, $\ell$ and $m$. In the
following we assume $m=O(n^{-1})$. This assumption can be understood by
observing the spherical harmonics. At large $D$ the spherical harmonics on
$S^{n+1}$, $\mathbb{Y}_{j\ell m_{\Phi}}(z)$, in the metric (A.3) is reduced to
[13]

$\displaystyle\mathbb{Y}_{j\ell m_{\Phi}}(z)\propto
e^{im_{\Phi}\Phi}\left(\sin{z}\right)^{|m_{\Phi}|+2k_{S}}\cos^{j}{z}$ (A.15)

where $j$ is the quantum number on $S^{n}$. $k_{S}$ is the quantum number
along $z$-direction. Comparing eqs. (A.14) and (A.15), we see that $\ell$ is
given by

$\displaystyle\ell=|m_{\Phi}|+2k_{S}.$ (A.16)

In this paper we do not consider the excitation on $S^{n}$, so our boundary
condition corresponds to one of $j=0$ in [13]. Note that, since we rescaled
$\phi$ coordinate by eq. (2.3), the quantum number $m_{\Phi}$ associated with
$\partial_{\Phi}$ is related with the quantum number $m$ of $\partial_{\phi}$
by $m_{\Phi}=\sqrt{n}m$. Thus the spherical harmonics has $\sqrt{n}m$
dependence, not $m$ dependence. If we assume $m=O(1)$, the spherical harmonics
is dominated by the quantum number $m$ as

$\displaystyle\mathbb{Y}_{j\ell m_{\Phi}}(z)\propto(\sin{z})^{\sqrt{n}m}.$
(A.17)

The analysis of this function becomes involved since the $z$-derivative can be
very large and diverging in $n$. To avoid this complexity we assume
$m=O(1/n)$. Then the perturbations can satisfy the boundary condition (A.14)
without diverging terms in $n$. Thus, in the perturbation (A.13), we introduce
new $O(1)$ quantity $\bar{m}$ by

$\displaystyle m=\frac{\bar{m}}{n}.$ (A.18)

Then the quasinormal mode frequencies are written by $\ell$, $\bar{m}$ and
$\hat{a}$. We can extend the all calculations up to $O(1/n)$ by including
$1/n$ corrections. The obtained quasinormal modes up to $O(1/n)$ is

$\displaystyle\omega_{\pm}=\omega_{\pm}^{(0)}+\frac{\omega_{\pm}^{(1)}}{n}+O(n^{-2}),$
(A.19)

where

$\displaystyle\omega_{\pm}^{(0)}=\pm\sqrt{\ell-1}-i(\ell-1)$ (A.20)

and

$\displaystyle\omega_{\pm}^{(1)}=\pm\Biggl{(}\frac{\sqrt{\ell-1}(3\ell-4)}{2}-\frac{\hat{a}^{2}(2\ell^{2}-9\ell+8)}{2\ell\sqrt{\ell-1}}+\frac{i\hat{a}\bar{m}\sqrt{\ell-1}}{\ell}\Biggr{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Biggl{(}i(\ell-1)(\ell-2)-\frac{i\hat{a}^{2}(\ell^{2}-5\ell+8)}{2\ell}-\frac{\hat{a}\bar{m}(\ell-1)}{\ell}\Biggr{)}.$
(A.21)

Actually this quasinormal mode reproduces the quasinormal mode of the
Schwarzschild black hole up to $1/n$ corrections for $\hat{a}=0$ [19] and one
of Myers-Perry black hole [13] with $a=\hat{a}/\sqrt{n}$ and
$m=\bar{m}/\sqrt{n}$.

###  A.2 Slowly boosted black string

We consider the embedding in another spherical coordinate, which is the
coordinate of the spacetime with one compact direction. The $D=n+4$
dimensional spacetime with one compact direction has the following metric in
the spherical coordinate

$\displaystyle
ds^{2}=-dt^{2}+d\Phi^{2}+dr^{2}+r^{2}\left(dz^{2}+\sin^{2}{z}~{}d\Omega^{2}_{n}\right).$
(A.22)

$\Phi$ is a coordinate of the compact direction. The embedding $r=r_{0}$ of
the leading order metric (A.1) into eq. (A.22) gives following identifications

$\displaystyle G(z)=1,~{}~{}H(z)=\sin{z},$ (A.23)

where we set to $r_{0}=1$. The embedded solution has $S^{1}\times S^{n+1}$
horizon topology in the spacetime with one compact dimension. So the solution
of the effective equations describe non-linear dynamical deformations of the
black string.

#### Effective equations

The effective equations in this embedding become

$\displaystyle\partial_{v}p_{v}-\cot{z}~{}\partial_{z}p_{v}-\partial_{\phi}^{2}p_{v}+\partial_{\phi}p_{\phi}+p_{z}\cot{z}=0,$
(A.24)
$\displaystyle\partial_{v}p_{\phi}-\cot{z}~{}\partial_{z}p_{\phi}-\partial_{\phi}^{2}p_{\phi}-\partial_{\phi}p_{v}+\partial_{\phi}\Biggl{[}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}-\cot{z}\frac{p_{z}p_{\phi}}{p_{v}}=0,$
(A.25)

and

$\displaystyle\partial_{v}p_{z}-\cot{z}~{}\partial_{z}p_{z}-\partial_{\phi}^{2}p_{z}+\partial_{z}p_{v}+\partial_{z}\Biggl{[}\frac{p_{z}p_{\phi}}{p_{v}}\Biggr{]}-\cot{z}\frac{p_{z}^{2}}{p_{v}}-\frac{\cos{2{z}}}{\sin^{2}{z}}p_{z}=0.$
(A.26)

#### Stationary solutions

We have two stationary solutions of the effective equations. One is the
boosted black string. The ansatz for the boosted black string is

$\displaystyle
p_{v}(z)=e^{P(z)},~{}~{}p_{\phi}=p_{\phi}(z),~{}~{}p_{z}=p_{z}(z).$ (A.27)

Then the effective equations give the solution

$\displaystyle
p_{v}(z)=e^{p_{0}+d_{0}\cos{z}},~{}~{}p_{\phi}=\hat{\sigma}p_{v}(z),~{}~{}p_{z}=p_{v}^{\prime}(z).$
(A.28)

$\hat{\sigma}$ is an integration constant, and it describes the boost
parameter of the boosted black string with the boost parameter

$\displaystyle\sinh{\alpha}=\frac{\hat{\sigma}}{\sqrt{n}},$ (A.29)

where the boost transformation of the black string and definition of $\alpha$
are given in eq. (3.28). The integration constants $p_{0}$ and $d_{0}$ are
horizon size and position of the black string origin respectively. Thus we can
set to $p_{0}=0$ and $d_{0}=0$.

Another stationary solution is the non-uniform black string. This solution is
inhomogeneous along $\Phi$ direction. So $\partial_{\Phi}$ is not the Killing
vector. The ansatz for the non-uniform black string is

$\displaystyle p_{v}=p_{v}(\phi),~{}~{}p_{\phi}=p_{\phi}(\phi),~{}~{}p_{z}=0.$
(A.30)

The condition $p_{z}=0$ reflects the fact that the Gregory-Laflamme mode of
the black string exists only in the S-wave sector. This ansatz gives the
equation for the non-uniform black string at large $D$ [29] as

$\displaystyle p_{\phi}=\partial_{\phi}p_{v}+P_{\phi}$ (A.31)

and

$\displaystyle\partial_{\phi}^{3}p_{v}+\partial_{\phi}p_{v}-\frac{2\partial_{\phi}p_{v}\partial_{\phi}^{2}p_{v}}{p_{v}}+\frac{(\partial_{\phi}p_{v})^{3}}{p_{v}^{2}}=0.$
(A.32)

$P_{\phi}$ is an integration constant describing the momentum along $\phi$
direction, and we set to $P_{\phi}=0$ in eq. (A.32). To solve this equation we
need the numerical treatment. In this paper we consider only the boosted black
string solution.

#### Quasinormal modes

Considering the perturbations around the stationary solution (A.28)

$\displaystyle p_{v}(v,z,\phi)=1+\epsilon F_{v}(z)e^{-i\omega
v}e^{i\hat{k}\phi},$ $\displaystyle
p_{\phi}(v,z,\phi)=\hat{\sigma}\left(1+\epsilon F_{\phi}(z)e^{-i\omega
v}e^{i\hat{k}\phi}\right),$ $\displaystyle p_{z}(v,z,\phi)=\epsilon
F_{z}(z)e^{-i\omega v}e^{i\hat{k}\phi},$ (A.33)

and boundary conditions at $\cos{z}=0$ given by

$\displaystyle F_{v}(z)\propto(\cos{z})^{\ell}\left(1+O(\cos{z})\right),$
(A.34)

we obtain the quasinormal mode condition as

$\displaystyle\frac{1}{\hat{k}+\hat{\sigma}^{2}\hat{k}+\hat{\sigma}(i\ell+i\hat{k}^{2}+\omega)}\Bigl{[}\omega^{3}+(-2i+3i\ell+3\hat{\sigma}\hat{k}+3i\hat{k}^{2})\omega^{2}$
$\displaystyle~{}~{}~{}+(-3\ell^{2}+\ell(3+6i\hat{\sigma}\hat{k}-6\hat{k}^{2})+\hat{k}(5\hat{k}+3\hat{\sigma}^{2}\hat{k}-3\hat{k}^{2}+2i\hat{\sigma}(3\hat{k}^{2}-2)))\omega$
$\displaystyle~{}~{}~{}-i(\ell^{3}+\ell^{2}(-1-3i\hat{\sigma}\hat{k}+3\hat{k}^{2})+k\ell(3i\hat{\sigma}-4\hat{k}-3\hat{\sigma}^{2}\hat{k}-6i\hat{\sigma}\hat{k}+3\hat{k}^{2})$
$\displaystyle~{}~{}~{}~{}+\hat{k}^{2}(2+i\hat{\sigma}^{3}\hat{k}-3\hat{k}^{2}+\hat{k}^{4}+\hat{\sigma}^{2}(2-3\hat{k}^{2})-i\hat{\sigma}\hat{k}(3\hat{k}^{2}-5))\Bigr{]}=0.$
(A.35)

This condition can be solved for $\ell=0$ mode in a simple form by

$\displaystyle\omega^{(\ell=0)}_{\pm}=\hat{\sigma}\hat{k}\pm
i\hat{k}(1\mp\hat{k}).$ (A.36)

This is the quasinormal modes of the S-wave sector of the scalar type
gravitational perturbation of the boosted black string.
$\omega^{(\ell=0)}_{+}$ shows the Gregory-Laflamme instability [16] for
$\hat{k}<1$. For $\hat{k}=0$ mode the quasinormal mode condition gives

$\displaystyle\omega^{(\hat{k}=0)}_{\pm}=\pm\sqrt{\ell-1}-i(\ell-1).$ (A.37)

Although, for general modes $\ell\neq 0$ and $\hat{k}\neq 0$, we cannot obtain
a simple solution of eq. (A.35), it can be seen that the instability mode
exists only in the S-wave ($\ell=0$) sector. This is consistent with the
result in [30] that the black string is unstable only for the S-wave sector.

We can also find $1/n$ corrections to the quasinormal modes. For the S-wave
sector, the quasinormal modes are

$\displaystyle\omega^{(\ell=0)}_{\pm}=\hat{\sigma}\hat{k}\pm
i\hat{k}(1\mp\hat{k})$
$\displaystyle~{}~{}~{}~{}+\frac{\hat{k}}{2n}\Bigl{[}i(\mp 1-2\hat{k}\pm
2\hat{k}^{2})-2\hat{\sigma}(1\mp 3\hat{k}+2\hat{k}^{2})+i\hat{\sigma}^{2}(\mp
2+3\hat{k})\Bigr{]}+O(n^{-2}).$ (A.38)

If we take $\hat{\sigma}=0$, the result reproduces the quasinormal modes of
the black string obtained in [11, 31]. The relation of the boosted black
string and black ring become clear by observing eq. (A.38). Actually, if we
take the large radius limit of the quasinormal modes $\omega^{(\ell=0)}_{\pm}$
of the black ring in eq. (3.41), we obtain

$\displaystyle\omega^{(\ell=0)}_{\pm}=\hat{m}\pm i\hat{m}(1\mp\hat{m})$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{\hat{m}}{2n}\Bigl{[}1\mp
6\hat{m}+4\hat{m}^{2}+i(\mp 3+\hat{m}\pm 2\hat{m}^{2})\Bigr{]}.$ (A.39)

Comparing eqs. (A.38) and (A.39) by identifying $\hat{k}=\hat{m}$, we can see
that the large radius limit of the black ring corresponds to the boosted black
string with the boost parameter

$\displaystyle\sinh{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\hat{\sigma}}{\sqrt{n}}$ (A.40) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n+1}},$

up to $O(1/n)$. This boost parameter corresponds to eq. (3.29), and this is
consistent with the result in [6] 171717 In [6] they did not use $1/n$
expansion, so the relation (3.29) obtained in [6] should be valid at all order
in $1/n$ at the large radius limit. . The Gregory-Laflamme mode
$\hat{k}_{\text{GL}}$ of the boosted black string, for which the imaginary
part of $\omega^{(\ell=0)}_{+}$ vanishes, is obtained as

$\displaystyle\hat{k}_{\text{GL}}=1-\frac{1-\hat{\sigma}^{2}}{2n}+O(n^{-2}).$
(A.41)

This Gregory-Laflamme mode of $\hat{\sigma}=1$ reproduces the threshold wave
number (3.43) for the black ring.

For $\hat{k}=0$ modes, the quasinormal modes up to $1/n$ corrections are

$\displaystyle\omega^{(\hat{k}=0)}_{\pm}=\pm\sqrt{\ell-1}-i(\ell-1)$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{1}{2n}\Bigl{[}\pm(3\ell-4-\hat{\sigma}^{2})\sqrt{\ell-1}-i(\ell-1)(2\ell-4-\hat{\sigma}^{2})\Bigr{]}+O(n^{-2}).$
(A.42)

Using eq. (A.40), the quasinormal mode (A.42) reproduces eq. (3.48) at the
large radius limit $R=\infty$.

##  Appendix B Trivial perturbations in ring coordinate

In this appendix we study trivial perturbations in the $D=n+4$ dimensional
ring coordinate given by

$\displaystyle
ds^{2}=-dt^{2}+\frac{R^{2}}{(R+r\cos{\theta})^{2}}\Biggl{[}\frac{R^{2}dr^{2}}{R^{2}-r^{2}}+(R^{2}-r^{2})d\Phi^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\Omega^{2}_{n})\Biggr{]}.$
(B.1)

There are two trivial perturbations, axisymmetric and non-axisymmetric trivial
perturbations.

#### Axisymmetric trivial perturbation

The axisymmetric trivial perturbation is the redefinition of the ring radius
$R$. The perturbations

$\displaystyle R\rightarrow R+\delta R,$ (B.2)

and

$\displaystyle r\rightarrow r+\delta
R\left(\frac{r}{R}+\frac{R^{2}-r^{2}}{R^{2}}\cos{\theta}\right),~{}~{}\theta\rightarrow\theta-\delta
R\frac{\sin{\theta}}{r}$ (B.3)

do not change the form of the metric (B.1) up to $O(\delta R)$.

As another trivial axisymmetric perturbation, there is a perturbation which
becomes trivial at the large $D$ limit. If we consider the perturbation,

$\displaystyle\Phi\rightarrow\Phi+\delta\Phi$ (B.4)

and

$\displaystyle r\rightarrow
r+\delta\Phi\frac{(R^{2}-r^{2})(r+R\cos{\theta})}{R(R+r\cos{\theta})},~{}~{}\theta\rightarrow\theta-\delta\Phi\frac{(R^{2}-r^{2})\sin{\theta}}{r(R+r\cos{\theta})},$
(B.5)

the metric does change its form by

$\displaystyle\delta(ds^{2})=-\delta\Phi\frac{2R^{2}}{(R^{2}-r^{2})(R+r\cos{\theta})^{4}}\Bigl{(}R^{2}\cos{\theta}dr+r(Rdr-(R^{2}-r^{2})\sin{\theta}d\theta)\Bigr{)}^{2}.$
(B.6)

Thus this transformation is not a trivial perturbation. However, if we
consider the large $D$ limit, this transformation becomes trivial. Taking
$\delta\Phi=\hat{\delta\Phi}/n$ and using $\mathsf{R}=(r/r_{0})^{n}$, the
metric deformation (B.6) becomes

$\displaystyle\delta(ds^{2})=\frac{\hat{\delta\Phi}}{n}\frac{2R^{2}(R^{2}-1)}{(R+\cos{\theta})^{4}}d\theta^{2}+O(n^{-2}),$
(B.7)

where we set to $r_{0}=1$, and we neglect $dr^{2}$ and $drdz$ terms since
$dr\simeq d\mathsf{R}/(n\mathsf{R})$ is higher order in $1/n$. The deformation
(B.7) can be absorbed into the $1/n$ redefinition of $\theta$. So the
transformation (B.5) becomes trivial at the large $D$ limit. We can see that
$\mathsf{R}=(r/r_{0})^{n}$ changes its form by the transformation (B.5) to

$\displaystyle\mathsf{R}\rightarrow\mathsf{R}~{}\text{exp}\Biggl{[}\hat{\delta\Phi}\left(R^{2}-1-\frac{(R^{2}-1)^{2}}{R(R+\cos{\theta})}\right)\Biggr{]}.$
(B.8)

So, observing the solution eq. (3.13), we find that the integration constants
$p_{0}$ and $d_{0}$ in eq. (3.13) represent the transformation (B.5) and the
$O(1/n)$ redefinition of $r_{0}$. So $p_{0}$ and $d_{0}$ in eq. (3.13) do not
have physical degree of freedom.

#### Non-axisymmetric trivial perturbations

Let us consider the perturbation given by

$\displaystyle r\rightarrow r+\epsilon
e^{im_{\Phi}\Phi}\frac{\sqrt{R^{2}-r^{2}}}{R}(r+R\cos{\theta}),~{}~{}z\rightarrow
z+\epsilon e^{im_{\Phi}\Phi}\frac{\sqrt{R^{2}-r^{2}}}{r}\sin{\theta}.$ (B.9)

For $m_{\Phi}=1$, the metric does not change the form under this non-
axisymmetric perturbation up to $O(\epsilon)$. So the stationary deformation
of the black ring with $m_{\Phi}=1$ is just a trivial deformation. Thus it
might be reasonable to regard that the perturbations with $m_{\Phi}=1$ and
$\ell=0$ does not have physical degree of freedom.

##  Appendix C $1/D$ corrections of effective equations

In this appendix we show $1/n$ corrections to the effective equations (2.16),
(2.17) and (2.18). The effective equations up to $O(1/n)$ are

$\displaystyle\partial_{v}p_{v}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{v}-\frac{\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}G(z)^{2}}+\frac{\partial_{\phi}p_{\phi}}{G(z)^{2}}+\frac{H^{\prime}(z)}{H(z)}p_{z}+\frac{\Delta_{v}}{n}=0,$
(C.1)
$\displaystyle\partial_{v}p_{\phi}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{\phi}-\frac{\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}G(z)^{2}}+\frac{1}{G(z)^{2}}\partial_{\phi}\Biggl{[}\frac{p_{\phi}^{2}}{p_{v}}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{4\hat{\kappa}^{2}G(z)H(z)^{2}+2G^{\prime}(z)H(z)H^{\prime}(z)}{4\hat{\kappa}^{2}G(z)H(z)^{2}}\partial_{\phi}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{H^{\prime}(z)}{H(z)}\frac{p_{z}p_{\phi}}{p_{v}}+\frac{G^{\prime}(z)H^{\prime}(z)}{\hat{\kappa}G(z)H(z)}p_{\phi}+\frac{\Delta_{\phi}}{n}=0,$
(C.2)

and

$\displaystyle\partial_{v}p_{z}-\frac{H^{\prime}(z)}{2\hat{\kappa}H(z)}\partial_{z}p_{z}-\frac{\partial_{\phi}^{2}p_{z}}{2\hat{\kappa}G(z)^{2}}+\partial_{z}p_{v}+\frac{1}{G(z)^{2}}\partial_{\phi}\Biggl{[}\frac{p_{\phi}p_{z}}{p_{v}}\Biggr{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{H^{\prime}(z)}{H(z)}\frac{p^{2}_{z}}{p_{v}}-\frac{G^{\prime}(z)}{G(z)^{3}}\frac{p^{2}_{\phi}}{p_{v}}+\frac{G^{\prime}(z)}{\hat{\kappa}G(z)^{3}}\partial_{\phi}p_{\phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{H(z)G^{\prime}(z)^{2}H^{\prime}(z)+G(z)(G^{\prime}(z)-H(z)H^{\prime}(z)G^{\prime\prime}(z))}{4\hat{\kappa}^{2}G(z)^{2}H(z)^{2}}p_{v}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\frac{1-2H^{\prime}(z)^{2}}{2\hat{\kappa}H(z)^{2}}p_{z}+\frac{\Delta_{z}}{n}=0.$
(C.3)

The corrections terms $\Delta_{v}$, $\Delta_{\phi}$ and $\Delta_{z}$ are given
by

$\displaystyle\Delta_{v}=-\frac{\partial_{\phi}^{2}p_{v}\left(5G^{\prime}H^{\prime}+4\hat{\kappa}^{2}GH\right)}{8\hat{\kappa}^{3}G^{3}H}-\frac{\partial_{z}p_{v}\left(G^{\prime}-12\hat{\kappa}^{2}GHH^{\prime}\right)}{8\hat{\kappa}^{3}GH^{2}}$
(C.4)
$\displaystyle+\frac{1}{p_{v}}\Biggl{(}-\frac{G^{\prime}H^{\prime}p_{\phi}^{2}}{\hat{\kappa}G^{3}H}+p_{\phi}\left(\frac{H^{\prime}\partial_{z}p_{\phi}}{2\hat{\kappa}G^{2}H}+\frac{H^{\prime}\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}H}+\frac{3\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}G^{4}}\right)+\frac{H^{\prime}\partial_{\phi}p_{\phi}p_{z}}{\hat{\kappa}G^{2}H}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{3(\partial_{\phi}p_{\phi})^{2}}{2\hat{\kappa}G^{4}}+\left(\frac{1}{2\hat{\kappa}H^{2}}-2\hat{\kappa}\right)p_{z}^{2}\Biggr{)}+\frac{\partial_{\phi}p_{\phi}\left(\frac{2G^{\prime}H^{\prime}}{\hat{\kappa}^{2}H}-G\right)}{G^{3}}$
$\displaystyle+p_{z}\left(\frac{G^{\prime}}{G}+\frac{\left(\frac{1}{\hat{\kappa}^{2}}-10H^{2}\right)H^{\prime}}{2H^{3}}\right)-\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}-\frac{H^{\prime}\partial_{\phi}^{2}p_{z}}{2\hat{\kappa}^{2}G^{2}H}+\frac{3(\partial_{\phi}p_{v})^{2}p_{\phi}^{2}}{2\hat{\kappa}G^{4}p_{v}^{3}}$
$\displaystyle+\frac{-\frac{H^{\prime}\partial_{\phi}p_{v}p_{\phi}p_{z}}{\hat{\kappa}G^{2}H}+p_{\phi}^{2}\left(-\frac{H^{\prime}\partial_{z}p_{v}}{4\hat{\kappa}G^{2}H}-\frac{3\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}G^{4}}\right)-\frac{3\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}p_{\phi}}{\hat{\kappa}G^{4}}}{p_{v}^{2}}+\left(3-\frac{1}{2\hat{\kappa}^{2}H^{2}}\right)\partial_{z}p_{z}$
$\displaystyle-\frac{\hat{\kappa}G^{4}\partial_{z}^{2}p_{v}+\partial_{\phi}^{3}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}p_{v}\left(H(G^{\prime})^{2}H^{\prime}+G\left(G^{\prime}-HG^{\prime\prime}H^{\prime}\right)\right)}{4\hat{\kappa}^{3}G^{2}H^{3}},$
(C.5)
$\displaystyle\Delta_{\phi}=\frac{G^{\prime\prime}}{\hat{\kappa}G}p_{\phi}-\frac{4GHG^{\prime}H^{\prime}\hat{\kappa}^{2}+8GH^{2}G^{\prime\prime}\hat{\kappa}^{2}-3\left(4\hat{\kappa}^{2}H^{2}-1\right)(G^{\prime})^{2}}{16\hat{\kappa}^{4}G^{2}H^{2}}\partial_{\phi}p_{v}$
$\displaystyle-\frac{12\hat{\kappa}^{2}\left((G^{\prime})^{2}-GG^{\prime\prime}\right)H^{3}+8\hat{\kappa}^{2}GG^{\prime}H^{\prime}H^{2}-3\left((G^{\prime})^{2}-GG^{\prime\prime}\right)H-3GG^{\prime}H^{\prime}}{8\hat{\kappa}^{3}G^{2}H^{3}}p_{\phi}\log{p_{v}}$
$\displaystyle+\frac{3p_{\phi}^{3}\log{p_{v}}(\partial_{\phi}p_{v})^{2}}{\hat{\kappa}G^{4}p_{v}^{4}}-\frac{p_{\phi}^{3}(\partial_{\phi}p_{v})^{2}}{\hat{\kappa}G^{4}p_{v}^{4}}+\frac{3p_{\phi}^{2}\log{p_{v}}(\partial_{\phi}p_{v})^{3}}{2\hat{\kappa}^{2}G^{4}p_{v}^{4}}+\frac{p_{\phi}^{2}(\partial_{\phi}p_{v})^{3}}{4\hat{\kappa}^{2}G^{4}p_{v}^{4}}+\frac{\left(\frac{H^{\prime}}{H}-\frac{2G^{\prime}}{G}\right)\partial_{\phi}p_{z}}{2\hat{\kappa}}$
$\displaystyle-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{3}G^{3}H}+\frac{G^{\prime}\partial_{z}p_{\phi}}{2\hat{\kappa}G}-\frac{\partial_{z}^{2}p_{\phi}}{2\hat{\kappa}}-\frac{3G^{\prime}H^{\prime}\partial_{v}p_{\phi}}{4\hat{\kappa}^{2}GH}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{\left(4\hat{\kappa}^{2}\left((G^{\prime})^{2}-GG^{\prime\prime}\right)H^{3}+\left(GG^{\prime\prime}-(G^{\prime})^{2}\right)H-GG^{\prime}H^{\prime}\right)\partial_{\phi}p_{v}}{8\hat{\kappa}^{4}G^{2}H^{3}}$
$\displaystyle~{}~{}~{}~{}+\frac{\left(\left(4\hat{\kappa}^{2}H^{2}-1\right)G^{\prime}-2\hat{\kappa}^{2}GHH^{\prime}\right)\partial_{\phi}p_{z}}{4\hat{\kappa}^{3}GH^{2}}-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}^{3}G^{3}H}+\frac{H^{\prime}\partial_{z}p_{\phi}}{2\hat{\kappa}H}\Biggr{)}$
$\displaystyle+\frac{\Delta_{\phi}^{(3)}}{p_{v}^{3}}+\frac{\Delta_{\phi}^{(2)}}{p_{v}^{2}}+\frac{\Delta_{\phi}^{(1)}}{p_{v}},$
(C.6)

and

$\displaystyle\Delta_{z}=\left(2-\frac{G^{\prime}H^{\prime}}{4\hat{\kappa}^{2}GH}\right)\partial_{v}p_{z}-\frac{\partial_{z}^{2}p_{z}}{2\hat{\kappa}}+p_{z}\Biggl{(}\frac{-G^{2}+H^{2}G^{\prime\prime}G+H^{2}(G^{\prime})^{2}}{2\hat{\kappa}G^{2}H^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{\log{p_{v}}\left(-4\hat{\kappa}^{2}H(H^{\prime})^{2}G^{2}+\left(G^{\prime}H^{\prime}-H(H^{\prime})^{2}G^{\prime\prime}\right)G+H(H^{\prime})^{2}(G^{\prime})^{2}\right)}{4\hat{\kappa}^{3}G^{2}H^{3}}\Biggr{)}$
$\displaystyle+p_{v}\Biggl{(}\frac{\log{p_{v}}G^{\prime}\left(-4\hat{\kappa}^{2}\left((G^{\prime})^{2}-GG^{\prime\prime}\right)H^{3}+\left((G^{\prime})^{2}-GG^{\prime\prime}\right)H+GG^{\prime}H^{\prime}\right)}{8\hat{\kappa}^{4}G^{3}H^{3}}$
$\displaystyle+\frac{-2\hat{\kappa}^{2}\left(2(G^{\prime})^{3}-3GG^{\prime\prime}G^{\prime}+G^{2}G^{(3)}(z)\right)H^{3}+\left((G^{\prime})^{3}-GG^{\prime}G^{\prime\prime}\right)H+G(G^{\prime})^{2}H^{\prime}}{8\hat{\kappa}^{4}G^{3}H^{3}}\Biggr{)}$
$\displaystyle+\frac{\left(3HH^{\prime}(G^{\prime})^{2}+G\left(\left(4\hat{\kappa}^{2}H^{2}+1\right)G^{\prime}-HH^{\prime}G^{\prime\prime}\right)\right)\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{3}G^{4}H^{2}}+\frac{\left(GH^{\prime}-HG^{\prime}\right)\partial_{z}p_{z}}{2\hat{\kappa}GH}$
$\displaystyle+\frac{\left(4GHG^{\prime}H^{\prime}\hat{\kappa}^{2}-8GH^{2}G^{\prime\prime}\hat{\kappa}^{2}+\left(4\hat{\kappa}^{2}H^{2}-1\right)(G^{\prime})^{2}\right)\partial_{z}p_{v}}{16\hat{\kappa}^{4}G^{2}H^{2}}-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{3}G^{3}H}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{\left(7HH^{\prime}(G^{\prime})^{2}+3G\left(G^{\prime}-HH^{\prime}G^{\prime\prime}\right)\right)\partial_{\phi}p_{\phi}}{8\hat{\kappa}^{3}G^{4}H^{2}}-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{\phi}^{2}p_{z}}{4\hat{\kappa}^{3}G^{3}H}$
$\displaystyle+\frac{H^{\prime}\partial_{z}p_{z}}{\hat{\kappa}H}+\frac{\partial_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}\Biggr{)}+\frac{\Delta_{z}^{(4)}}{p_{v}^{4}}+\frac{\Delta_{z}^{(3)}}{p_{v}^{3}}+\frac{\Delta_{z}^{(2)}}{p_{v}^{2}}+\frac{\Delta_{z}^{(1)}}{p_{v}}.$
(C.7)

The coefficients $\Delta_{\phi}^{(1,2,3)}$ and $\Delta_{z}^{(1,2,3,4)}$ have
messy forms as

$\displaystyle\Delta_{\phi}^{(3)}=\left(-\frac{\log{p_{v}}\partial_{\phi}^{2}p_{v}}{\hat{\kappa}G^{4}}-\frac{H^{\prime}\partial_{z}p_{v}}{2\hat{\kappa}G^{2}H}\right)p_{\phi}^{3}+\Biggl{(}\partial_{\phi}p_{v}\left(\frac{3\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{4}}-\frac{\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}^{2}G^{4}}\right)-\frac{2\partial_{\phi}p_{v}\partial_{v}p_{v}}{\hat{\kappa}G^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\log{p_{v}}\left(\partial_{\phi}p_{v}\left(-\frac{6\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{4}}-\frac{3\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}^{2}G^{4}}\right)-\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{v}}{2\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{\phi}p_{v}\partial_{v}p_{v}}{\hat{\kappa}G^{2}}\right)\Biggr{)}p_{\phi}^{2}$
$\displaystyle~{}~{}~{}~{}~{}+p_{z}\left(p_{\phi}^{2}\left(\frac{2H^{\prime}\partial_{\phi}p_{v}}{\hat{\kappa}G^{2}H}-\frac{2\log{p_{v}}H^{\prime}\partial_{\phi}p_{v}}{\hat{\kappa}G^{2}H}\right)-\frac{\log{p_{v}}p_{\phi}H^{\prime}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{2}H}\right)$
$\displaystyle~{}~{}~{}~{}~{}+\left(-\frac{3\log{p_{v}}\partial_{\phi}p_{\phi}(\partial_{\phi}p_{v})^{2}}{\hat{\kappa}^{2}G^{4}}-\frac{\partial_{\phi}p_{\phi}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{4}}\right)p_{\phi},$
(C.8)
$\displaystyle\Delta_{\phi}^{(2)}=\left(-\frac{\log{p_{v}}G^{\prime}H^{\prime}}{2\hat{\kappa}G^{3}H}-\frac{G^{\prime}H^{\prime}}{2\hat{\kappa}G^{3}H}\right)p_{\phi}^{3}+\Biggl{(}\frac{\left(\frac{3G^{\prime}H^{\prime}}{\hat{\kappa}^{2}H}-4G\right)\partial_{\phi}p_{v}}{4G^{3}}-\frac{3H^{\prime}\partial_{\phi}p_{z}}{4\hat{\kappa}G^{2}H}+\frac{\partial_{\phi}^{3}p_{v}}{8\hat{\kappa}^{2}G^{4}}$
$\displaystyle+\frac{3H^{\prime}\partial_{z}p_{\phi}}{4\hat{\kappa}G^{2}H}+\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{v}}{8\hat{\kappa}^{2}G^{2}H}+\frac{3\partial_{v}\partial_{\phi}p_{v}}{4\hat{\kappa}G^{2}}+\log{p_{v}}\Biggl{(}-\frac{3G^{\prime}H^{\prime}\partial_{\phi}p_{v}}{2\hat{\kappa}^{2}G^{3}H}+\frac{H^{\prime}\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}H}+\frac{3\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}G^{4}}$
$\displaystyle+\frac{\partial_{\phi}^{3}p_{v}}{4\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{v}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{v}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}p_{\phi}^{2}+\Biggl{(}2\hat{\kappa}+\left(\frac{1}{2\hat{\kappa}H^{2}}-2\hat{\kappa}\right)\log{p_{v}}-\frac{1}{2H^{2}\hat{\kappa}}\Biggr{)}p_{z}^{2}p_{\phi}$
$\displaystyle+\Biggl{(}\frac{3\partial_{\phi}p_{\phi}\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{4}}+\frac{\partial_{\phi}p_{v}\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}\partial_{\phi}p_{\phi}\partial_{z}p_{v}}{4\hat{\kappa}^{2}G^{2}H}-\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\frac{3\partial_{\phi}p_{\phi}\partial_{v}p_{v}}{2\hat{\kappa}G^{2}}+\frac{5\partial_{\phi}p_{v}\partial_{v}p_{\phi}}{2\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{3(\partial_{\phi}p_{\phi})^{2}}{\hat{\kappa}G^{4}}+\frac{3\partial_{\phi}^{2}p_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}\partial_{z}p_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}\left(\frac{H^{\prime}\partial_{\phi}p_{z}}{2\hat{\kappa}^{2}G^{2}H}+\frac{3\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}\right)$
$\displaystyle-\frac{\partial_{v}p_{v}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}}+\partial_{\phi}p_{v}+\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{\phi}p_{v}\partial_{v}p_{\phi}}{\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}p_{\phi}+\frac{3\log{p_{v}}\partial_{\phi}p_{v}(\partial_{\phi}p_{\phi})^{2}}{2\hat{\kappa}^{2}G^{4}}$
$\displaystyle+\frac{\partial_{\phi}p_{v}(\partial_{\phi}p_{\phi})^{2}}{4\hat{\kappa}^{2}G^{4}}+p_{z}\Biggl{(}\frac{\log{p_{v}}H^{\prime}\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}+p_{\phi}\Biggl{(}-\frac{3H^{\prime}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}H}-\partial_{z}p_{v}+\frac{H^{\prime}\partial_{v}p_{v}}{\hat{\kappa}H}$
$\displaystyle+\log{p_{v}}\left(\frac{2H^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}H}+\frac{H^{\prime}\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{2}H}+\left(\frac{1}{4\hat{\kappa}^{2}H^{2}}-1\right)\partial_{z}p_{v}-\frac{H^{\prime}\partial_{v}p_{v}}{2\hat{\kappa}H}\right)\Biggr{)}\Biggr{)},$
(C.9)
$\displaystyle\Delta_{\phi}^{(1)}=-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{3}G^{3}H}-\frac{3\partial_{\phi}^{2}p_{\phi}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}-\frac{H^{\prime}\partial_{z}p_{\phi}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}-\frac{3\partial_{v}p_{\phi}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\left(\partial_{\phi}p_{\phi}\left(-\frac{H^{\prime}\partial_{\phi}p_{z}}{2\hat{\kappa}^{2}G^{2}H}-\frac{3\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}\right)-\frac{H^{\prime}\partial_{\phi}p_{\phi}\partial_{z}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{\phi}p_{\phi}\partial_{v}p_{\phi}}{\hat{\kappa}G^{2}}\right)$
$\displaystyle+p_{z}\Biggl{(}p_{\phi}\left(\frac{\left(16\hat{\kappa}^{2}H^{2}-3\right)G^{\prime}}{4\hat{\kappa}^{2}GH^{2}}+\frac{\log{p_{v}}\left(-8\hat{\kappa}^{2}G^{\prime}H^{3}-8\hat{\kappa}^{2}GH^{\prime}H^{2}+2G^{\prime}H+GH^{\prime}\right)}{4\hat{\kappa}^{2}GH^{3}}\right)$
$\displaystyle+\partial_{z}p_{\phi}-\frac{H^{\prime}\partial_{v}p_{\phi}}{\hat{\kappa}H}+\log{p_{v}}\left(-\frac{H^{\prime}\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\left(1-\frac{1}{4\hat{\kappa}^{2}H^{2}}\right)\partial_{z}p_{\phi}+\frac{H^{\prime}\partial_{v}p_{\phi}}{2\hat{\kappa}H}\right)\Biggr{)}$
$\displaystyle+p_{\phi}\Biggl{(}\frac{\left(4G+\frac{G^{\prime}H^{\prime}}{\hat{\kappa}^{2}H}\right)\partial_{\phi}p_{\phi}}{2G^{3}}-\frac{\partial_{\phi}^{3}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}+\partial_{z}p_{z}-\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}-\frac{H^{\prime}\partial_{v}p_{z}}{\hat{\kappa}H}-\frac{3\partial_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{3G^{\prime}H^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}^{2}G^{3}H}-\frac{H^{\prime}\partial_{\phi}^{2}p_{z}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{\phi}^{3}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+\left(1-\frac{1}{4\hat{\kappa}^{2}H^{2}}\right)\partial_{z}p_{z}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}+\frac{H^{\prime}\partial_{v}p_{z}}{2\hat{\kappa}H}+\frac{\partial_{v}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)},$
(C.10)
$\displaystyle\Delta_{z}^{(4)}=\frac{3\partial_{\phi}p_{v}\partial_{z}p_{v}p_{\phi}^{3}}{2\hat{\kappa}G^{4}}+\frac{3\partial_{\phi}p_{v}^{2}\partial_{z}p_{v}p_{\phi}^{2}}{4\hat{\kappa}^{2}G^{4}}$
$\displaystyle+p_{z}\left(\left(\frac{3\log{p_{v}}(\partial_{\phi}p_{v})^{2}}{\hat{\kappa}G^{4}}-\frac{5\partial_{\phi}p_{v}^{2}}{2\hat{\kappa}G^{4}}\right)p_{\phi}^{2}+\left(\frac{3\log{p_{v}}(\partial_{\phi}p_{v})^{3}}{2\hat{\kappa}^{2}G^{4}}-\frac{(\partial_{\phi}p_{v})^{3}}{2\hat{\kappa}^{2}G^{4}}\right)p_{\phi}\right),$
(C.11)
$\displaystyle\Delta_{z}^{(3)}=\left(\frac{2\log{p_{v}}G^{\prime}\partial_{\phi}p_{v}}{\hat{\kappa}G^{5}}+\frac{G^{\prime}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{5}}-\frac{\partial_{z}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{4}}\right)p_{\phi}^{3}+\Biggl{(}\frac{G^{\prime}\partial_{\phi}p_{v}^{2}}{2\hat{\kappa}^{2}G^{5}}-\frac{3\partial_{z}p_{\phi}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{4}}$
$\displaystyle+\left(\frac{3\partial_{\phi}p_{z}}{2\hat{\kappa}G^{4}}-\frac{\partial_{z}\partial_{\phi}p_{v}}{2\hat{\kappa}^{2}G^{4}}\right)\partial_{\phi}p_{v}-\frac{H^{\prime}\partial_{z}p_{v}^{2}}{4\hat{\kappa}^{2}G^{2}H}+\log{p_{v}}\left(\frac{3G^{\prime}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{5}}-\frac{2\partial_{\phi}p_{v}\partial_{\phi}p_{z}}{\hat{\kappa}G^{4}}\right)$
$\displaystyle+\left(-\frac{3\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{4}}-\frac{\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{4}}\right)\partial_{z}p_{v}+\frac{\partial_{z}p_{v}\partial_{v}p_{v}}{2\hat{\kappa}G^{2}}\Biggr{)}p_{\phi}^{2}+\Biggl{(}-\frac{3\log{p_{v}}\partial_{\phi}p_{z}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{4}}+\frac{\partial_{\phi}p_{z}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{4}}$
$\displaystyle-\frac{\partial_{z}p_{\phi}\partial_{\phi}p_{v}^{2}}{2\hat{\kappa}^{2}G^{4}}-\frac{\partial_{\phi}p_{\phi}\partial_{z}p_{v}\partial_{\phi}p_{v}}{\hat{\kappa}^{2}G^{4}}\Biggr{)}p_{\phi}$
$\displaystyle+p_{z}^{2}\left(p_{\phi}\left(\frac{5H^{\prime}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{2}H}-\frac{2\log{p_{v}}H^{\prime}\partial_{\phi}p_{v}}{\hat{\kappa}G^{2}H}\right)-\frac{\log{p_{v}}H^{\prime}(\partial_{\phi}p_{v})^{2}}{2\hat{\kappa}^{2}G^{2}H}\right)$
$\displaystyle+p_{z}\Biggl{(}\left(-\frac{\log{p_{v}}\partial_{\phi}^{2}p_{v}}{\hat{\kappa}G^{4}}+\frac{\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}G^{4}}-\frac{H^{\prime}\partial_{z}p_{v}}{\hat{\kappa}G^{2}H}\right)p_{\phi}^{2}+\Biggl{(}\partial_{\phi}p_{v}\left(\frac{3\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{4}}+\frac{\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{4}}\right)+\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{v}}{4\hat{\kappa}^{2}G^{2}H}$
$\displaystyle-\frac{5\partial_{\phi}p_{v}\partial_{v}p_{v}}{2\hat{\kappa}G^{2}}+\log{p_{v}}\left(\partial_{\phi}p_{v}\left(-\frac{4\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{4}}-\frac{3\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}^{2}G^{4}}\right)-\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{v}}{2\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{\phi}p_{v}\partial_{v}p_{v}}{\hat{\kappa}G^{2}}\right)\Biggr{)}p_{\phi}$
$\displaystyle-\frac{3\log(p_{v})(\partial_{\phi}p_{v})^{2}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+\frac{\partial_{\phi}p_{v}^{2}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}\Biggr{)},$
(C.12)
$\displaystyle\Delta_{z}^{(2)}=\left(2\hat{\kappa}+\left(\frac{1}{2\hat{\kappa}H^{2}}-2\hat{\kappa}\right)\log{p_{v}}-\frac{1}{2H^{2}\hat{\kappa}}\right)p_{z}^{3}+\Biggl{(}-\frac{H^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}H}-\partial_{z}p_{v}+\frac{H^{\prime}\partial_{v}p_{v}}{\hat{\kappa}H}$
$\displaystyle+\log{p_{v}}\left(\frac{H^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}H}+\frac{H^{\prime}\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{2}H}+\left(\frac{1}{4\hat{\kappa}^{2}H^{2}}-1\right)\partial_{z}p_{v}-\frac{H^{\prime}\partial_{v}p_{v}}{2\hat{\kappa}H}\right)\Biggr{)}p_{z}^{2}$
$\displaystyle+\Biggl{(}\left(\frac{G\left(4\hat{\kappa}^{2}H^{2}-1\right)+HG^{\prime}H^{\prime}}{2\hat{\kappa}G^{3}H^{2}}-\frac{3\log{p_{v}}G^{\prime}H^{\prime}}{2\hat{\kappa}G^{3}H}\right)p_{\phi}^{2}+\frac{3\partial_{\phi}p_{v}\partial_{v}p_{\phi}}{2\hat{\kappa}G^{2}}$
$\displaystyle+\Biggl{(}\frac{\left(-16\hat{\kappa}^{2}GH^{2}+3G^{\prime}H^{\prime}H+G\right)\partial_{\phi}p_{v}}{4\hat{\kappa}^{2}G^{3}H^{2}}-\frac{2H^{\prime}\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}H}-\frac{\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}G^{4}}+\frac{H^{\prime}\partial_{z}p_{\phi}}{\hat{\kappa}G^{2}H}+\frac{\partial_{v}\partial_{\phi}p_{v}}{\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}-\frac{\left(-8\hat{\kappa}^{2}GH^{2}+6G^{\prime}H^{\prime}H+G\right)\partial_{\phi}p_{v}}{4\hat{\kappa}^{2}G^{3}H^{2}}+\frac{2H^{\prime}\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}H}+\frac{\partial_{\phi}^{2}p_{\phi}}{\hat{\kappa}G^{4}}+\frac{\partial_{\phi}^{3}p_{v}}{4\hat{\kappa}^{2}G^{4}}$
$\displaystyle+\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{v}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{v}\partial_{\phi}p_{v}}{2\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}p_{\phi}-\frac{(\partial_{\phi}p_{\phi})^{2}}{2\hat{\kappa}G^{4}}-\frac{\partial_{\phi}p_{v}\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}-\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{\phi}p_{\phi}\partial_{v}p_{v}}{\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{(\partial_{\phi}p_{\phi})^{2}}{\hat{\kappa}G^{4}}+\frac{3\partial_{\phi}^{2}p_{v}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}\partial_{z}p_{v}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{v}p_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}+\partial_{\phi}p_{v}\left(\frac{H^{\prime}\partial_{\phi}p_{z}}{\hat{\kappa}^{2}G^{2}H}+\frac{3\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}\right)$
$\displaystyle+\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{\phi}p_{v}\partial_{v}p_{\phi}}{2\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}p_{z}+\frac{3\log{p_{v}}\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}\partial_{\phi}p_{z}}{2\hat{\kappa}^{2}G^{4}}-\frac{\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}\partial_{\phi}p_{z}}{2\hat{\kappa}^{2}G^{4}}+\frac{(\partial_{\phi}p_{\phi})^{2}\partial_{z}p_{v}}{4\hat{\kappa}^{2}G^{4}}$
$\displaystyle+\frac{\partial_{\phi}p_{v}\partial_{\phi}p_{\phi}\partial_{z}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+p_{\phi}\Biggl{(}\left(\frac{3\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{4}}+\frac{\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{4}}\right)\partial_{z}p_{\phi}+\partial_{z}p_{v}\left(\frac{\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}+\frac{H^{\prime}\partial_{z}p_{\phi}}{2\hat{\kappa}^{2}G^{2}H}\right)$
$\displaystyle-\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{z}}{4\hat{\kappa}^{2}G^{2}H}+\partial_{\phi}p_{\phi}\left(\frac{\partial_{z}\partial_{\phi}p_{v}}{2\hat{\kappa}^{2}G^{4}}-\frac{\partial_{\phi}p_{z}}{\hat{\kappa}G^{4}}\right)+\partial_{\phi}p_{v}\left(-\frac{G^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}^{2}G^{5}}-\frac{\partial_{\phi}^{2}p_{z}}{4\hat{\kappa}^{2}G^{4}}+\frac{\partial_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}\right)$
$\displaystyle+\left(\frac{\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}}-\frac{\partial_{z}p_{\phi}}{2\hat{\kappa}G^{2}}\right)\partial_{v}p_{v}-\frac{\partial_{z}p_{v}\partial_{v}p_{\phi}}{2\hat{\kappa}G^{2}}+\frac{3\partial_{\phi}p_{v}\partial_{v}p_{z}}{2\hat{\kappa}G^{2}}+\log{p_{v}}\Biggl{(}\frac{2\partial_{\phi}p_{\phi}\partial_{\phi}p_{z}}{\hat{\kappa}G^{4}}+\frac{3\partial_{\phi}^{2}p_{v}\partial_{\phi}p_{z}}{4\hat{\kappa}^{2}G^{4}}$
$\displaystyle+\frac{H^{\prime}\partial_{z}p_{v}\partial_{\phi}p_{z}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{v}p_{v}\partial_{\phi}p_{z}}{2\hat{\kappa}G^{2}}+\partial_{\phi}p_{v}\left(\frac{3\partial_{\phi}^{2}p_{z}}{4\hat{\kappa}^{2}G^{4}}-\frac{3G^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}^{2}G^{5}}\right)+\frac{H^{\prime}\partial_{\phi}p_{v}\partial_{z}p_{z}}{4\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{\phi}p_{v}\partial_{v}p_{z}}{2\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}$
$\displaystyle+p_{\phi}^{2}\Biggl{(}-\frac{3G^{\prime}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{5}}-\frac{G^{\prime}\partial_{\phi}^{2}p_{v}}{2\hat{\kappa}^{2}G^{5}}-\frac{\partial_{\phi}^{2}p_{z}}{4\hat{\kappa}G^{4}}+\frac{\left(G\left(H^{\prime
4}+6\hat{\kappa}^{2}H^{2}-1\right)-12\hat{\kappa}^{2}H^{3}G^{\prime}H^{\prime}\right)\partial_{z}p_{v}}{16\hat{\kappa}^{4}G^{3}H^{4}}$
$\displaystyle+\frac{H^{\prime}\partial_{z}p_{z}}{2\hat{\kappa}G^{2}H}+\frac{3\partial_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}G^{4}}+\frac{H^{\prime}\partial_{z}^{2}p_{v}G^{2}+Hm^{(0,1,2)}(u,z,x)}{8\hat{\kappa}^{2}G^{4}H}-\frac{G^{\prime}\partial_{v}p_{v}}{2\hat{\kappa}G^{3}}$
$\displaystyle+\log{p_{v}}\Biggl{(}-\frac{3G^{\prime}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{5}}-\frac{3G^{\prime}\partial_{\phi}^{2}p_{v}}{4\hat{\kappa}^{2}G^{5}}+\frac{\partial_{\phi}^{2}p_{z}}{2\hat{\kappa}G^{4}}-\frac{G^{\prime}H^{\prime}\partial_{z}p_{v}}{4\hat{\kappa}^{2}G^{3}H}+\frac{G^{\prime}\partial_{v}p_{v}}{2\hat{\kappa}G^{3}}\Biggr{)}-\frac{\partial_{v}\partial_{z}p_{v}}{4\hat{\kappa}G^{2}}\Biggr{)},$
(C.13)
$\displaystyle\Delta_{z}^{(1)}=\Biggl{(}\frac{4G\left(\left(2\hat{\kappa}^{2}H^{2}-1\right)G^{\prime}+HH^{\prime}G^{\prime\prime}\right)-5H(G^{\prime})^{2}H^{\prime}}{4\hat{\kappa}^{2}G^{4}H^{2}}$
$\displaystyle+\frac{\log{p_{v}}\left(G\left(\left(4\hat{\kappa}^{2}H^{2}-1\right)G^{\prime}+HH^{\prime}G^{\prime\prime}\right)-3H(G^{\prime})^{2}H^{\prime}\right)}{2\hat{\kappa}^{2}G^{4}H^{2}}\Biggr{)}p_{\phi}^{2}$
$\displaystyle+\Biggl{(}\frac{\left(3HH^{\prime}(G^{\prime})^{2}+G\left(\left(3-8\hat{\kappa}^{2}H^{2}\right)G^{\prime}-3HH^{\prime}G^{\prime\prime}\right)\right)\partial_{\phi}p_{v}}{8\hat{\kappa}^{3}G^{4}H^{2}}+\frac{\left(8G-\frac{G^{\prime}H^{\prime}}{\hat{\kappa}^{2}H}\right)\partial_{\phi}p_{z}}{4G^{3}}$
$\displaystyle+\frac{G^{\prime}\partial_{\phi}^{2}p_{\phi}}{\hat{\kappa}^{2}G^{5}}+\frac{\left(-8\hat{\kappa}^{2}GH^{2}+6G^{\prime}H^{\prime}H+G\right)\partial_{z}p_{\phi}}{4\hat{\kappa}^{2}G^{3}H^{2}}-\frac{H^{\prime}\partial_{z}^{2}p_{\phi}G^{2}+H\partial_{z}\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}H}+\frac{G^{\prime}\partial_{v}p_{\phi}}{\hat{\kappa}G^{3}}$
$\displaystyle-\frac{\partial_{v}\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}}+\log{p_{v}}\Biggl{(}\frac{\left(-8\hat{\kappa}^{2}GH^{2}+6G^{\prime}H^{\prime}H+G\right)\partial_{\phi}p_{z}}{4\hat{\kappa}^{2}G^{3}H^{2}}+\frac{3G^{\prime}\partial_{\phi}^{2}p_{\phi}}{2\hat{\kappa}^{2}G^{5}}-\frac{\partial_{\phi}^{3}p_{z}}{4\hat{\kappa}^{2}G^{4}}+\frac{G^{\prime}H^{\prime}\partial_{z}p_{\phi}}{2\hat{\kappa}^{2}G^{3}H}$
$\displaystyle-\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{z}}{4\hat{\kappa}^{2}G^{2}H}-\frac{G^{\prime}\partial_{v}p_{\phi}}{\hat{\kappa}G^{3}}+\frac{\partial_{v}\partial_{\phi}p_{z}}{2\hat{\kappa}G^{2}}\Biggr{)}+\frac{\partial_{v}\partial_{z}p_{\phi}}{2\hat{\kappa}G^{2}}\Biggr{)}p_{\phi}+\frac{G^{\prime}(\partial_{\phi}p_{\phi})^{2}}{\hat{\kappa}^{2}G^{5}}-\frac{H^{\prime}(\partial_{z}p_{\phi})^{2}}{4\hat{\kappa}^{2}G^{2}H}$
$\displaystyle+p_{z}^{2}\left(\frac{\left(8-\frac{1}{\hat{\kappa}^{2}H^{2}}\right)G^{\prime}}{4G}-\frac{\left(8\hat{\kappa}^{2}H^{2}-1\right)\log{p_{v}}H^{\prime}}{2\hat{\kappa}^{2}H^{3}}+\frac{2H^{\prime}}{H}\right)-\frac{\left(2GH\hat{\kappa}^{2}+G^{\prime}H^{\prime}\right)\partial_{\phi}p_{v}\partial_{\phi}p_{z}}{4\hat{\kappa}^{3}G^{3}H}$
$\displaystyle+\left(\frac{\partial_{\phi}p_{v}}{2\hat{\kappa}G^{2}}-\frac{\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}\right)\partial_{z}p_{\phi}-\frac{\partial_{\phi}p_{\phi}\partial_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}^{2}G^{4}}+\left(\frac{\partial_{z}p_{\phi}}{2\hat{\kappa}G^{2}}-\frac{\partial_{\phi}p_{z}}{\hat{\kappa}G^{2}}\right)\partial_{v}p_{\phi}-\frac{\partial_{\phi}p_{\phi}\partial_{v}p_{z}}{\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{3G^{\prime}(\partial_{\phi}p_{\phi})^{2}}{2\hat{\kappa}^{2}G^{5}}-\frac{3\partial_{\phi}^{2}p_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}-\frac{H^{\prime}\partial_{z}p_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{v}p_{z}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}-\frac{H^{\prime}(\partial_{\phi}p_{z})^{2}}{2\hat{\kappa}^{2}G^{2}H}-\frac{3\partial_{\phi}p_{z}\partial_{\phi}^{2}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}$
$\displaystyle-\frac{H^{\prime}\partial_{\phi}p_{z}\partial_{z}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\frac{\partial_{\phi}p_{z}\partial_{v}p_{\phi}}{2\hat{\kappa}G^{2}}\Biggr{)}+p_{z}\Biggl{(}\frac{\left(8G-\frac{G^{\prime}H^{\prime}}{\hat{\kappa}^{2}H}\right)\partial_{\phi}p_{\phi}}{4G^{3}}+2\partial_{z}p_{z}-\frac{2H^{\prime}\partial_{v}p_{z}}{\hat{\kappa}H}-\frac{\partial_{v}\partial_{\phi}p_{\phi}}{\hat{\kappa}G^{2}}$
$\displaystyle+\log{p_{v}}\Biggl{(}\frac{\left(-8\hat{\kappa}^{2}GH^{2}+6G^{\prime}H^{\prime}H+G\right)\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{3}H^{2}}-\frac{H^{\prime}\partial_{\phi}^{2}p_{z}}{2\hat{\kappa}^{2}G^{2}H}-\frac{\partial_{\phi}^{3}p_{\phi}}{4\hat{\kappa}^{2}G^{4}}+\left(2-\frac{1}{2\hat{\kappa}^{2}H^{2}}\right)\partial_{z}p_{z}$
$\displaystyle-\frac{H^{\prime}\partial_{z}\partial_{\phi}p_{\phi}}{4\hat{\kappa}^{2}G^{2}H}+\frac{H^{\prime}\partial_{v}p_{z}}{\hat{\kappa}H}+\frac{\partial_{v}\partial_{\phi}p_{\phi}}{2\hat{\kappa}G^{2}}\Biggr{)}\Biggr{)}.$
(C.14)

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