KEK-TH-1818, OCU-PHYS-423, AP-GR-123

Stationary black holes: Large $D$ analysis

Ryotaku Suzukia, Kentaro Tanabeb

aDepartment of Physics, Osaka City University, Osaka 558-8585, Japan

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

Tsukuba, Ibaraki, 305-0801, Japan

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

We consider the effective theory of large $D$ stationary black holes. By
solving the Einstein equations with a cosmological constant using the $1/D$
expansion in near zone of the black hole we obtain the effective equation for
the stationary black hole. The effective equation describes the Myers-Perry
black hole, bumpy black holes and, possibly, the black ring solution as its
solutions. In this effective theory the black hole is represented as an
embedded membrane in the background, e.g., Minkowski or Anti-de Sitter
spacetime and its mean curvature is given by the surface gravity redshifted by
the background gravitational field and the local Lorentz boost. The local
Lorentz boost property of the effective equation is observed also in the
metric itself. In fact we show that the leading order metric of the Einstein
equation in the $1/D$ expansion is generically regarded as a Lorentz boosted
Schwarzschild black hole. We apply this Lorentz boost property of the
stationary black hole solution to solve perturbation equations. As a result we
obtain an analytic formula for quasinormal modes of the singly rotating Myers-
Perry black hole in the $1/D$ expansion.

###### Contents

  1. 1 Introduction
  2. 2 Large $D$ stationary black holes
    1. 2.1 Set up
    2. 2.2 Solving the leading order equation
    3. 2.3 Effective equation
      1. 2.3.1 Ellipsoidal embedding
    4. 2.4 Boost representation
  3. 3 Perturbation
    1. 3.1 Schwarzschild black hole
    2. 3.2 Stationary solution
  4. 4 QNMs of Myers-Perry black hole
    1. 4.1 QNM frequency
      1. 4.1.1 Scalar type perturbation
      2. 4.1.2 Vector type perturbation
  5. 5 Summary
  6. A Singly rotating AdS Myers-Perry black hole
    1. A.1 Ellipsoidal embedding
  7. B Next-to-leading order analysis
  8. C Spheroidal harmonics at large $D$
  9. D Detail analysis on the vector type perturbation
  10. E Explicit forms of $c_{i}(\theta)$ and $d_{i}(\theta)$
  11. F Explicit form of $\delta^{S}_{\omega}$

##  1 Introduction

The infinite dimensional limit of General Relativity gives not only various
interesting pictures of black hole physics, but also new useful analytic
method to solve gravitational problem [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The
perturbation problem, such as quasinormal modes (QNMs), has been solved for
general static black holes [7, 8] and the Myers-Perry black hole with equal
spin [6]. As for non-linear problem the effective theory for the large $D$
black hole has been considered recently in [9, 10] and some non-trivial static
solutions were constructed. In this paper we study the effective theory of the
large $D$ stationary black hole in asymptotically flat or Anti-de Sitter (AdS)
background.

The reason for the simplification of the gravity at large $D$ is simple [2].
The radial gradient of the gravitational field around black holes becomes very
large with $O(D/r_{0})$ at large $D$. $r_{0}$ is the black hole size. So the
tangential dynamics along the horizon becomes sub-dominant, and the system
reduces to the ordinarily differential equation system with respect to the
radial direction. In addition the appearance of the hierarchical sale,
$r_{0}/D$ and $r_{0}$, gives two separated excitations around the black hole,
that is, the non-decoupled mode with the frequency $\omega r_{0}=O(D)$ and
decoupled mode $\omega r_{0}=O(D^{0})$. While the non-decoupled mode is
universal for black holes [5], the decoupled mode has a particular and
interesting feature for each black hole. For instance the instability of the
ultraspinning back holes and black branes are belonging to this decoupled
sector [2, 6]. These two facts, the dominance of the radial dynamics and
appearance of two hierarchical scales, make General Relativity simple and
analytically solvable, but still non-trivial at large $D$.

Achievements in QNMs by large $D$ method motivate us to construct the
effective theory for general black hole solutions in the $1/D$ expansion as a
next step. The gravitational field of the black hole is localized in the near
region of the horizon due to its large gradient at $D\rightarrow\infty$. Then
the black hole can be regarded as a membrane $\Sigma_{B}$ in the background
spacetime like the membrane paradigm [11, 12, 13] at this limit. The physical
feature of $\Sigma_{B}$ is determined by solving Einstein equations in near
zone of the black hole. If we consider the decoupled mode dynamics of the
black hole, the Einstein equations can be solved consistently only in near
zone. The obtained solution gives the boundary condition for $\Sigma_{B}$ as
physical properties. Then we can construct the effective theory of the large
$D$ black hole as the membrane $\Sigma_{B}$ in the background spacetime. This
construction of the effective theory has been considered in [9] for static
black holes and for more general solutions with time dependences in [10].

Here we study the effective theory for general large $D$ stationary black
holes in the Minkowski or AdS background such as the (AdS)Myers-Perry black
hole, bumpy black holes and black ring. These solutions have been known as
exact solutions [14, 15, 16] or numerical solutions [17, 18, 19]. We obtain
the general analytic metric for the stationary black holes in the $1/D$
expansion, which describes these solutions in the unified manner. The solution
gives the equation for the embedding of $\Sigma_{B}$ in the background. This
embedding determines the horizon topology and its geometric shape. Then the
effective equation for the embedding of $\Sigma_{B}$ is given by the equation
for the mean curvature $K$ of $\Sigma_{B}$ as

$\displaystyle\gamma^{-1}K\Big{|}_{\Sigma_{B}}=2\kappa,$ (1.1)

where $\gamma$ is a redshift factor containing both the gravitational redshift
on the membrane from the background geometry and the Lorentz redshift from the
local rotation along the membrane, i.e.,

$\displaystyle\gamma^{-1}=\sqrt{-g_{tt}(1-\Omega_{H}^{2}\mathcal{R}_{\phi}^{2})}.$
(1.2)

$\kappa$ and $\Omega_{H}$ are the surface gravity and horizon angular velocity
of the black hole respectively. $\mathcal{R}_{\phi}$ is the rotation radius of
$\Sigma_{B}$ defined by

$\displaystyle\mathcal{R}_{\phi}=\sqrt{-\frac{g_{\phi\phi}}{g_{tt}}}\Bigg{|}_{\Sigma_{B}}.$
(1.3)

A solution of this effective equation gives the black hole metric which solves
the leading order Einstein equation in the $1/D$ expansion. This effective
equation is one of main results of this paper. In the static case this
effective equation reduces to one of static black holes [9].

The effective equation implies that the mean curvature of $\Sigma_{B}$ is
regarded as the surface gravity redshifted by the local Lorentz boost. This
local Lorentz boost property of the effective equation turns out to be also
the property of the leading order metric itself. Actually we find that the
leading order metric in near zone is obtained as a Lorentz boosted
Schwarzschild black hole. This property of the leading order solution would
not hold if we go to the higher order structure in the $1/D$ expansion.
However it can still play a crucial role in solving the Einstein equations. As
one interesting example, we show that perturbation equations on the obtained
leading order general stationary black hole can be explicitly solved by using
the solution of one of the Schwarzschild black hole. This perturbation
solution, of course, contains the solution of perturbation equations of the
Myers-Perry black hole. Going to the higher order of perturbation equations in
the $1/D$ expansion we obtain an analytic formula for QNMs frequency of the
singly rotating Myers-Perry black hole both for the axisymmetric and non-
axisymmetric perturbations. As a result the threshold angular momentum of the
ultraspinning instability of the singly rotating Myers-Perry black hole is
found to be

$\displaystyle\frac{a^{2}}{r^{2}_{0}}=2k_{S}-1,~{}~{}k_{S}=1,2,3,...$ (1.4)

where $r_{0}$ is the horizon radius and $k_{S}$ is a positive integer. This
result can be obtained in a much easier and simpler manner by examining the
effective equation (1.1) for the embedding of the bumpy black hole. Note that
there is no dynamical instability at the threshold angular momentum of the
case $k_{S}=1$. The threshold for $k_{S}=1$ does not imply any existence of a
bifurcation zero-mode. There is only a trivial perturbation along the Myers-
Perry solution branch.

The remaining of this paper is organized as follows. In section 2 we consider
the effective theory for stationary black holes at large $D$. We give the
leading order solution and its boost representation. The effective equation
for the large $D$ stationary black holes is derived from this leading order
metric. We study a simple solution and non-trivial solutions of the effective
equation there. In section 3 we apply the boost property of the leading order
solution to solve perturbation equations on the general stationary solutions.
The section 4 pursues the consideration for the perturbation in the previous
section in more detail. We concentrate on the perturbation of the Myers-Perry
black hole and obtain the QNMs frequency analytically. We close this paper by
discussion and outlook in section 5. The appendices contain some technical
details of calculations for the main part.

##  2 Large $D$ stationary black holes

We study the $D$ dimensional stationary black hole solution and its large $D$
effective theory. The following analysis is performed in the similar way with
static case [9] 111 But the index notation is changed from [9] . We use the
small expansion parameter $1/n$ where

$\displaystyle n=D-3,$ (2.1)

instead of $1/D$.

###  2.1 Set up

The large $D$ black hole has a very large gradient along the radial direction,
$\rho$, with $O(n)$. Then our metric ansatz for the $D$ dimensional stationary
black hole is given by

$\displaystyle
ds^{2}=\frac{N(\rho,z)^{2}}{n^{2}}d\rho^{2}+g_{ab}dx^{a}dx^{b}.$ (2.2)

$z$ is one of $D-1$ dimensional coordinate $x^{a}$. In this paper we consider
the metric which has only one inhomogeneous coordinate, $z$, other than $\rho$
for simplicity. The generalization to several inhomogeneous coordinate case
would be straightforward as done in [9]. Then the Einstein equation with a
cosmological constant

$\displaystyle\Lambda=-\frac{(D-1)(D-2)}{L^{2}}$ (2.3)

is decomposed on $\rho=\text{constant}$ surface to

$\displaystyle-R+K^{2}-K^{a}_{~{}b}K^{b}_{~{}a}-\frac{(D-1)(D-2)}{L^{2}}=0,$
(2.4) $\displaystyle\nabla_{a}K^{a}_{~{}b}-\nabla_{b}K=0,$ (2.5)
$\displaystyle\frac{n}{N}\partial_{\rho}K^{a}_{~{}b}=-KK^{a}_{~{}b}+R^{a}_{~{}b}+\frac{D-1}{L^{2}}\delta^{a}_{~{}b}-\frac{1}{N}\nabla^{a}\nabla_{b}N,$
(2.6)

and

$\displaystyle K^{a}_{~{}b}=\frac{n}{2N(\rho,z)}g^{ac}\partial_{\rho}g_{cb}.$
(2.7)

$R_{ab}$ and $\nabla_{a}$ is the Ricci curvature and covariant derivative of
$D-1$ dimensional metric $g_{ab}$ respectively. $K_{ab}$ is the extrinsic
curvature of $D-1$ dimensional $\rho=\text{constant}$ surface.

We consider stationary solutions with a rotation. Then the $D-1$ dimensional
metric ansatz for $g_{ab}$ is

$\displaystyle ds^{2}_{D-1}$ $\displaystyle=$ $\displaystyle
g_{ab}dx^{a}dx^{b}$ (2.8) $\displaystyle=$
$\displaystyle-A(\rho,z)^{2}dt^{2}+F(\rho,z)^{2}(d\phi-W(\rho,z)dt)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G(\rho,z)^{2}dz^{2}+H(\rho,z)^{2}q_{AB}dx^{A}dx^{B},$

where $x^{A}$ and $q_{AB}$ are a coordinate and the standard metric on
$S^{D-4}$ respectively. We solve the Einstein equations for $K_{ab}$ and
metric functions $A(\rho,z)$, $F(\rho,z)$, $W(\rho,z)$, $G(\rho,z)$ and
$H(\rho,z)$ by using the $1/D$ expansion. In the following we specify large
$D$ behaviors of these functions and boundary conditions as a set up.

##### Large $D$ behavior

At first each metric functions should be $O(n^{0})$. In particular

$\displaystyle g_{zz}=O(n^{0})$ (2.9)

implies that the derivative with respect to $z$ is $O(n^{0})$. The
hierarchical scaling between $\rho-$ and $z-$ dependence makes the leading
order equation at $D\rightarrow\infty$ be ordinary differential equations with
respect to $\rho$. Next we assume that the leading order of $G(\rho,z)$ and
$H(\rho,z)$ has only $z$-dependence and no $\rho$-dependence. This assumption
is motivated by the exact solution of the Myers-Perry black hole (see Appendix
A) and the fact that the large dimensionality of the sphere gives another $D$
factor in $\rho$-dependent terms through the trace on $S^{D-4}$. Thus we
assume

$\displaystyle K^{z}_{~{}z},K^{A}_{~{}B}=O(n^{0}),$ (2.10)

while the leading order of $A(\rho,z)$, $F(\rho,z)$ and $W(\rho,z)$ has
$\rho$-dependences as

$\displaystyle K^{t}_{~{}t},K^{t}_{~{}\phi},K^{\phi}_{~{}\phi},K=O(n).$ (2.11)

From these assumptions we can derive the scaling law of various geometric
quantities at large $D$. For example, the Ricci scalar $R$ of $g_{ab}$ is
$O(n^{2})$.

##### Boundary condition

The boundary conditions are imposed on the horizon and at the asymptotic
infinity of near zone defined by $n\gg\rho\gg 1$ 222 The effective theory of
the large $D$ black holes is described as the membrane physics seen from the
far zone [9]. In the overlap zone $n\gg\rho\gg 1$ we can obtain physical
properties of the membrane by the near zone solution as the boundary
condition. . At the horizon the extrinsic curvature has a singular behavior
essentially only in one component. This condition guarantees the regularity of
following quantities

$\displaystyle
K-K^{t}_{~{}t}-K^{\phi}_{~{}\phi},~{}K^{z}_{~{}z},~{}K^{A}_{~{}B},$ (2.12)

on the horizon. For the static case $K^{\phi}_{~{}\phi}$ should be regular on
the horizon. In this paper we consider the asymptotically flat or AdS black
hole. Then the boundary condition at asymptotic infinity of near zone is

$\displaystyle
A(\rho,z)=\mathsf{V}_{0}(z)+O(1/n,e^{-2\rho}),~{}~{}W(\rho,z)=O(e^{-2\rho}).$
(2.13)

Note that the Newton potential of the black hole is $O(e^{-2\rho})$ in our
gauge. So this asymptotic boundary condition means the leading order metric
becomes the flat or AdS metric other than the potential term at $\rho\gg 1$.
While $g_{t\phi}$ should be damping exponentially in $\rho$ at all order of
$1/D$, $g_{tt}$ has some terms which does not decay at the asymptotic infinity
in the higher order solution in the $1/D$ expansion when we consider the
asymptotically AdS case (See Appendix A). Since we solve the leading order
equations in this section, we do not explore this structure in detail. The
boundary condition (2.13) is enough for the leading order solutions.

###  2.2 Solving the leading order equation

We solve the leading order equation of the Einstein equation in the $1/D$
expansion by performing $\rho$ integrations. This integration gives free
functions of $z$ as integration functions. They consist of the effective
theory of large $D$ stationary black holes.

Our large $D$ scaling assumptions give following large $D$ behaviors

$\displaystyle N(\rho,z)=N_{0}(z)+O(1/n),$ (2.14) $\displaystyle
G(\rho,z)=1+O(1/n),$ (2.15) $\displaystyle
H(\rho,z)=\mathcal{R}_{\text{H}}(z)+O(1/n),$ (2.16)

where $\rho$-dependence of the leading order of $N(\rho,z)$ and $z$-dependence
of the leading order of $G(\rho,z)$ have been absorbed into the definition of
$\rho$ and $z$ coordinate respectively. The leading order of the Ricci scalar
of $g_{ab}$ can be calculated from this ansatz as

$\displaystyle
R=n^{2}\left(\frac{1}{L^{2}}+\frac{1}{\mathcal{R}_{\text{H}}(z)^{2}}-\frac{\mathcal{R}^{\prime}_{\text{H}}(z)^{2}}{\mathcal{R}_{\text{H}}(z)^{2}}\right),$
(2.17)

where here and in the following we omit $O(1/n)$ symbols showing the existence
of subleading corrections in $1/D$ for the simplicity in the representation.
Then the trace part of eq. (2.6) is integrated to

$\displaystyle
K=\frac{n}{r_{0}(z)}\coth{\left(\frac{N_{0}(z)}{r_{0}(z)}(\rho-\rho_{0}(z))\right)},$
(2.18)

where we defined $r_{0}(z)$ by

$\displaystyle\frac{1}{r_{0}(z)^{2}}=\frac{1}{L^{2}}+\frac{1}{\mathcal{R}_{\text{H}}(z)^{2}}-\frac{\mathcal{R}^{\prime}_{\text{H}}(z)^{2}}{\mathcal{R}_{\text{H}}(z)^{2}}.$
(2.19)

$\rho_{0}(z)$ is the horizon position $\rho=\rho_{0}(z)$, so we can set to
$\rho_{0}(z)=0$ by a gauge choice. The residual gauge in $\rho$ coordinate can
be used to have

$\displaystyle N_{0}(z)=2r_{0}(z).$ (2.20)

Note that this choice for $N_{0}(z)$ is different from [9] by factor two. This
is just a gauge choice and it does not affect physical properties. The
equations for $K^{t}_{~{}t}$, $K^{t}_{~{}\phi}$ and $K^{\phi}_{~{}\phi}$ in
eq. (2.6) are solved by

$\displaystyle
K^{t}_{~{}t}=n\frac{C_{tt}(z)}{r_{0}(z)\sinh{2\rho}},~{}K^{t}_{~{}\phi}=n\frac{C_{t\phi}(z)}{r_{0}(z)\sinh{2\rho}},~{}K^{\phi}_{~{}\phi}=n\frac{C_{\phi\phi}(z)}{r_{0}(z)\sinh{2\rho}}.$
(2.21)

$C_{tt}(z)$, $C_{t\phi}(z)$ and $C_{\phi\phi}(z)$ are integration functions
with respect to the $\rho$ integration. The boundary condition on the horizon
(2.12) gives

$\displaystyle 1-C_{tt}(z)-C_{\phi\phi}(z)=0.$ (2.22)

In the following we eliminate $C_{\phi\phi}(z)$ by this condition. Then we can
integrate other components of eqs. (2.6) and (2.7), and obtain the following
leading order solutions

$\displaystyle
A(\rho,z)^{2}=\frac{A_{0}(z)^{2}F_{0}(z)^{2}\tanh^{2}{\rho}}{F_{0}(z)^{2}-A_{0}(z)^{2}C_{t\phi}(z)^{2}\tanh^{2}{\rho}},$
(2.23) $\displaystyle
F(\rho,z)^{2}=F_{0}(z)^{2}-A_{0}(z)C_{t\phi}(z)^{2}\tanh^{2}{\rho},$ (2.24)
$\displaystyle
W(\rho,z)=\frac{F_{0}(z)^{2}(1-C_{tt}(z))-A_{0}(z)^{2}C_{t\phi}(z)^{2}C_{tt}(z)\tanh^{2}{\rho}}{C_{t\phi}(z)F_{0}(z)^{2}-A_{0}(z)^{2}C_{t\phi}(z)^{3}\tanh^{2}{\rho}},$
(2.25)

and

$\displaystyle
G(\rho,z)=1-\frac{2r_{0}(z)^{2}}{n}\left(\frac{\mathcal{R}_{\text{H}}^{\prime\prime}(z)}{\mathcal{R}_{\text{H}}(z)}-\frac{1}{L^{2}}\right)\log{(\cosh{\rho})},$
(2.26) $\displaystyle
H(\rho,z)=\mathcal{R}_{\text{H}}(z)\left(1+\frac{2}{n}\log{(\cosh{\rho})}\right).$
(2.27)

$A_{0}(z)$ and $F_{0}(z)$ are integration functions. The integration functions
of $G(\rho,z)$ and $H(\rho,z)$ are absorbed into the $O(1/n)$ redefinition of
$z$ coordinate and $\mathcal{R}_{\text{H}}(z)$. The scalar constraint equation
(2.4) could be satisfied by choosing the integration function in $W(\rho,z)$
properly. We can see that these solutions reduce to the static solution [9] if
we set to

$\displaystyle C_{tt}(z)=1,~{}~{}C_{t\phi}=0.$ (2.28)

We could solve the scalar constraint equation (2.4) and all evolution
equations of eqs. (2.6) and (2.7). The remaining equation is the vector
constraint equation (2.5). The vector constraint equation gives an additional
condition between integration functions as

$\displaystyle\frac{d}{dz}\log{\left(\frac{A_{0}(z)C_{tt}(z)}{r_{0}(z)}\right)}-(1-C_{tt}(z))\frac{d}{dz}\log{\left(\frac{C_{tt}(z)}{C_{t\phi}(z)}\right)}=0.$
(2.29)

This equation, at first glance, seems to be a non-trivial condition. However
we can make this equation trivial in the following sense. To realize this we
observe the surface gravity $\kappa$ and angular velocity $\Omega_{H}$ of the
horizon of the leading order solution. These quantities are read as

$\displaystyle\kappa=\frac{n}{2}\frac{A_{0}(z)}{r_{0}(z)},~{}~{}\Omega_{H}=\frac{1-C_{tt}(z)}{C_{t\phi}(z)}.$
(2.30)

These quantities should be constant on the horizon [20]. Then, using the
condition that $\kappa$ and $\Omega_{H}$ are constant, we can see that the
constraint equation (2.29) is satisfied automatically. Thus the vector
constraint equation becomes trivial under the constancy condition of the
surface gravity and angular velocity of the horizon. This is the same
situation with the static case [9]. In the static case the vector constraint
is equivalent to the constancy condition of the surface gravity. In stationary
case the vector constraint can be satisfied if we use the constancy condition
of not only the surface gravity but also the horizon angular velocity. One may
feel that this statement is strange since originally the constancy of the
surface gravity and angular velocity of the horizon was shown by using the
Einstein equations [20]. It means that the constancy condition of the surface
gravity and horizon angular velocity should be derived conditions, not
additionally imposed one to satisfy the Einstein equation. In fact the
constancy conditions (2.30) can be derived if we use eq. (2.29) and another
equation which is obtained at the next-to-leading order of $1/D$ expansions.
The regularity condition of the next-to-leading order correction of
$K^{z}{}_{z}$ on the horizon requires one additional condition between
$A_{0}(z)$ and $F_{0}(z)$ (see eq. (B.17)). Then these two equations give the
constancy condition of the surface gravity and horizon angular velocity as eq.
(2.30). The detail of this argument will be given in the appendix B. So, here,
we assume the constancy conditions (2.30) on the leading order solution in
advance. If constancy conditions (2.30) are not satisfied, the next-to-leading
order solutions becomes singular at the horizon although the leading order
solutions still regular. So we can replace $A_{0}(z)$ and $C_{t\phi}(z)$ by
$C_{tt}(z)$, $\kappa$ and $\Omega_{\text{H}}$ using eq. (2.30).

Finally we impose the boundary condition at the asymptotic infinity (2.13) on
the leading order solutions. Then we find

$\displaystyle
C_{tt}(z)=\frac{\mathsf{V}_{0}(z)^{2}}{4\hat{\kappa}^{2}r_{0}(z)^{2}},~{}~{}F_{0}(z)^{2}=\frac{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}{4\hat{\kappa}^{2}\Omega_{H}^{2}r_{0}(z)^{2}},$
(2.31)

where we defined the reduced surface gravity $\hat{\kappa}$ by

$\displaystyle\kappa=n\hat{\kappa}.$ (2.32)

As a result we obtain regular leading order solutions as

$\displaystyle
A(\rho,z)^{2}=\frac{4\hat{\kappa}^{2}r_{0}(z)^{2}\mathsf{V}_{0}(z)^{2}\sinh^{2}{\rho}}{\mathsf{V}_{0}(z)^{2}+4\hat{\kappa}^{2}r_{0}(z)^{2}\sinh^{2}{\rho}},$
(2.33) $\displaystyle
F(\rho,z)^{2}=(\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2})~{}\frac{\mathsf{V}_{0}(z)^{2}+4\hat{\kappa}^{2}r_{0}(z)^{2}\sinh^{2}{\rho}}{4\Omega_{H}^{2}\hat{\kappa}^{2}r_{0}(z)^{2}\cosh^{2}{\rho}},$
(2.34) $\displaystyle
W(\rho,z)=\frac{\Omega_{H}}{\mathsf{V}_{0}(z)^{2}+4\hat{\kappa}^{2}r_{0}(z)^{2}\sinh^{2}{\rho}},$
(2.35)

and eqs. (2.26) and (2.27) for $G(\rho,z)$ and $H(\rho,z)$. One can check that
the (AdS)Myers-Perry black hole is described by this leading order solution as
seen in Appendix A.

###  2.3 Effective equation

In the same spirit with [9] we can obtain the effective equation for the
stationary black hole from the leading order solution of near zone obtained
above. The near zone metric is matched with the far zone metric at overlap
zone $n\gg\rho\gg 1$ under the boundary condition (2.13). The near zone metric
on $\rho=\text{constant}$ surface $\Sigma_{B}$ at overlap zone is

$\displaystyle
ds^{2}|_{\Sigma_{B}}=-\mathsf{V}_{0}(z)^{2}dt^{2}+\frac{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}{\Omega^{2}_{H}}d\phi^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+dz^{2}+\mathcal{R}_{\text{H}}(z)^{2}q_{AB}dx^{A}dx^{B},$
(2.36)

where we neglect the potential terms $O(e^{-2\rho})$. Notice that the trace of
the extrinsic curvature, $K$, is read as

$\displaystyle K\Big{|}_{\Sigma_{B}}=\frac{n}{r_{0}(z)},$ (2.37)

where $r_{0}(z)$ is given in eq. (2.19). The gravitational redshift factor of
$\Sigma_{B}$ by the background spacetime, $g_{tt}$, is

$\displaystyle\sqrt{-g_{tt}}|_{\Sigma_{B}}=\mathsf{V}_{0}(z).$ (2.38)

Using eqs. (2.37) and (2.38), we can see that the leading order solution in
near zone gives the following embedding equation for $\Sigma_{B}$ at overlap
zone as

$\displaystyle\sqrt{-g_{tt}(1-\Omega_{H}^{2}\mathcal{R}_{\phi}^{2})}~{}K\Big{|}_{\Sigma_{B}}=2\kappa,$
(2.39)

where we defined the rotational radius $\mathcal{R}_{\phi}$ by

$\displaystyle\mathcal{R}_{\phi}=\sqrt{-\frac{g_{\phi\phi}}{g_{tt}}}~{}\Bigg{|}_{\Sigma_{B}}.$
(2.40)

The effective equation (2.39) gives the mean curvature of $\Sigma_{B}$ as the
surface gravity redshifted by the gravitational effect and the local Lorentz
boost effect with the boost velocity $v_{\phi}$ defined by

$\displaystyle v_{\phi}=\Omega_{H}\mathcal{R}_{\phi}.$ (2.41)

The regular leading order solution contains arbitrary functions,
$\mathsf{V}_{0}(z)$ and $\mathcal{R}_{\text{H}}(z)$. These functions can be
determined by specifying the embedding into the background. If one consider
the embedding into the Minkowski background, we have $\mathsf{V}_{0}(z)=1$ and
there is only one remaining function $\mathcal{R}_{\text{H}}(z)$ which should
satisfy eq. (2.39). On the other hand, if one considers the embedding into AdS
background, the embedding would give a non-trivial redshift factor
$\mathsf{V}_{0}(z)$ of $\Sigma_{B}$. Thus the embedding into the AdS
background should satisfy eqs. (2.38) and (2.39) for $\mathsf{V}_{0}(z)$ and
$\mathcal{R}_{\text{H}}(z)$.

The effective equation (2.39) for the stationary black hole can be still
interpreted by a soap-film equation as one for the static case [9] 333 This
interpretation of the effective equation (2.39) is due to Roberto Emparan. We
thank him for sharing it with us. . The membrane (soap-film) between two
fluids satisfies the following equation for the mean curvature $K$ of the
membrane (see eq. (2.15) in [21])

$\displaystyle K=\frac{\gamma T\Delta s-\Delta\rho}{\sigma},$ (2.42)

where $\sigma$ is the surface tension, $T$ is the temperature, $\Delta s$ and
$\Delta\rho$ are the differences between the entropy density and the energy
density of the fluids at both sides of the membrane. $\gamma$ is the same
redshift factor as in eq. (1.2): in eq. (2.14) in [21], one just has to
extract $\xi^{2}=-g_{tt}$. If we now consider a case in which $\Delta\rho$ is
negligible, then the equation (2.42) becomes

$\displaystyle\gamma^{-1}K=\frac{T\Delta s}{\sigma},$ (2.43)

so if we identify $2\kappa=T\Delta s/\sigma$, then this is the same as eq.
(2.39) (of course in this identification $T$, $\Delta s$, and $\sigma$ are not
independently determined, instead only their combination into $\kappa$ is
fixed).

There are two remarks on the metric (2.36) at $\Sigma_{B}$. One is that the
metric (2.36) does not need to have a component of $g_{t\phi}$. The metric
(2.36) describes the background metric where the solution is embedded, and it
does not need to be rotating itself although the solution is rotating. Thus
the metric (2.36) does not need to have $g_{t\phi}$ component. Another remark
is about the static limit. The static limit corresponds to the zero horizon
angular velocity $\Omega_{H}=0$. Then the metric (2.36) requires

$\displaystyle\mathsf{V}_{0}(z)^{2}=4\hat{\kappa}^{2}r_{0}(z)^{2}$ (2.44)

at the static limit. One may think that the appearance of this additional
condition is curious. However one can immediately find that the condition
(2.44) is nothing but the effective equation of general static solutions [9]

$\displaystyle\sqrt{-g_{tt}}K\Big{|}_{\Sigma_{B}}=2\kappa,$ (2.45)

by using eqs. (2.37) and (2.38). This embedding equation can be obtained also
by the static limit of eq. (2.39). So there is no appearance of new additional
condition at the static limit.

The embedding in the Minkowski background has a constant redshift factor as

$\displaystyle\mathsf{V}_{0}(z)=1.$ (2.46)

For the static embedding in this Minkowski background we have only one unique
embedding by a round sphere corresponding to the Schwarzschild black hole [9,
22]. This fact is consistent with the uniqueness theorem of the static
solution [23]. On the other hand, however, the embedding of the stationary
solution allows various embeddings as seen below. This is due to the
appearance of new degree of freedom in the horizon deformation described by
$\Omega_{\text{H}}\mathcal{R}_{\phi}$ in the effective equation.

####  2.3.1 Ellipsoidal embedding

As one application of the effective equation (2.39) we consider an ellipsoidal
embedding in the Minkowski background. We embed the leading order solution by
$r=r(\theta)$ in the Minkowski background using the ellipsoidal coordinate.
The flat metric in the ellipsoidal coordinate is

$\displaystyle
ds^{2}=-dt^{2}+\frac{r^{2}+a^{2}\cos^{2}{\theta}}{r^{2}+a^{2}}dr^{2}+(r^{2}+a^{2}\cos^{2}{\theta})d\theta^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(r^{2}+a^{2})\sin^{2}{\theta}d\phi^{2}+r^{2}\cos^{2}{\theta}d\Omega^{2}_{D-4}.$
(2.47)

$a$ is a parameter describing the oblateness of the ellipsoidal. We set

$\displaystyle\hat{\kappa}=\frac{1}{2},$ (2.48)

by using the normalization of near zone $t$-coordinate for simplicity. The
relation $z$ and $\theta$ can be seen from eqs. (2.36) and (2.47) with
$r=r(\theta)$ as

$\displaystyle\frac{dz}{d\theta}=\sqrt{\frac{(\mathcal{R}_{\text{H}}(\theta)^{2}+a^{2}\cos^{4}{\theta})(\mathcal{R}_{\text{H}}(\theta)^{2}+(\mathcal{R}^{\prime}_{\text{H}}(\theta)+\tan{\theta}\mathcal{R}_{\text{H}}(\theta))^{2}+a^{2}\cos^{2}{\theta})}{\cos^{2}{\theta}(\mathcal{R}_{\text{H}}(\theta)^{2}+a^{2}\cos^{2}{\theta})}},$
(2.49)

where we used the relation

$\displaystyle r(\theta)\cos{\theta}=\mathcal{R}_{\text{H}}(\theta)$ (2.50)

in the leading order solution. Then the effective equation (2.39) becomes

$\displaystyle\Omega_{H}^{2}\tan^{2}{\theta}(a^{2}\cos^{4}{\theta}+\mathcal{R}_{\text{H}}(\theta)^{2}+\cos{\theta}\sin{\theta}\mathcal{R}_{\text{H}}(\theta)\mathcal{R}^{\prime}_{\text{H}}(\theta))^{2}(a^{2}\cos^{2}{\theta}+\mathcal{R}_{\text{H}}(\theta)^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}+(\mathcal{R}_{\text{H}}(\theta)^{2}-1)(a^{2}\cos^{4}{\theta}+\mathcal{R}_{\text{H}}(\theta)^{2})^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}+2\cos{\theta}\sin{\theta}\mathcal{R}_{\text{H}}(\theta)\mathcal{R}^{\prime}_{\text{H}}(\theta)(\mathcal{R}_{\text{H}}(\theta)^{2}-1)(a^{2}\cos^{4}{\theta}+\mathcal{R}_{\text{H}}(\theta)^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}+\cos^{2}{\theta}\mathcal{R}_{\text{H}}(\theta)^{2}\mathcal{R}^{\prime}_{\text{H}}(\theta)^{2}(a^{2}\cos^{4}{\theta}+\mathcal{R}_{\text{H}}(\theta)^{2}-\sin^{2}{\theta})=0.$
(2.51)

We solve this equation for $\mathcal{R}_{\text{H}}(\theta)$ in some cases
below.

##### Myers-Perry solution

The Myers-Perry solution is described by the embedding

$\displaystyle\mathcal{R}_{\text{H}}(\theta)=\cos{\theta},~{}~{}\Omega_{H}=\frac{a}{1+a^{2}},$
(2.52)

in the ellipsoidal embedding for arbitrary $a$.

##### Bumpy black holes

The Myers-Perry black hole has marginally stable quasinormal modes $\omega=0$
in the ultraspinning region [24, 25, 26]. The marginally stable modes suggest
the existences of new deformed solution branching off the Myers-Perry solution
branch. These solutions, so called bumpy black holes, have been constructed
numerically in [17, 18]. Here we consider this solution by using the effective
equation (2.51). To study this we perform the perturbative analysis of the
effective equation (2.51) around the Myers-Perry black hole. We expand the
embedding function $\mathcal{R}_{\text{H}}(\theta)$ around the Myers-Perry
black hole as

$\displaystyle\mathcal{R}_{\text{H}}(\theta)=\cos{\theta}\Bigl{[}1+\hat{\mathcal{R}}(\theta)\epsilon+O\left(\epsilon^{2}\right)\Bigr{]},$
(2.53)

where

$\displaystyle\epsilon=\Omega_{H}-\frac{a}{1+a^{2}}.$ (2.54)

Then, perturbing eq. (2.51) with respect to $\epsilon$, we find the
perturbative solution

$\displaystyle\hat{\mathcal{R}}(\theta)=\frac{a(1+a^{2})^{2}\sin^{2}{\theta}}{(1-a^{2})(1+a^{2}\cos^{2}{\theta})}+A\frac{\sin^{1+a^{2}}{\theta}}{1+a^{2}\cos^{2}{\theta}},$
(2.55)

where $A$ is an integration constant describing the perturbation amplitude.
The first part describes just the trivial perturbation adding the angular
momentum along the Myers-Perry black hole solution branch. The second part is
the non-trivial deformation of the solution into the bumpy black hole solution
branch. For general $a$ and $A\neq 0$, we have a non-analytic behavior in the
solution at $\theta=0$. The regularity of the solution at $\theta=0$ requires
444 Here the regularity means that any derivatives of
$\hat{\mathcal{R}}(\theta)$ should be finite at $\theta=0$ and
$\hat{\mathcal{R}}(\theta)$ be an even function around $\theta=0$. To satisfy
this requirement $a$ should be discretized by $k$ as given in eq. (2.56). As
we can see in appendix C this regularity condition at $\theta=0$ is equivalent
to the condition for the spheroidal harmonics at large $D$. Furthermore this
discretization parameter $k$ corresponds to the angular momentum number
$\ell$.

$\displaystyle a_{c}^{2}=2k-1,~{}~{}~{}k=1,2,3,...$ (2.56)

For $k=1$, i.e., $a_{c}=1$, the solution is same with the first part of eq.
(2.55), so it corresponds to the Myers-Perry black hole with a different
angular momentum. The value $a_{c}$ for $k=1$ is known analytically in all $D$
by the thermodynamic argument [24] as

$\displaystyle\left(\frac{a}{r_{0}}\right)^{D-3}=\frac{D-3}{2(D-4)}\left(\frac{D-3}{D-5}\right)^{(D-5)/2}.$
(2.57)

$r_{0}$ is the horizon radius of the Myers-Perry black hole (see Appendix A).
At the large $D$ limit the formula (2.57) gives same value with eq. (2.56) for
$k=1$. For $k>1$ we have non-trivial bumpy black hole solutions by eq. (2.55).
Eq. (2.56) gives the threshold angular momentum of the ultraspinning
instability of the Myers-Perry black hole. In Table 1 we show the comparison
of eq. (2.56) with numerical results in [26]. Actually we have good agreements
within the expected error $O(1/D)$. This threshold angular momentum will be
reproduced also by observing the quasinormal mode frequency directly in
section 4.

In Appendix A we generalize the above analysis to the AdS Myers-Perry black
hole. Then we obtain the threshold angular momentum for the ultraspinning
instability of the AdS Myers-Perry black hole in the $1/D$ expansion.

| $a_{c}|_{k=1}$ | $a_{c}|_{k=2}$ | $a_{c}|_{k=3}$ | $a_{c}|_{k=4}$
---|---|---|---|---
eq. (2.56) | $1$ | $\sqrt{3}=1.732$ | $\sqrt{5}=2.236$ | $\sqrt{7}=2.646$
numerical $(D=6)$ | 1.097 | $1.572$ | $1.849$ | $2.036$
numerical $(D=7)$ | 1.075 | $1.714$ | $2.141$ | $2.487$
numerical $(D=8)$ | 1.061 | $1.770$ | $2.275$ | $2.725$
numerical $(D=9)$ | 1.051 | $1.792$ | $2.337$ | $2.807$
numerical $(D=10)$ | 1.042 | $1.795$ | $2.361$ | $2.855$
numerical $(D=11)$ | 1.035 | $1.798$ | $2.373$ | $2.879$
Table 1: Comparison of our analytic formula (2.56) for the threshold of the
instability of the Myers-Perry black hole with numerical results in [26]. The
analytic formula shows quite good agreements with numerical results within the
expected error $O(1/D)$. The value of $a_{c}$ for $k=1$ by [26] has perfect
agreements with eq. (2.57).

###  2.4 Boost representation

One important property of the leading order solution in near zone is its local
Lorentz boost representation. The effective equation (2.39) has the
interpretation that the embedded surface $\Sigma_{B}$ is the locally Lorentz
boosted membrane. This local Lorentz boost property holds also for the metric
itself in near zone. To see this we rewrite the $(t,\phi)$ part of the
obtained leading order metric in near zone as

$\displaystyle
ds^{2}_{(t,\phi)}=-\mathsf{V}_{0}(z)^{2}dt^{2}+\frac{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}{\Omega_{H}^{2}}d\phi^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{4\hat{\kappa}^{2}r_{0}(z)^{2}\cosh^{2}{\rho}}\left(\mathsf{V}_{0}(z)^{2}dt-\frac{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}^{2}}{\Omega_{H}}d\phi\right)^{2}.$
(2.58)

By introducing the boost parameter $\sigma(z)$ defined by

$\displaystyle\cosh{\sigma(z)}=\frac{\mathsf{V}_{0}(z)}{2\hat{\kappa}r_{0}(z)}$
(2.59)

and new local frame $dx^{p}$ by

$\displaystyle
dx^{p}(z)=\left(\mathsf{V}_{0}(z)dt,~{}\frac{\sqrt{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}}{\Omega_{H}}d\phi\right),$
(2.60)

the leading order metric is written in a very simple form as

$\displaystyle
ds^{2}=\frac{4r_{0}(z)^{2}}{n^{2}}d\rho^{2}+\left(\eta_{pq}+\frac{u_{p}u_{q}}{\cosh^{2}{\rho}}\right)dx^{p}dx^{q}+dz^{2}+\mathcal{R}_{\text{H}}(z)^{2}d\Omega^{2}_{D-4},$
(2.61)

where $\eta_{pq}$ is the flat metric on two dimensional spacetime. The fluid
velocity $u_{p}$ is

$\displaystyle
u_{p}dx^{p}=\mathsf{V}_{0}(z)\cosh{\sigma(z)}dt-\frac{\sqrt{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}}{\Omega_{H}}\sinh{\sigma(z)}d\phi.$
(2.62)

Note that the leading order metric of the Schwarzschild black hole is (see
Appendix A)

$\displaystyle
ds^{2}=\frac{4}{n^{2}}d\rho^{2}-\tanh^{2}{\rho}~{}dt^{2}+dz^{2}+\sin^{2}{\theta}d\phi^{2}+\cos^{2}{\theta}d\Omega^{2}_{D-4}.$
(2.63)

Thus the stationary solution can be represented as the locally boosted
Schwarzschild black hole under the following boost transformation

$\displaystyle
dt\rightarrow\mathsf{V}_{0}(z)dt\cosh{\sigma(z)}-\frac{\sqrt{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)^{2}}}{\Omega_{H}}d\phi\sinh{\sigma(z)},$
(2.64) $\displaystyle
d\phi\rightarrow\frac{\sqrt{\mathsf{V}_{0}(z)^{2}-4\hat{\kappa}^{2}r_{0}(z)}}{\Omega_{H}}d\phi\cosh{\sigma(z)}-\mathsf{V}_{0}(z)dt\sinh{\sigma(z)}.$
(2.65)

This boost representation is known for the Myers-Perry black hole [3, 6].
However our leading order solution covers much wider class of solutions
including the asymptotically AdS black holes. This boost representation is the
universal feature for the large $D$ stationary black holes under our ansatz.

The boost parameter, $\sigma(z)$, is not constant and the boost transformation
is inhomogeneous in $z$-direction. This $z$-dependence also appears as the
horizon radius inhomogeneity in $g_{\rho\rho}$. Thus the identification with
the Schwarzschild black hole by the boost transformation is valid only locally
in $z$-direction. However this inhomogeneity along $z$-direction is not so
crucial since the dynamics along $z$-direction is sub-dominant compared with
the radial dynamics. This boost transformation is the property of the radial
dynamics, which is the leading order dynamics of large $D$ black holes. The
local effect by $z$-dependence would be introduced at the sub-leading order in
$1/D$ expansion as corrections. Such $z$-dependent dynamics in the higher
order structure of the $1/D$ expansion contains the essential effects by the
rotation and horizon topology.

##  3 Perturbation

As one interesting application of the boost property in the leading order
solution we consider the perturbation problem. By using the boost
transformation we can obtain the solution of the perturbation equation of the
stationary black hole from one of the Schwarzschild black hole. In the
following we investigate the decoupled mode perturbation, whose the frequency
is $\omega=O(D^{0})$ [7], of the asymptotically flat black hole given by

$\displaystyle\mathsf{V}_{0}(z)=1,$ (3.1)

in the leading order metric in section 2. The extension to the asymptotically
AdS black hole is straightforward.

###  3.1 Schwarzschild black hole

We study the perturbation on the Schwarzschild black hole. Next we perform the
boost transformation on the perturbation solutions to obtain the perturbation
solution of the stationary solution. There are various ways in the
representation of the perturbation around the Schwarzschild black hole [27,
28]. Here we fix the gauge for the perturbation, $h_{\mu\nu}$, to solve the
perturbation equation. Especially we use the transverse traceless gauge as

$\displaystyle\hat{\nabla}^{\mu}h_{\mu\nu}=0,~{}~{}\hat{g}^{\mu\nu}h_{\mu\nu}=0,$
(3.2)

where $\hat{\nabla}_{\mu}$ is the covariant derivative of the $D$ dimensional
background metric $\hat{g}_{\mu\nu}$. Then the perturbation equation becomes
equivalent to the Lichnerowicz equation given by

$\displaystyle\hat{\nabla}^{\rho}\hat{\nabla}_{\rho}h_{\mu\nu}+2\hat{R}_{\mu\rho\nu\sigma}h^{\rho\sigma}=0,$
(3.3)

where $\hat{R}_{\mu\nu\rho\sigma}$ is a background Riemann tensor. The
explicit components of the perturbation equation are complicate so we do not
show them here. The background spacetime is the $D=n+3$ dimensional
Schwarzschild black hole given by

$\displaystyle
ds^{2}=-f(r)^{-1}dt^{2}+f(r)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}+\cos^{2}{\theta}d\Omega^{2}_{D-4}\right),$
(3.4)

where

$\displaystyle f(r)=1-\left(\frac{r_{0}}{r}\right)^{n}\equiv
1-\frac{1}{\mathsf{R}}.$ (3.5)

Here we introduced the new radial coordinate $\mathsf{R}=(r/r_{0})^{n}$. This
radial coordinate $\mathsf{R}$ is related with the radial coordinate $\rho$
which we used in the previous section by

$\displaystyle\mathsf{R}=\cosh^{2}{\rho}.$ (3.6)

The perturbation is decomposed into the scalar, vector and tensor type with
respect to $S^{D-4}$ in the metric 555 Note that the usual scalar, vector and
tensor type perturbation of the Schwarzschild black hole is based on the
decomposition on $S^{D-2}$ [27]. Then, for example, our scalar type
perturbation on $S^{D-4}$ consists of the scalar, vector and tensor type
perturbation on $S^{D-2}$. . In this paper we investigate the scalar and
vector type perturbation on $S^{D-4}$. Using the vielbein

$\displaystyle
e^{(0)}_{\text{Sch}}=\sqrt{f(r)}dt,~{}e^{(1)}_{\text{Sch}}=\frac{dr}{\sqrt{f(r)}},~{}e^{(2)}_{\text{Sch}}=rd\theta,~{}e^{(3)}_{\text{Sch}}=r\sin{\theta}d\phi,~{}e^{(A)}_{\text{Sch}}=r\cos{\theta}\hat{e}^{(A)},$
(3.7)

where $\hat{e}^{(A)}$ is a vielbein on the unit sphere $S^{D-4}$, the scalar
type perturbation, $h^{(S)}_{\mu\nu}$, on $S^{D-4}$ can be written by

$\displaystyle h^{(S)}_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f^{(S)\text{Sch}}_{IJ}\mathbb{Y}^{(S)}_{j}e^{(I)}_{\text{Sch}}e^{(J)}_{\text{Sch}}+f^{(S)\text{Sch}}_{I}\mathsf{D}_{A}\mathbb{Y}^{(S)}_{j}e^{(I)}_{\text{Sch}}e^{(A)}_{\text{Sch}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}\cos^{2}{\theta}(H^{(S)\text{Sch}}_{T}\mathbb{Y}^{(S)}_{j}\gamma_{AB}+H^{(S)\text{Sch}}_{L}\mathbb{Y}^{(S)j}_{AB})dx^{A}dx^{B}\Bigr{]},$
(3.8)

where $I,J=0,1,2,3$. $\mathbb{Y}^{(S)}_{j}$ is the scalar harmonics on
$S^{D-4}$ satisfying

$\displaystyle(\mathsf{D}^{2}+\lambda_{S})\mathbb{Y}^{(S)}_{j}=0,$ (3.9)

where the eigenvalue $\lambda_{S}$ is given by

$\displaystyle\lambda_{S}=j(j+D-5).$ (3.10)

The scalar derived tensor harmonics $\mathbb{Y}^{(S)j}_{AB}$ is defined by

$\displaystyle\mathbb{Y}^{(S)j}_{AB}=\mathsf{D}_{A}\mathsf{D}_{B}\mathbb{Y}^{(S)}_{j}+\frac{\lambda_{S}}{D-4}\mathbb{Y}^{(S)}_{j}q_{AB}.$
(3.11)

The vector type perturbation, $h^{(V)}_{\mu\nu}$, on $S^{D-4}$ is given by

$\displaystyle h^{(V)}_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f^{(V)\text{Sch}}_{I}\mathbb{Y}^{(V)j}_{A}e^{(I)}e^{(A)}+H^{(V)}_{L}\mathbb{Y}^{(V)j}_{AB}e^{(A)}e^{(B)}\Bigr{]},$
(3.12)

where $\mathbb{Y}^{(V)j}_{A}$ and $\mathbb{Y}^{(V)j}_{AB}$ are the vector
harmonics on $S^{D-4}$ defined by

$\displaystyle(\mathsf{D}^{2}+\lambda_{V})\mathbb{Y}^{(V)j}_{A}=0,~{}~{}\mathsf{D}^{A}\mathbb{Y}^{(V)j}_{A}=0,~{}~{}\mathbb{Y}^{(V)j}_{AB}=\mathsf{D}_{(A}\mathbb{Y}^{(V)j}_{B)},$
(3.13)

with the eigenvalue $\lambda_{V}$ given by

$\displaystyle\lambda_{V}=j(j+D-5)-1.$ (3.14)

The perturbation equation is a coupled PDE system for scalar and vector type
perturbations. At large $D$ the situation is changed and the perturbation
equations become decoupled ODE system. We can find decoupled perturbation
variables $F^{(S)\text{Sch}}_{IJ}$ for the scalar type perturbation easily as

$\displaystyle
F^{(S)\text{Sch}}_{00}=f^{(S)\text{Sch}}_{00}+f^{(S)\text{Sch}}_{11},~{}~{}F^{(S)\text{Sch}}_{11}=f^{(S)\text{Sch}}_{00}-f^{(S)\text{Sch}}_{11},~{}~{}F^{(S)\text{Sch}}_{01}=f^{(S)\text{Sch}}_{01},$
$\displaystyle
F^{(S)\text{Sch}}_{02}=f^{(S)\text{Sch}}_{02},~{}~{}F^{(S)\text{Sch}}_{03}=f^{(S)\text{Sch}}_{03},~{}~{}F^{(S)\text{Sch}}_{12}=f_{12}^{(S)\text{Sch}},~{}~{}F^{(S)\text{Sch}}_{13}=f_{13}^{(S)\text{Sch}},$
$\displaystyle
F^{(S)\text{Sch}}_{22}=f_{22}^{(S)\text{Sch}},~{}~{}F^{(S)\text{Sch}}_{23}=f_{23}^{(S)\text{Sch}},~{}~{}F^{(S)\text{Sch}}_{33}=f_{33}^{(S)\text{Sch}},$
(3.15)

and

$\displaystyle
F^{(S)\text{Sch}}_{I}=f^{(S)\text{Sch}}_{I},~{}~{}F_{T,L}^{(S)\text{Sch}}=H^{(S)\text{Sch}}_{T,L}.$
(3.16)

The vector type perturbation has following decoupled perturbation variables at
large $D$

$\displaystyle
F^{(V)\text{Sch}}_{I}=f^{(V)\text{Sch}}_{I},~{}~{}F^{(V)\text{Sch}}_{L}=H^{(V)\text{Sch}}_{L}.$
(3.17)

Then each decoupled perturbation variables are expanded at large $D$ as

$\displaystyle F^{(S)\text{Sch}}_{ab}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(S)\text{Sch}}_{ab}}{n^{k}},~{}F^{(S)\text{Sch}}_{a}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(S)\text{Sch}}_{a}}{n^{k}},~{}F_{T,L}^{(S)\text{Sch}}=\sum_{k\geq
0}\frac{{}^{(k)}F_{T,L}^{(S)\text{Sch}}}{n^{k}},$ (3.18)

for the scalar type perturbation and

$\displaystyle F^{(V)\text{Sch}}_{I}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(V)\text{Sch}}_{I}}{n^{k}},~{}F_{L}^{(V)\text{Sch}}=\sum_{k\geq
0}\frac{{}^{(k)}F_{L}^{(V)\text{Sch}}}{n^{k}},$ (3.19)

for the vector type perturbation. By using transverse traceless gauge
conditions we can obtain $F^{(S)\text{Sch}}_{I}$, $F_{T,L}^{(S)\text{Sch}}$
and $F^{(V)\text{Sch}}_{L}$ from $F^{(S)\text{Sch}}_{IJ}$ and
$F^{(V)\text{Sch}}_{I}$. Thus the equations to be solved are the equations of
$F^{(S)\text{Sch}}_{IJ}$ for the scalar type perturbation and
$F^{(V)\text{Sch}}_{I}$ for the vector type perturbation. The perturbation
equations for $F_{IJ}^{(S)\text{Sch}}$ are

$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{00}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{00}^{(S)\text{Sch}}}{\mathsf{R}-1}={}^{(k)}\mathcal{S}^{(S)}_{00},$
(3.20)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{01}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{01}^{(S)\text{Sch}}}{\mathsf{R}-1}={}^{(k)}\mathcal{S}^{(S)}_{01},$
(3.21)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{02}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{02}^{(S)\text{Sch}}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(S)}_{02},$
(3.22)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{03}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{03}^{(S)\text{Sch}}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(S)}_{03},$
(3.23)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{11}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{11}^{(S)\text{Sch}}}{\mathsf{R}}={}^{(k)}\mathcal{S}^{(S)}_{11},$
(3.24)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{12}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{12}^{(S)\text{Sch}}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(S)}_{12},$
(3.25)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{13}^{(S)\text{Sch}}-\frac{{}^{(k)}F_{13}^{(S)\text{Sch}}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(S)}_{13},$
(3.26)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{22}^{(S)\text{Sch}}={}^{(k)}\mathcal{S}^{(S)}_{22},$
(3.27)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{23}^{(S)\text{Sch}}={}^{(k)}\mathcal{S}^{(S)}_{23},$
(3.28)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F_{33}^{(S)\text{Sch}}={}^{(k)}\mathcal{S}^{(S)}_{33}.$
(3.29)

The $k-$th order source term ${}^{(k)}\mathcal{S}^{(S)}_{ab}$ is coming from
the lower order solutions and the source term does not have the decoupling
property. The perturbation variables, ${}^{(k)}F^{(S)\text{Sch}}_{IJ}$, depend
on $\mathsf{R}$ and $\theta$ 666 The Schwarzschild black hole has a spherical
symmetry. So the perturbation equation can be reduced to the ordinarily
differential equation in principle. Here, for the usefulness in the following
discussion, we use the harmonics on $S^{D-4}$ and the perturbation equation
becomes the apparent partial differential equation. . However the perturbation
equation becomes ordinarily differential equation with respect to $\mathsf{R}$
at large $D$. Furthermore each $k$-th order variables
${}^{(k)}F_{IJ}^{(S)\text{Sch}}$ are decoupling and we can solve the solution
by the straightforward integration procedure at each order. The vector type
perturbation has the same property. The perturbation equation for
$F^{(V)\text{Sch}}_{I}$ is

$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)\text{Sch}}_{0}-\frac{{}^{(k)}F^{(V)\text{Sch}}_{0}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(V)}_{0},$
(3.30)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)\text{Sch}}_{1}-\frac{{}^{(k)}F^{(V)\text{Sch}}_{1}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(V)}_{1},$
(3.31)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)\text{Sch}}_{2}={}^{(k)}\mathcal{S}^{(V)}_{2},$
(3.32)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)\text{Sch}}_{3}={}^{(k)}\mathcal{S}^{(V)}_{3}.$
(3.33)

The leading order solution satisfying the regularity condition on the horizon
and at infinity is given by

$\displaystyle{}^{(0)}F_{00}^{(S)\text{Sch}}=\frac{A(\theta)}{\mathsf{R}-1},~{}{}^{(0)}F_{01}^{(S)\text{Sch}}=\frac{A(\theta)}{\mathsf{R}-1},~{}{}^{(0)}F_{02}^{(S)\text{Sch}}=\frac{B(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
(3.34)
$\displaystyle{}^{(0)}F_{03}^{(S)\text{Sch}}=\frac{C(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}{}^{(0)}F_{11}^{(S)\text{Sch}}=\frac{D(\theta)}{\mathsf{R}},~{}{}^{(0)}F_{12}^{(S)\text{Sch}}=\frac{B(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
(3.35)
$\displaystyle{}^{(0)}F_{13}^{(S)\text{Sch}}=\frac{C(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}{}^{(0)}F_{22}^{(S)\text{Sch}}=0,~{}{}^{(0)}F_{23}^{(S)\text{Sch}}=0,~{}{}^{(0)}F_{33}^{(S)\text{Sch}}=0,$
(3.36)

for the scalar type perturbation and

$\displaystyle{}^{(0)}F_{0}^{(V)\text{Sch}}=\frac{W(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}{}^{(0)}F_{1}^{(V)\text{Sch}}=\frac{W(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},{}^{(0)}F_{2}^{(V)\text{Sch}}={}^{(0)}F_{3}^{(V)\text{Sch}}=0,$
(3.37)

for the vector type perturbation. The leading order perturbation solution
contains undetermined integration functions $A(\theta)$, $B(\theta)$,
$C(\theta)$, $D(\theta)$ and $W(\theta)$ for each type perturbation. If we go
to the higher order of $k$ we can obtain the non-trivial relation between
integration functions. Especially the solution at $k=1$ and $k=2$ order gives
quasinormal mode frequency of the Schwarzschild black hole for the vector and
scalar type perturbation respectively. The detail analysis and results of this
will be given in section 4 and Appendix D.

###  3.2 Stationary solution

We apply the boost transformation to the perturbation solution of the
Schwarzschild black hole. At first, before taking large $D$ limit, we give the
general procedure for the perturbation of the stationary black hole solution
given by

$\displaystyle
ds^{2}=-A(\rho,z)^{2}dt^{2}+B(\rho.z)^{2}d\rho^{2}+F(\rho,z)^{2}(d\phi-w(\rho,z)dt)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G(\rho,z)^{2}dz^{2}+H(\rho,z)^{2}d\Omega^{2}_{D-4}.$
(3.38)

We consider the scalar and vector type perturbation on $S^{D-4}$. To give
perturbation variables we use the following vielbein

$\displaystyle
e^{(0)}=A(\rho,z)dt,~{}~{}e^{(1)}=B(\rho,z)d\rho,~{}~{}e^{(2)}=F(\rho,z)(d\phi-w(\rho,z)dt)$
(3.39)

and

$\displaystyle e^{(3)}=G(\rho,z)dz,~{}~{}e^{(A)}=H(\rho,z)\hat{e}^{(A)},$
(3.40)

where $\hat{e}^{(A)}$ is the vielbein on the unit sphere $S^{D-4}$ again. Then
the perturbed metric $h_{\mu\nu}$ is given by the components with respect to
these vielbeins as

$\displaystyle h^{(S)}_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f^{(S)}_{IJ}\mathbb{Y}^{(S)}_{j}e^{(I)}e^{(J)}+f^{(S)}_{I}\mathsf{D}_{A}\mathbb{Y}^{(S)}_{j}e^{(I)}e^{(A)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+H(\rho,z)^{2}(H^{(S)}_{T}\mathbb{Y}^{(S)}_{j}q_{AB}+H^{(S)}_{L}\mathbb{Y}^{(S)j}_{AB})dx^{A}dx^{B}\Bigr{]},$
(3.41)

for scalar type perturbation and

$\displaystyle h^{(V)}_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f^{(V)}_{I}\mathbb{Y}^{(V)j}_{A}e^{(I)}e^{(A)}+H^{(V)}_{L}\mathbb{Y}^{(V)j}_{AB}e^{(A)}e^{(B)}\Bigr{]},$
(3.42)

for the vector type perturbation. Then we perform the boost transformation on
these perturbed metrics of the leading order solution obtained in section 2.
The boost transformation gives the relation between vielbeins as

$\displaystyle
e^{(0)}_{\text{Sch}}\rightarrow\frac{2-r_{0}(z)^{2}(1-\sinh^{2}{\rho})}{r_{0}(z)^{2}\cosh{\rho}\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}e^{(0)}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}e^{(2)},$
(3.43) $\displaystyle
e^{(2)}_{\text{Sch}}\rightarrow\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}e^{(2)}-\frac{2+r_{0}(z)^{2}\sinh^{2}{\rho}}{r_{0}\sinh{\rho}}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)\sinh^{2}{\rho}}}e^{(0)},$
(3.44)

where we used the vielbein of the Schwarzschild black hole (3.7) and of the
stationary solution (3.39) for the leading order metric (2.61) 777 We consider
the asymptotically flat black hole $\mathsf{V}_{0}(z)=1$. . This relation
between vielbeins gives also the relation between the perturbation variables
of the Schwarzschild black hole and general stationary solutions. Since we
know the decoupled perturbation variables on the Schwarzschild black hole, we
can obtain the following decoupled perturbation variables, $F^{(S)}_{IJ}$,
$F^{(S)}_{I}$, $F^{(S)}_{T,L}$, $F^{(V)}_{I}$ and $F^{(V)}_{L}$ of the general
stationary solution at large $D$ as

$\displaystyle
F^{(S)}_{00}=\frac{\cosh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{00}-\frac{2\sinh{\rho}\cosh{\rho}\sqrt{1-r_{0}(z)^{2}}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{02}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(1-r_{0}(z)^{2})\sinh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{22}+f^{(S)}_{11},$
(3.45) $\displaystyle
F^{(S)}_{11}=\frac{\cosh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{00}-\frac{2\sinh{\rho}\cosh{\rho}\sqrt{1-r_{0}(z)^{2}}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{02}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(1-r_{0}(z)^{2})\sinh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{22}-f^{(S)}_{11},$
(3.46) $\displaystyle
F^{(S)}_{02}=\frac{\cosh^{2}{\rho}-(1-r_{0}(z)^{2})\sinh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{02}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\sqrt{1-r_{0}(z)^{2}}\sinh{\rho}\cosh{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}(f^{(S)}_{00}+f^{(S)}_{02}),$
(3.47) $\displaystyle
F^{(S)}_{22}=\frac{\cosh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{22}-\frac{2\sinh{\rho}\cosh{\rho}\sqrt{1-r_{0}(z)^{2}}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{02}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(1-r_{0}(z)^{2})\sinh^{2}{\rho}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}f^{(S)}_{00},$
(3.48) $\displaystyle
F^{(S)}_{01}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{01}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{12},$
(3.49) $\displaystyle
F^{(S)}_{12}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{12}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{01},$
(3.50) $\displaystyle
F^{(S)}_{03}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{03}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{23},$
(3.51) $\displaystyle
F^{(S)}_{23}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{23}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{03},$
(3.52) $\displaystyle
F^{(S)}_{0}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{0}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{2},$
(3.53) $\displaystyle
F^{(S)}_{2}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{2}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(S)}_{0},$
(3.54)

and

$\displaystyle
F^{(S)}_{13}=f^{(S)}_{13},~{}~{}F^{(S)}_{33}=f^{(S)}_{33},~{}~{}F^{(S)}_{1}=f^{(S)}_{1},~{}~{}F^{(S)}_{3}=f^{(S)}_{3},~{}~{}F^{(S)}_{T,L}=H^{(S)}_{T,L},$
(3.55)

for the scalar type perturbation and

$\displaystyle
F^{(V)}_{0}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{0}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{2},$
(3.56) $\displaystyle
F^{(V)}_{2}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{2}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{0},$
(3.57)

and

$\displaystyle~{}F^{(V)}_{1}=f^{(V)}_{1},~{}~{}F^{(V)}_{3}=f^{(V)}_{3},~{}~{}F^{(V)}_{L}=H^{(V)}_{L},$
(3.58)

for the vector type perturbation. Each perturbation variables are expanded at
large $D$ as

$\displaystyle F^{(S)}_{IJ}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(S)}_{IJ}}{n^{k}},~{}F^{(S)}_{I}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(S)}_{I}}{n^{k}},~{}F^{(S)}_{T,L}=\sum_{k\geq
0}\frac{{}^{(k)}F_{T,L}^{(S)}}{n^{k}},$ (3.59)

and

$\displaystyle F^{(V)}_{I}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(V)}_{I}}{n^{k}},~{}F^{(V)}_{T,L}=\sum_{k\geq
0}\frac{{}^{(k)}F_{L}^{(V)}}{n^{k}}.$ (3.60)

Then the perturbation equation for these decoupling variables can be obtained
from the Lichnerowicz equation under the transverse traceless gauge condition.
Now we have obtained the leading order metric of the stationary black hole in
the $1/D$ expansion in section 2. Then the perturbation equation for $k=0$
order is also obtained by using the leading order metric. The perturbation
equation of the decoupling variables can be found to satisfy same equations
with one of the Schwarzschild black hole at the leading order by changing the
radial coordinate from $\rho$ to $\mathsf{R}$ by 888 This relation can be
understood by the horizon position. The Schwarzschild black hole has the
horizon at $\mathsf{R}=1$ and the general stationary black hole horizon
position is set to be $\rho=0$.

$\displaystyle\mathsf{R}=\cosh^{2}{\rho}.$ (3.61)

At higher orders the perturbation equation of the stationary solution becomes
different from the boosted one of the Schwarzschild black hole in source
terms. This is because the boost relation is valid only in the leading order
metric, and higher order structure gives the essential effect coming from the
difference from the Schwarzschild black holes. However the homogeneous part of
the equation at each order is always same with one of the Schwarzschild black
hole. So, in principle, the perturbation equation is a set of decoupled
ordinarily differential equations and we can solve it by a straightforward
integration method with the Green’s function.

The regular leading order solution of the perturbation equation for the
stationary black hole is

$\displaystyle{}^{(0)}F_{00}^{(S)}=\frac{A(\theta)}{\mathsf{R}-1},~{}{}^{(0)}F_{01}^{(S)}=\frac{A(\theta)}{\mathsf{R}-1},~{}{}^{(0)}F_{02}^{(S)}=\frac{B(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
(3.62)
$\displaystyle{}^{(0)}F_{03}^{(S)}=\frac{C(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}{}^{(0)}F_{11}^{(S)}=\frac{D(\theta)}{\mathsf{R}},~{}{}^{(0)}F_{12}^{(S)}=\frac{B(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
(3.63)
$\displaystyle{}^{(0)}F_{13}^{(S)}=\frac{C(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}{}^{(0)}F_{22}^{(S)}=0,~{}{}^{(0)}F_{23}^{(S)}=0,~{}{}^{(0)}F_{33}^{(S)}=0,$
(3.64)

for the scalar type perturbation and

$\displaystyle{}^{(0)}F_{0}^{(V)}=\frac{W(\theta)}{\mathsf{R}},~{}{}^{(0)}F_{1}^{(V)}=\frac{W(\theta)}{\mathsf{R}},{}^{(0)}F_{2}^{(V)}={}^{(0)}F_{3}^{(V)}=0,$
(3.65)

for the vector type perturbation. In the derivation of the perturbation
equations and solutions we need only the leading order solution of general
stationary solutions given in section 2. To go to more higher order
perturbation equations, the higher order solutions of stationary solutions are
required. In the next section we consider the higher order perturbation
equation of the Myers-Perry black hole, which is a known exact solution.

Note that the quasinormal mode frequency of the decoupled mode perturbation is
obtained by solving higher order perturbation equation in $1/D$. The higher
order structure of the perturbation equation reflects the higher order
structure of the background metric, which does not have the boost property.
Hence the frequency does not have the boost property either. This implies that
the quasinormal mode frequency of the Myers-Perry black hole cannot be
obtained by the boost transformation from one of the Schwarzschild black hole.
This non-boost property allows the existence of the instability mode in the
decoupled sector of the perturbation of the Myers-Perry black hole. In
contrast the quasinormal mode of the non-decoupled sector perturbation is
obtained in the leading order structure and we expect that such mode has the
boost property as the universal feature [5].

##  4 QNMs of Myers-Perry black hole

We pursue the decoupling property of perturbation variables at large $D$
expansion shown in the previous section to more higher order structure to
obtain the quasinormal mode (QNM) frequency of the $D=n+3$ dimensional singly
rotating Myers-Perry black hole. This was calculated numerically in [29] for
non-axisymmetric perturbations. Here we give the analytic formula of the QNM
frequency both for the axisymmetric and non-axisymmetric perturbations. The
metric of the Myers-Perry black hole is given in the Appendix A and it is
written by the following parameter setting

$\displaystyle\mathsf{V}_{0}(z)=1,~{}~{}\hat{\kappa}=\frac{1}{2},~{}~{}\Omega_{H}=\frac{a}{1+a^{2}},~{}~{}R(\theta)=\cos{\theta},~{}~{}r_{0}(\theta)^{-2}=\frac{1+a^{2}}{1+a^{2}\cos^{2}{\theta}},$
(4.1)

in the leading order metric obtained in section 2. Our detail analysis in this
section concentrates on the scalar type perturbation since the analysis of the
vector type perturbation is performed in the same manner. The only result of
the QNM frequency for the vector type perturbation will be given later. The
scalar type perturbation variables are defined in the same way with one in
section 3 as

$\displaystyle h_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f_{ab}\mathbb{Y}^{(S)}_{j}e^{(a)}e^{(b)}+f_{a}\mathsf{D}_{A}\mathbb{Y}^{(S)}_{j}e^{(a)}e^{(A)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}\cos^{2}{\theta}(H_{T}\mathbb{Y}^{(S)}_{j}q_{AB}+H_{L}\mathbb{Y}^{(S)j}_{AB})dx^{A}dx^{B}\Bigr{]},$
(4.2)

where the vielbein was defined in section 3 999 We omit the index (S)
representing the scalar type perturbation in section 3. . So the perturbation
is parameterized by $\omega$, $m$ and $j$. $j$ describes the deformation
parameter on $S^{D-4}$. Thus $j=0$ correspond to the perturbation without the
deformation on $S^{D-4}$. Then the decoupled perturbation variables can be
constructed and expanded at large $D$ as

$\displaystyle F_{IJ}=\sum_{k\geq
0}\frac{F^{(k)}_{IJ}}{n^{k}},~{}~{}F_{a}=\sum_{k\geq
0}\frac{F^{(k)}_{a}}{n^{k}},~{}~{}F_{T,L}=\sum_{k\geq
0}\frac{F_{T,L}^{(k)}}{n^{k}},$ (4.3)

where $D=n+3$. The boundary condition for the perturbation on the horizon is
the ingoing boundary condition [29] as

$\displaystyle
h_{\mu\nu}(r,\theta)=\hat{h}_{\mu\nu}(r,\theta)(r-r_{+})^{-i(\omega-m\Omega_{H})/2\kappa},$
(4.4)

where $r_{+}$ is the horizon position and $\kappa$ is a surface gravity. The
component of $\hat{h}_{\mu\nu}$ in the Eddington-Finkelstein ingoing
coordinate should be regular on the horizon. Here we omit the $t$, $\phi$ and
$x^{A}$ dependence just for the simplicity 101010 $x^{A}$ is a coordinate on
$S^{D-4}$. . The boundary condition at infinity $\mathsf{R}\gg 1$ is the
decoupled mode condition [7] as

$\displaystyle h_{\mu\nu}=O(\mathsf{R}^{-1}).$ (4.5)

This condition is equivalent to the outgoing wave boundary condition at
infinity for the decoupled mode defined by $\omega r_{0}=O(D^{0})$.

In the following we give the brief summary of perturbation solutions of the
scalar type perturbation up to $k=2$ order. At $k=2$ we obtain QNM frequency
for the first time.

##### Leading order solution $(k=0)$

The leading order solution satisfying the boundary conditions is

$\displaystyle
F_{00}^{(0)}=\frac{A_{(0)}(\theta)}{\mathsf{R}-1},~{}F_{01}^{(0)}=\frac{A_{(0)}(\theta)}{\mathsf{R}-1},~{}F_{02}^{(0)}=\frac{B_{(0)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
$\displaystyle
F_{03}^{(0)}=\frac{C_{(0)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}F_{11}^{(0)}=\frac{D_{(0)}(\theta)}{\mathsf{R}},~{}F_{12}^{(0)}=\frac{B_{(0)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},$
(4.6) $\displaystyle
F_{13}^{(0)}=\frac{C_{(0)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}F_{22}^{(0)}=0,~{}F_{23}^{(0)}=0,~{}F_{33}^{(0)}=0.$

At leading order we have four independent integration functions, which cannot
be determined by the boundary condition. The solution for $F^{(0)}_{I}$ and
$F^{(0)}_{T,L}$ can be obtained by the transverse traceless gauge condition,
and we omit them since it is not so important.

##### Next to leading order solution $(k=1)$

The next to leading order solution of $F_{IJ}$ is represented by

$\displaystyle
F_{00}^{(1)}=\frac{A_{(1)}(\theta)}{\mathsf{R}-1}+\hat{F}_{00}^{(1)},~{}F_{01}^{(1)}=\frac{A_{(1)}(\theta)}{\mathsf{R}-1}+\hat{F}_{01}^{(1)},~{}$
$\displaystyle
F_{02}^{(1)}=\frac{B_{(1)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+\hat{F}_{02}^{(1)},~{}F_{03}^{(1)}=\frac{C_{(1)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+\hat{F}_{03}^{(1)},~{}$
$\displaystyle
F_{11}^{(1)}=\frac{D_{(1)}(\theta)}{\mathsf{R}}+\hat{F}_{11}^{(1)},~{}F_{12}^{(1)}=\frac{B_{(1)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+\hat{F}_{12}^{(1)},$
$\displaystyle
F_{13}^{(1)}=\frac{C_{(1)}(\theta)}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+\hat{F}_{13}^{(1)},~{}F_{22}^{(1)}=\hat{F}_{22}^{(1)},~{}F_{23}^{(1)}=0,~{}F_{33}^{(1)}=\hat{F}_{33}^{(1)},$
(4.7)

where the solution with the hat comes from the integration of the source term
composed by the leading order solution. At this order we have new four
integration functions $A_{(1)}(\theta)$, $B_{(1)}(\theta)$, $C_{(1)}(\theta)$
and $D_{(1)}(\theta)$. To satisfy boundary conditions we obtain some non-
trivial relations for $A_{(0)}(\theta)$, $B_{(0)}(\theta)$, $C_{(0)}(\theta)$
and $D_{(0)}(\theta)$. Conditions we find are

$\displaystyle B_{(0)}(\theta)=0,~{}~{}C_{(0)}(\theta)=0,$ (4.8)

and

$\displaystyle\cos{\theta}(1+a^{2}\cos^{2}{\theta})\left((1+a^{2})\cos{\theta}A_{(0)}(\theta)-\sin{\theta}D_{(0)}^{\prime}(\theta)\right)$
$\displaystyle~{}~{}~{}~{}+\bigl{[}j+(-1+iam+a^{2}(-3+2j-i\omega)-i\omega)\cos^{2}{\theta}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+a^{2}(1+iam+a^{2}(j-1-i\omega)-i\omega)\cos^{4}{\theta}\bigr{]}D_{(0)}(\theta)=0.$
(4.9)

Using this condition we can eliminate $A_{(0)}(\theta)$. To obtain QNM
frequency we should specify the function $D_{(0)}(\theta)$, and we can do it
at next to next to leading order.

##### Next-to-next-to leading order solution $(k=2)$

At this order we can obtain the solutions in the same way with the next to
leading order. We find additional non-trivial condition for integration
functions. By eliminating $A_{(0)}(\theta)$, the additional conditions are
written as

$\displaystyle
c_{1}(\theta)C_{(1)}^{\prime\prime}(\theta)+c_{2}(\theta)C_{(1)}^{\prime}(\theta)+c_{3}(\theta)C_{(1)}(\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c_{4}(\theta)D_{(0)}^{\prime\prime}(\theta)+c_{5}(\theta)D_{(0)}^{\prime}(\theta)+c_{6}(\theta)D_{(0)}(\theta)=0,$
(4.10)

and

$\displaystyle
d_{1}(\theta)D_{(0)}^{\prime\prime\prime}(\theta)+d_{2}(\theta)D_{(0)}^{\prime\prime}(\theta)+d_{3}(\theta)D_{(0)}^{\prime}(\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+d_{4}(\theta)D_{(0)}(\theta)+d_{5}(\theta)C_{(1)}^{\prime}(\theta)+d_{6}(\theta)C_{(1)}(\theta)=0.$
(4.11)

Each functions $c_{i}(\theta)$ and $d_{i}(\theta)$ have messy forms so we do
not show them here. They are given in the Appendix E. In general it seems that
we should solve these differential equations to obtain the QNM frequency.
However, we find that the QNM frequency can be obtained only from the boundary
conditions 111111 Actually we can confirm that eqs. (4.10) and (4.11) can be
solved regularly under the QNM frequency obtained below. Thus there may be
some non-trivial structure in the solutions of eqs. (4.10) and (4.11) and the
boundary conditions.  . At $\theta=\pi/2$ the boundary condition is read from
eqs (4.10) and (4.11) as

$\displaystyle
C_{(1)}(\theta)|_{\theta=\pi/2}=\cos^{j-3}{\theta},~{}~{}D_{(0)}(\theta)|_{\theta=\pi/2}=\cos^{j}{\theta}.$
(4.12)

On the other hand the behavior at $\theta=0$ is

$\displaystyle
C_{(1)}(\theta)|_{\theta=0}=\sin^{\delta^{S}_{\omega}-1}{\theta},~{}~{}D_{(0)}(\theta)|_{\theta=0}=\sin^{\delta^{S}_{\omega}}{\theta},$
(4.13)

where $\delta^{S}_{\omega}$ depends on $a$, $m$, $j$ and $\omega$. To see
which value $\delta^{S}_{\omega}$ should take, we observe the spherical
harmonics, $\mathbb{S}^{\ell}(\theta)e^{im\phi}\mathbb{Y}^{(S)}_{j}$, on
$S^{D-2}$, defined by

$\displaystyle\bigl{[}\Delta_{S^{D-2}}+\ell(\ell+D-3)\bigr{]}\mathbb{S}^{\ell}e^{im\phi}\mathbb{Y}^{(S)}_{j}$
$\displaystyle=\Biggl{[}\frac{1}{\sin{\theta}\cos^{D-4}{\theta}}\frac{d}{d\theta}\left(\sin{\theta}\cos^{D-4}{\theta}\frac{d}{d\theta}\mathbb{S}^{\ell}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{m^{2}}{\sin^{2}{\theta}}\mathbb{S}^{\ell}-\frac{j(j+D-5)}{\cos^{2}{\theta}}\mathbb{S}^{\ell}+\ell(\ell+D-3)\mathbb{S}^{\ell}\Biggr{]}e^{im\phi}\mathbb{Y}^{(S)}_{j}=0.$
(4.14)

The solution of this equation can be written by the hypergeometric function as

$\displaystyle\mathbb{S}^{\ell}=\cos^{j}{\theta}~{}\sin^{|m|}{\theta}~{}{}_{2}F_{1}(-k_{S},k_{S}+|m|+j+n/2,|m|+1;\sin^{2}{\theta}),$
(4.15)

where $k_{S}$ is a non-negative integer satisfying

$\displaystyle\ell=j+|m|+2k_{S}.$ (4.16)

At large $D$ this solution becomes

$\displaystyle\mathbb{S}^{\ell}$ $\displaystyle=$
$\displaystyle\sin^{\ell-j}{\theta}\cos^{j}{\theta}+O(1/D)$ (4.17)
$\displaystyle=$
$\displaystyle\sin^{|m|+2k_{S}}{\theta}\cos^{j}{\theta}+O(1/D).$

$k_{S}$ describes the overtone number of $\mathbb{S}^{\ell}$ along $\theta$
direction, and this overtone number can be observed from the behavior of
$\mathbb{S}^{\ell}$ at $\theta=0$ at large $D$ limit as seen eq. (4.17). From
this observation on the spherical harmonics we impose the following condition
on $\delta^{S}_{\omega}$ as the harmonics condition

$\displaystyle\delta^{S}_{\omega}$ $\displaystyle=$ $\displaystyle\ell-j$
(4.18) $\displaystyle=$ $\displaystyle 2k_{s}+|m|,$

where $\ell$ is parametrized as eq. (4.16) by non-negative integer $k_{S}$.
Then eqs. (4.10) and (4.11) under the condition (4.18) give one non-trivial
algebraic condition on $\omega$. This non-trivial condition on $\omega$ can be
regarded as the QNM condition. In the following we solve the condition eq.
(4.18) for $\omega$ in some cases. The explicit form of the QNM condition
(4.18) is given in the Appendix F.

Note that the condition (4.18) is equivalent to the regularity condition to
derive eq. (2.56) in section 2. The condition (4.18) is suggested by the
analysis for spherical harmonics at large $D$. However, as we can see in
Appendix C, the analysis of the spheroidal harmonics also gives same condition
(4.18). Thus we expect that the condition (4.18) would hold also in the
gravitational perturbation of the Myers-Perry black hole.

###  4.1 QNM frequency

We show some explicit results of the QNM frequency of the scalar and vector
type perturbations by solving the QNM condition (4.18). The instability exists
only in the scalar type perturbation and the vector type perturbation is
always stable for the decoupled sector perturbation 121212 The instability
mode is conjectured to exist only in the decoupled sector (saturated mode)
[29]. The tensor type perturbation of the singly rotating Myers-Perry black
hole does not have the decoupled sector and it was shown to be stable [31]. .
We set the horizon radius to unity as

$\displaystyle r_{0}=1,$ (4.19)

by fixing the unit.

####  4.1.1 Scalar type perturbation

The QNM frequency is given by the solution of the algebraic equation (4.18),
which can be solved analytically. We solve eq. (4.18) for some cases below.
Since the $j\neq 0$ modes do not show any instability, we consider only $j=0$
modes. The perturbation with $j\neq 0$ describes the deformation of $S^{D-4}$
of the Myers-Perry black hole. So we might be able to say that the
axisymmetric instability occurs only in the ”S-wave” sector in the
perturbation similar to the Gregory-Laflamme instability of the black brane
[32, 33].

Then, for this mode, the quasinormal mode condition (4.18) can be written in a
relatively simple form as

$\displaystyle(1+a^{2})^{3}\omega^{3}-(1+a^{2})^{2}(4i-3\ell+3am)\omega^{2}$
$\displaystyle~{}~{}~{}~{}+(1+a^{2})(-4+7\ell-3\ell^{2}-6iam(\ell-1)+a^{2}-4+3\ell+3m^{2})\omega$
$\displaystyle~{}~{}~{}~{}+(-i\ell^{3}+\ell^{2}(3i+ia^{2}+3am)-\ell(2i+5am+a^{3}m-ia^{2}(-2+3m^{2}))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+am(2-2iam-a^{2}(m^{2}-2)))=0.$
(4.20)

At $a=0$ the scalar type perturbation has the physical degree of freedom for
$\ell\geq 2$ [30] and, hence, we study the modes of $\ell\geq 2$.

##### Schwarzschild black hole

At $a=0$ where the solution is the Schwarzschild black hole we can solve eq.
(4.20) explicitly by

$\displaystyle\omega=\pm\sqrt{\ell-1}-i(\ell-1),~{}~{}\omega=-i\ell.$ (4.21)

These modes correspond to the decoupled scalar and vector type mode of
Schwarzschild black hole on $S^{D-2}$ [7].

##### Axisymmetric perturbation

Next we consider the axisymmetric perturbation $m=0$ with $a\neq 0$. In this
perturbation we find that there are stationary perturbations $\omega=0$ at
$a_{c}$ where

$\displaystyle a^{2}_{c}$ $\displaystyle=$ $\displaystyle\ell-1$ (4.22)
$\displaystyle=$ $\displaystyle 2k_{S}-1,$

and we used eq. (4.16) with $j=m=0$. The assumption of $\ell\geq 2$ implies
$k_{S}\geq 1$ for $j=m=0$, and this result reproduces the result (2.56)
derived from the effective theory obtained in section 2. In figure 1 we give
the plot of the QNM frequency with $(\ell,m,j)=(4,0,0)$. The black thick line,
purely imaginary frequency mode, shows the instability mode and it becomes the
marginally stable mode at $a^{2}_{c}=3$.

Figure 1: The QNM frequency for $(\ell,m,j)=(4,0,0)$. The black thick line is
the pure imaginary mode and it shows the instability at $a^{2}>3$. The gray
thick line is the stable mode.

For the axisymmetric perturbation with $\omega=0$ and $j=0$ we can solve eqs.
(4.10) and (4.11) explicitly by

$\displaystyle
C_{0}(\theta)=0,~{}~{}D_{0}(\theta)=\tilde{D}_{0}\frac{\sin^{1+a^{2}}{\theta}}{1+a^{2}\cos^{2}{\theta}},$
(4.23)

where $\tilde{D}_{0}$ is a constant. Then we can see that this stationary
solution is equivalent to the perturbation solution (2.55) obtained in the
effective theory. This equivalence suggests the validity of our QNM condition
(4.18).

##### Non-axisymmetric perturbation

The non-axisymmetric mode perturbation also shows the instability at
$a^{2}>a^{2}_{c}$ where $a_{c}$ is given in eq. (4.22) as

$\displaystyle a^{2}_{c}$ $\displaystyle=$ $\displaystyle\ell-1$ (4.24)
$\displaystyle=$ $\displaystyle|m|+2k_{S}-1.$

At the critical rotation the instability mode satisfies the superradiance
condition as

$\displaystyle\omega|_{a=a_{c}}=\frac{a_{c}m}{1+a^{2}_{c}}=m\Omega_{H}(a_{c}).$
(4.25)

Hence the dynamical instability mode shows the superradiance instability at
the same time. This coincidence of the onsets of dynamical and superradiant
instability has been also observed in the QNMs of the Myers-Perry black hole
with equal spin [6] up to $O(1/D)$ correction. The numerical results [34] also
suggest the coincidence for the equal spin case. On the other hand the
numerical analysis of QNMs of the singly rotating Myers-Perry black hole does
not show such coincidence [29, 35]: the superradiant instability appears at
slower rotation than the critical rotation of the dynamical instability. Thus
our coincidence is just the property only at large $D$ limit. If one considers
$1/D$ corrections of QNMs of singly rotating Myers-Perry black hole, there
would be a difference between the onset of the superradiant and dynamical
instability.

For the bar mode defined by $m=\ell$, we can solve eq. (4.20) explicitly by

$\displaystyle\omega_{\pm}=\frac{\pm\sqrt{m-1}+a(m-1)}{1+a^{2}}-i\frac{m-1\mp
a\sqrt{m-1}}{1+a^{2}},~{}~{}\omega_{(0)}=\frac{am}{1+a^{2}}-i\frac{m}{1+a^{2}}.$
(4.26)

$\omega_{+}$ shows the instability at $a>a_{c}$ and saturates the
superradiance condition at $a=a_{c}$. In figure 2 we show the plot of the bar
mode QNM frequency for $m=2$.

Figure 2: The plot of the bar mode QNM frequency of $(\ell,m,j)=(2,2,0)$ is
shown. The black dashed line is the superradiance condition. The onset of the
dynamically unstable mode is same as the one of the superradiance instability.

The QNM frequency for the bar mode was obtained numerically in [29]. The
behavior of the frequency shows good agreements with our analytic results.

The non-axisymmetric perturbation with $\ell>m$ also shows the instability.
But its origin is different from one of the bar mode. The instability mode of
the bar mode perturbation comes from the scalar type perturbation of the
Schwarzschild black hole at $a=0$ where the frequency is complex (see eq.
(4.21)). On the other hand non-axisymmetric mode with $\ell>m$ has an
instability in the mode coming from the vector type perturbation of the
Schwarzschild black hole, whose frequency is a pure imaginary as (4.21). This
is same situation with the perturbation of the Myers-Perry black hole with
equal spin [6]. In figure 3 we give the plot of QNM frequency for $\ell=4$,
$m=2$. As we can see, the mode showing instability becomes purely imaginary at
$a=0$, which corresponds to the vector type perturbation of the Schwarzschild
black hole.

Figure 3: The plot of the QNM frequency of $(\ell,m,j)=(4,2,0)$ is shown. The
black dashed line is the superradiance condition. The gray thick line is the
stable mode $\omega_{-}$, while the black thick line shows the unstable mode
$\omega_{+}$. The dynamical instability appears exactly when the superradiance
condition is satisfied.

####  4.1.2 Vector type perturbation

The QNM frequency of the vector type perturbation can be given as

$\displaystyle\omega^{V}=\omega^{V}_{\text{LO}}+\frac{\omega^{V}_{\text{NLO}}}{n}+O(1/n^{2}),$
(4.27)

where

$\displaystyle\omega^{V}_{\text{LO}}=\frac{am}{1+a^{2}}-i\left(j-1+\frac{\ell-j}{1+a^{2}}\right),$
(4.28)

and

$\displaystyle\omega^{V}_{\text{NLO}}=\frac{2am}{1+a^{2}}\Bigl{(}\ell-1+a^{2}(j-1)+\log{(1+a^{2})}\Bigr{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-i\Biggl{(}(j-1)^{2}+\frac{(1+a^{2})(\ell-j)(\ell+j-2)-2a^{2}m^{2}}{(1+a^{2})^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left(j-1-\frac{\ell-j}{1+a^{2}}+\frac{2(\ell-j)}{(1+a^{2})^{2}}\right)\log{(1+a^{2})}\Biggr{)}.$
(4.29)

We have solved the perturbation equation up to $k=2$ order and obtained the
QNM frequency up to $1/D$ correction. This QNM frequency is obtained by
imposing the same harmonics condition with the scalar type perturbation on the
behavior at $\theta=0$ motivated by eq. (4.17). The QNM frequency (4.28)
reduces to the vector type perturbation on $S^{D-2}$ of the Schwarzschild
black hole [7] at $a=0$ as

$\displaystyle\omega=-i(\ell-1)\left(1+\frac{\ell-1}{n}+O(1/n^{2})\right).$
(4.30)

##  5 Summary

The large $D$ expansion method has been found to be useful to solve the
gravitational problem as shown in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In this
paper we have performed the next step for the further developing of the large
$D$ expansion method by constructing the effective theory of the large $D$
stationary black hole in asymptotically flat or AdS spacetime. Considering the
additional degree of freedom of the horizon, rotation radius of the black
hole, the effective theory becomes non-trivial and allows the existence of
various solutions compared to the static case in the asymptotically flat
spacetime which has only one unique solution, Schwarzschild black hole [9].
This new degree of freedom appears as the Lorentz boost of the black hole
along the rotational direction, and we can rewrite the solution of the
effective equation in very simple and illuminative expression by using this
boost property.

As applications of the effective theory we have considered the ellipsoidal
embeddings and obtained the Myers-Perry black hole and bumpy black hole
solutions in perturbative manner. Then we have succeeded to derive the
threshold angular momentum of the instability of the Myers-Perry black hole.
The bumpy black hole construction and perturbation analysis of the Myers-Perry
black hole had been investigated numerically. These numerical studies of the
solution and perturbations need very advanced and sophisticated technique.
However the effective theory obtained in this paper is rather simple equation
and we can obtain the solution easily.

As another application we have obtained the quasinormal mode frequencies of
the singly rotating Myers-Perry black hole. Our effective theory is only for
the stationary black hole. But, if we consider the dynamics with $\omega
r_{0}=O(D^{0})$, the time-dependence becomes sub-dominant compared with the
radial dynamics. As a result we could solve the perturbation equation as the
ordinarily differential equation with respect to the radial direction.
Furthermore the leading order solution of our effective theory has a very
important property, the boost property. All stationary black hole solution is
represented as the boost transformation of the Schwarzschild black hole. Thus
various things which Schwarzschild black hole possess are common with
stationary black holes. Using this useful feature we have solved the
perturbation equation of the singly rotating Myers-Perry black hole explicitly
and obtained the quasinormal mode condition, which is just an algebraic
equation, analytically. Although originally the perturbation equation is the
partial differential equation system, the large radial gradient and boost
property of the large $D$ Myers-Perry black hole reduce the equation to the
analytically solvable ordinarily differential equation.

We can consider many directions of the extension of our work. In this paper we
consider the solution with only one angular momentum in vacuum. Thus it is
interesting to include more angular momentum, matter fields such as a gauge
field or some compact dimensions into the effective theory of stationary black
holes. These inclusions would give further additional degree of freedom for
the deformation of the horizon. So its dynamics becomes much richer. Another
direction of the extension is the more detail investigation of the effective
theory of stationary black hole by searching other solution such as black ring
solutions and going to more higher order nonlinear solutions. By constructing
such solution we can draw the phase diagram of black holes analytically and
compare with numerical results. Actually the higher order investigation of the
effective theory for the static solution [9] allows very interesting phenomena
such as the existence of the critical dimension of the non-uniform black
string [36].

## Acknowledgments

The authors are very grateful to Roberto Emparan and Hideo Kodama for valuable
comments on the draft and useful discussions. KT was supported by JSPS Grant-
in-Aid for Scientific Research No.26-3387.

##  Appendix A Singly rotating AdS Myers-Perry black hole

In this appendix we give the metric of the AdS Myers-Perry black hole and its
large $D$ limit in our notation. The $D=n+3$ dimensional singly rotating AdS
Myers-Perry black hole is given by [15, 16] 131313 We change the $\phi$
coordinate by  $\displaystyle\phi\rightarrow\phi-\frac{a}{L^{2}}dt$ from [15,
16] to satisfy the boundary condition (2.13).

$\displaystyle
ds^{2}=-A(r,\theta)^{2}dt^{2}+B(r,\theta)^{2}d\hat{\mathsf{R}}^{2}+F(r,\theta)^{2}(d\phi-w(r,\theta)dt)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G(r,\theta)^{2}d\theta^{2}+r^{2}\cos^{2}{\theta}d\Omega^{2}_{D-4},$
(A.1)

where

$\displaystyle
A(r,\theta)^{2}=\frac{\Delta_{\hat{\mathsf{R}}}\Delta_{\theta}(r^{2}+L^{2})(r^{2}+a^{2}\cos^{2}{\theta})}{L^{2}\Xi\Delta_{\hat{\mathsf{R}}}(r^{2}+a^{2}\cos^{2}{\theta})+r^{2}L^{2}\Delta_{\theta}(r^{2}+a^{2})},$
(A.2) $\displaystyle
B(r,\theta)^{2}=\frac{r^{2}}{n^{2}\hat{\mathsf{R}}^{2}}\frac{\hat{\mathsf{R}}(r^{2}+a^{2}\cos^{2}{\theta})}{\Delta_{\hat{\mathsf{R}}}},$
(A.3) $\displaystyle
F(r,\theta)^{2}=\sin^{2}{\theta}\left(\frac{r^{2}+a^{2}}{\Xi}+\frac{a^{2}r^{2}\sin^{2}{\theta}}{\Xi^{2}(r^{2}+a^{2}\cos^{2}{\theta})\hat{\mathsf{R}}}\right),$
(A.4) $\displaystyle
w(r,\theta)=\frac{ar^{2}\Delta_{\theta}}{L^{2}\Xi\Delta_{\hat{\mathsf{R}}}(r^{2}+a^{2}\cos^{2}{\theta})+r^{2}L^{2}\Delta_{\theta}(r^{2}+a^{2})},$
(A.5)

and

$\displaystyle
G(r,\theta)^{2}=\frac{r^{2}+a^{2}\cos^{2}{\theta}}{\Delta_{\theta}}.$ (A.6)

Here we defined

$\displaystyle\hat{\mathsf{R}}=\left(\frac{r}{r_{0}}\right)^{n},~{}~{}\Xi=1-\frac{a^{2}}{L^{2}},$
(A.7)

and following functions

$\displaystyle\Delta_{\hat{\mathsf{R}}}=r^{2}-(r^{2}+a^{2})\left(1+\frac{r^{2}}{L^{2}}\right)\hat{\mathsf{R}},~{}~{}\Delta_{\theta}=1-\frac{a^{2}}{L^{2}}\cos^{2}{\theta}.$
(A.8)

The horizon position, $r=r_{+}$, is defined by the positive real root of
$\Delta_{\hat{\mathsf{R}}}$ by

$\displaystyle\Delta_{\hat{\mathsf{R}}}(r_{+})=0.$ (A.9)

$L$ is the AdS curvature and the metric is the solution of the Einstein
equation with a cosmological constant

$\displaystyle\Lambda=-\frac{(D-1)(D-2)}{L^{2}}.$ (A.10)

At the large $D$ limit the metric is simplified to 141414 In the following we
set $r_{0}=1$.

$\displaystyle
ds^{2}=\frac{d\mathsf{R}^{2}}{n^{2}}\frac{L^{2}(1+a^{2}\cos^{2}{\theta})}{(1+a^{2})(1+L^{2})\mathsf{R}(\mathsf{R}-1)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\left(1-\frac{\cosh^{2}{\sigma(\theta)}}{\mathsf{R}}\right)d\hat{t}^{2}+\left(1+\frac{\sinh^{2}{\sigma(\theta)}}{\mathsf{R}}\right)d\hat{\phi}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{2\sinh{\sigma(\theta)}\cosh{\sigma(\theta)}}{\mathsf{R}}d\hat{t}~{}d\hat{\phi}+\frac{1+a^{2}\cos^{2}{\theta}}{\Delta_{\theta}}d\theta^{2}+\cos^{2}{\theta}d\Omega^{2}_{D-4},$
(A.11)

where we introduced

$\displaystyle\mathsf{R}=\frac{(1+a^{2})(1+L^{2})}{L^{2}}\hat{\mathsf{R}},~{}~{}\cosh^{2}{\sigma(\theta)}=\frac{(1+a^{2})\Delta_{\theta}}{(1+a^{2}\cos^{2}{\theta})\Xi}.$
(A.12)

$d\hat{t}$ and $d\hat{\phi}$ are defined by

$\displaystyle
d\hat{t}^{2}=\frac{(1+L^{2})\Delta_{\theta}}{L^{2}\Xi}dt^{2},~{}~{}d\hat{\phi}^{2}=\frac{1+a^{2}}{\Xi}\sin^{2}{\theta}d\phi^{2}.$
(A.13)

At the large $D$ limit the horizon is located on

$\displaystyle\mathsf{R}\Big{|}_{r=r_{+}}=1+O(1/n).$ (A.14)

This solution can be reproduced by the leading order solution in section 2 by

$\displaystyle\mathsf{V}_{0}(\theta)^{2}=\frac{(1+L^{2})\Delta_{\theta}}{L^{2}\Xi},~{}~{}\mathcal{R}_{\text{H}}(z)=\cos{\theta},$
(A.15)

and

$\displaystyle\frac{dz}{d\theta}=\sqrt{\frac{1+a^{2}\cos^{2}{\theta}}{\Delta_{\theta}}},~{}~{}\mathsf{R}=\cosh^{2}\rho.$
(A.16)

The surface gravity and horizon angular velocity at the leading order of large
$D$ limit are

$\displaystyle\hat{\kappa}=\frac{1}{2}\frac{1+L^{2}}{L^{2}},~{}~{}\Omega_{H}=\frac{a}{1+a^{2}}\frac{1+L^{2}}{L^{2}}.$
(A.17)

The metric (A.11) becomes the large $D$ metric of the Schwarzschild black hole
[3]

$\displaystyle
ds^{2}=\frac{4}{n^{2}}d\rho^{2}-\tanh^{2}{\rho}~{}dt^{2}+dz^{2}+\sin^{2}{\theta}d\phi^{2}+\cos^{2}{\theta}d\Omega^{2}_{D-4},$
(A.18)

when $a=0$ and $1/L=0$. We changed the coordinate by
$\mathsf{R}=\cosh^{2}{\rho}$.

##### Structure in $1/D$ correction

Let us consider the $1/D$ correction in $g_{tt}$ of the AdS Myers-Perry black
hole at $a=0$. Using

$\displaystyle r=1+\frac{\log{\hat{\mathsf{R}}}}{n}+O(1/n^{2})$ (A.19)

we find that $g_{tt}$ becomes

$\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle-A(r,\theta)^{2}$ (A.20)
$\displaystyle=$ $\displaystyle
1+\frac{1}{L^{2}}-\frac{1}{\hat{\mathsf{R}}}+\frac{2\log{\hat{\mathsf{R}}}}{nL^{2}}+O(1/n^{2}).$

At large $\mathsf{R}$ $g_{tt}$ has a linear term in $\rho$ since the relation
between $\rho$ and $\hat{\mathsf{R}}$ is

$\displaystyle\hat{\mathsf{R}}=\frac{L^{2}}{1+L^{2}}\cosh^{2}{\rho}.$ (A.21)

Thus, in general, the higher order corrections in $A(\rho,\theta)$ cannot be
assumed to be damping exponentially at the overlap region $n\gg\rho\gg 1$ in
the presence of the cosmological constant as mentioned in section 2.

###  A.1 Ellipsoidal embedding

The AdS-Myers-Perry black hole is described as the solution of the effective
equation (2.39) for the ellipsoidal embedding in the AdS background. The
ellipsoidal coordinate in AdS is given by 151515 The coordinate transformation
from the usual AdS coordinate to the ellipsoidal coordinate is [15]
$\displaystyle
r^{2}\sin^{2}{\theta}\rightarrow\Xi^{-1}(r^{2}+a^{2})\sin^{2}{\theta},~{}~{}r^{2}\cos^{2}{\theta}\rightarrow
r^{2}\cos^{2}{\theta}.$

$\displaystyle
ds^{2}=-\frac{\Delta_{\theta}(L^{2}+r^{2})}{L^{2}\Xi}dt^{2}+\frac{L^{2}(r^{2}+a^{2}\cos^{2}{\theta})}{(r^{2}+a^{2})(L^{2}+r^{2})}dr^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{r^{2}+a^{2}\cos^{2}{\theta}}{\Delta_{\theta}}d\theta^{2}+\frac{(r^{2}+a^{2})\sin^{2}{\theta}}{\Xi}d\phi^{2}+r^{2}\cos^{2}{\theta}d\Omega^{2}_{D-4}.$
(A.22)

The ellipsoidal embedding is defined by

$\displaystyle
r=r(\theta)=\frac{\mathcal{R}_{\text{H}}(\theta)}{\cos{\theta}},$ (A.23)

and the leading order metric of the AdS Myers-Perry black hole in the $1/D$
expansion is reproduced by $r(\theta)=1$ when we fix the horizon position by
$r_{+}=1$ with the surface gravity and horizon angular velocity given in eq.
(A.17). We consider the perturbation around this AdS Myers-Perry black hole
embedding. As done in section 2 the obtained perturbed solution would give the
bumpy black hole solution in AdS in the perturbative manner. At first we fix
the surface gravity to

$\displaystyle\hat{\kappa}=\frac{1+L^{2}}{2L^{2}},$ (A.24)

by using the time coordinate normalization in near zone. The perturbed
embedding is

$\displaystyle\mathcal{R}_{\text{H}}(\theta)=\cos{\theta}\Bigl{[}1+\epsilon\hat{\mathcal{R}}(\theta)+O\left(\epsilon^{2}\right)\Bigr{]},$
(A.25)

where

$\displaystyle\epsilon=\Omega_{H}-\frac{a}{1+a^{2}}\frac{1+L^{2}}{L^{2}}.$
(A.26)

Then, perturbing eq. (2.39) with respect to $\epsilon$, we find the
perturbative solution

$\displaystyle\hat{\mathcal{R}}(\theta)=\frac{a(1+a^{2})^{2}\sin^{2}{\theta}}{\Xi(1-a^{2})(1+a^{2}\cos^{2}{\theta})}+A\frac{(\sin{\theta})^{\delta_{\text{MP}}}\Delta_{\theta}^{1-\delta_{\text{MP}}/2}}{1+a^{2}\cos^{2}{\theta}},$
(A.27)

where

$\displaystyle\delta_{\text{MP}}=\frac{(1+a^{2})(L^{2}-1)}{L^{2}\Xi}.$ (A.28)

This perturbation solution becomes regular at $\theta=0$ when $a=a_{c}$ given
by

$\displaystyle a^{2}_{c}=\frac{(2k-1)L^{2}+1}{L^{2}+2k-1}$ (A.29)

where $k$ is a positive integer. At $L\rightarrow\infty$ eq. (A.29) reproduces
the threshold angular momentum (2.56). Hence we expect that eq. (A.29) gives
the threshold angular momentum of the AdS Myers-Perry black hole and branching
angular momentum for the bumpy black hole in AdS. It is interesting to check
this result by solving the perturbation equation of the AdS Myers-Perr black
hole and obtaining the QNM frequency as done for the Myers-Perry black hole in
section 4.

##  Appendix B Next-to-leading order analysis

Here we briefly give results for next-to-leading order solutions. In
particular we will see how the constancy condition of the surface gravity and
horizon angular velocity, eq. (2.30), can be obtained.

Our $D=n+3$ dimensional metric ansatz is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{N^{2}(\rho,z)}{n^{2}}d\rho^{2}-A(\rho,z)^{2}dt^{2}+F(\rho,z)^{2}(d\phi-W(\rho,z)dt)^{2}$
(B.1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G(\rho,z)^{2}dz^{2}+H(\rho,z)^{2}q_{AB}dx^{A}dx^{B}.$

We solve the Einstein equations for metric functions and the extrinsic
curvature on $\rho=\text{constant}$ surface defined by

$\displaystyle K^{a}_{~{}b}=\frac{n}{2N(\rho,z)}g^{ac}\partial_{\rho}g_{cb}.$
(B.2)

The Einstein equations are decomposed on $\rho=\text{constant}$ surface as

$\displaystyle-R+K^{2}-K^{a}_{~{}b}K^{b}_{~{}a}-\frac{(D-1)(D-2)}{L^{2}}=0,$
(B.3) $\displaystyle\nabla_{a}K^{a}_{~{}b}-\nabla_{b}K=0,$ (B.4)
$\displaystyle\frac{n}{N}\partial_{\rho}K^{a}_{~{}b}=-KK^{a}_{~{}b}+R^{a}_{~{}b}+\frac{D-1}{L^{2}}\delta^{a}_{~{}b}-\frac{1}{N}\nabla^{a}\nabla_{b}N.$
(B.5)

The leading order solutions which are regular on the horizon are

$\displaystyle
A(\rho,z)^{2}=\frac{A_{0}(z)^{2}F_{0}(z)^{2}\tanh^{2}{\rho}}{F_{0}(z)^{2}-A_{0}(z)^{2}C_{t\phi}(z)^{2}\tanh^{2}{\rho}},$
(B.6) $\displaystyle
F(\rho,z)^{2}=F_{0}(z)^{2}-A_{0}(z)C_{t\phi}(z)^{2}\tanh^{2}{\rho},$ (B.7)
$\displaystyle
W(\rho,z)=\frac{F_{0}(z)^{2}(1-C_{tt}(z))+A_{0}(z)^{2}C_{t\phi}(z)^{2}C_{tt}(z)\tanh^{2}{\rho}}{K_{t\phi}(z)F_{0}(z)^{2}-A_{0}(z)^{2}C_{t\phi}(z)^{3}\tanh^{2}{\rho}},$
(B.8)

and

$\displaystyle
G(\rho,z)=1-\frac{2r_{0}(z)^{2}}{n}\left(\frac{\mathcal{R}_{\text{H}}^{\prime\prime}(z)}{\mathcal{R}_{\text{H}}(z)}-\frac{1}{L^{2}}\right)\log{(\cosh{\rho})},$
(B.9) $\displaystyle
H(\rho,z)=\mathcal{R}_{\text{H}}(z)\left(1+\frac{2}{n}\log{(\cosh{\rho})}\right).$
(B.10)

To satisfy the boundary conditions at $\rho\gg 1$, $C_{tt}$ and $C_{t\phi}$
are found to be

$\displaystyle
C_{tt}(z)=\frac{1}{A_{0}(z)^{2}}~{}~{}C_{t\phi}(z)=\frac{F_{0}(z)\sqrt{1-A_{0}(z)^{2}}}{A_{0}(z)}.$
(B.11)

Then we can see that the leading order solutions satisfy the boundary
condition at $\rho\gg 1$ as

$\displaystyle A(\rho,z)=1+O(e^{-\rho}),~{}~{}W(\rho,z)=O(e^{-\rho}).$ (B.12)

Furthermore the functions in this leading order solution should satisfy one
equation coming from the vector constraint (B.4) given by

$\displaystyle\frac{d}{dz}\log{\left(\frac{A_{0}(z)C_{tt}(z)}{r_{0}(z)}\right)}-(1-C_{tt}(z))\frac{d}{dz}\log{\left(\frac{C_{tt}(z)}{C_{t\phi}(z)}\right)}=0.$
(B.13)

This equation is rewritten by using eq. (B.11) as

$\displaystyle\frac{d}{dz}\log{\left(A_{0}r_{0}(z)\right)}+\frac{1-A_{0}(z)^{2}}{A_{0}(z)^{2}}\frac{d}{dz}\log{\left(A_{0}(z)F_{0}(z)\sqrt{1-A_{0}(z)^{2}}\right)}=0.$
(B.14)

We consider the $1/D$ corrections to the leading order solutions. Especially,
by solving eq. (B.5), we found that the $1/D$ correction to $K^{z}{}_{z}$, say
${}^{(1)}K^{z}{}_{z}$, has the following form

$\displaystyle{}^{(1)}K^{z}{}_{z}=\frac{C^{(1)}_{zz}}{\sinh{\rho}}+\delta
K^{z}{}_{z}(\rho,z),$ (B.15)

where $C^{(1)}_{zz}$ is an integration function with respect to
$\rho$-integration. $\delta K^{z}{}_{z}(\rho,z)$ is a solution coming from the
integration of the source term at next-to-leading order. $C^{(1)}_{zz}$ is
used to eliminate the divergence of $O(\rho^{-1})$ in $\delta
K^{z}{}_{z}(\rho,z)$ at the horizon $\rho=0$. Even though the $O(\rho^{-1})$
divergence is eliminated by the integration function, ${}^{(1)}K^{z}{}_{z}$
cannot satisfy the regularity condition at the horizon. Actually $\delta
K^{z}{}_{z}(\rho,z)$ has the following behavior after eliminating
$O(\rho^{-1})$ divergence at the horizon

$\displaystyle\delta
K^{z}{}_{z}(\rho,z)=-\frac{r_{0}(z)\log{\rho}}{(1-A_{0}(z)^{2})A_{0}(z)^{2}F_{0}(z)^{2}\rho}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigl{[}F_{0}(z)A_{0}^{\prime}(z)+(1-A_{0}^{2})A_{0}(z)F_{0}^{\prime}(z)\Bigr{]}^{2}+O(1).$
(B.16)

To satisfy the regularity condition on the horizon we should eliminate also
this $O(\rho^{-1}\log{\rho})$ divergence. Then $A_{0}(z)$ and $F_{0}(z)$
should satisfy additional condition

$\displaystyle
F_{0}(z)A_{0}^{\prime}(z)+(1-A_{0}^{2})A_{0}(z)F_{0}^{\prime}(z)=0.$ (B.17)

This condition can be solved by

$\displaystyle
F_{0}(z)=\frac{\sqrt{1-A_{0}(z)^{2}}}{\Omega_{\text{H}}A_{0(z)}},$ (B.18)

where we introduce an integration constant $\Omega_{\text{H}}$. Using this
solution (B.18), eq. (B.14) is reduced to the equation only for $A_{0}(z)$,
and it can be solved by

$\displaystyle A_{0}(z)=2\hat{\kappa}r_{0}(z),$ (B.19)

where $\hat{\kappa}$ is an integration constant.

Let us summarize above results. We obtained additional condition (B.17) from
the boundary condition on $O(1/D)$ corrections. By combining this additional
condition with the leading order vector constraint (B.14) we get the
expressions of $A_{0}(z)$ and $F_{0}(z)$ in terms of $r_{0}(z)$ and
integration constants as eqs. (B.18) and (B.19). Using eq. (B.11) we can see
that eq. (2.30) is obtained by defining $\kappa=n\hat{\kappa}$.

##  Appendix C Spheroidal harmonics at large $D$

We give some analysis for the spheroidal harmonics in the $1/D$ expansion. Let
us consider the massless scalar field equation

$\displaystyle\Box\Psi=0,$ (C.1)

in the $D=n+3$ dimensional flat spacetime in the ellipsoidal coordinate given
by

$\displaystyle
ds^{2}=-dt^{2}+\frac{r^{2}+a^{2}\cos^{2}{\theta}}{r^{2}+a^{2}}dr^{2}+(r^{2}+a^{2}\cos^{2}{\theta})d\theta^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(r^{2}+a^{2})\sin^{2}{\theta}d\phi^{2}+r^{2}\cos^{2}{\theta}d\Omega^{2}_{D-4}.$
(C.2)

$a$ is an oblateness parameter. The scalar field can be decomposed as

$\displaystyle\Psi=e^{-i\omega t}e^{im\phi}\psi(r)S(\theta)\mathbb{Y}_{j}.$
(C.3)

$\mathbb{Y}_{j}$ is the spherical harmonics on $S^{D-4}$ with the angular
momentum number $j$. Eq. (C.1) becomes equations for $\psi(r)$ and $S(\theta)$
as

$\displaystyle\Biggl{[}\frac{1}{r^{n-1}}\partial_{r}r^{n-1}\partial_{r}+\frac{((r^{2}+a^{2})\omega+am)^{2}}{r^{2}+a^{2}}-\frac{a^{2}(j(j+n-2))}{r^{2}}-\Lambda\Biggr{]}\psi(r)=0,$
(C.4)

and

$\displaystyle\Biggl{[}\frac{1}{\sin{\theta}\cos^{n-1}{\theta}}\partial_{\theta}\sin{\theta}\cos^{n-1}{\theta}\partial_{\theta}-\frac{(a\omega\sin^{2}{\theta}+m)^{2}}{\sin^{2}{\theta}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{j(j+n-2)}{\cos^{2}{\theta}}+\Lambda\Biggr{]}S(\theta)=0.$
(C.5)

$\Lambda$ is the separation constant. The spheroidal harmonics, $S(\theta)$,
is defined by the solution of eq. (C.5). While there are numerical analysis
[37] to find solution and eigenvalue of eq. (C.5), but it is hard to obtain
analytic solutions of eq. (C.5). Here we apply the large $D$ expansion to
solve eq. (C.5). At first we assume that the parameters in eq. (C.5) have
following orders at large $D$

$\displaystyle\omega=O(1),~{}~{}a=O(1),~{}~{}m=O(1),~{}~{}j=O(1),~{}~{}\Lambda=O(n).$
(C.6)

The reason for $\Lambda=O(n)$ can be seen by considering $a=0$ limit. At $a=0$
eq. (C.5) is reduced to the equation for the spherical harmonics. The equation
at $a=0$ can be solved by $S^{(a=0)}_{\ell}(\theta)$

$\displaystyle
S^{(a=0)}(\theta)=\cos^{j}{\theta}~{}\sin^{|m|}{\theta}~{}{}_{2}F_{1}(-k_{S},k_{S}+|m|+j+n/2,|m|+1;\sin^{2}{\theta}),$
(C.7)

with the separation constant

$\displaystyle\Lambda^{(a=0)}=\ell(\ell+n).$ (C.8)

$k_{S}$ is given by

$\displaystyle k_{S}=\frac{\ell-j-|m|}{2}.$ (C.9)

Then, at $a=0$, the separation constant is $O(n)$. The spheroidal harmonics
becomes the spherical harmonics at $a=0$. Thus it is natural to assume
$\Lambda=O(n)$ also at $a\neq 0$. The important observation is that the
spherical harmonics (C.7) has the following large $D$ behaviors

$\displaystyle
S^{(a=0)}_{\ell}(\theta)=\sin^{\ell-j}{\theta}\cos{{}^{j}}{\theta}+O(1/n).$
(C.10)

Usually the spherical harmonics $S^{(a=0)}_{\ell}(\theta)$ is understood as
the function which has $\ell$ nodes. This $\ell$ is the overtone number along
$\theta$ direction. At large $D$ limit the situation is changed, and any nodes
would disappear according to the large $D$ limit solution (C.10). Instead the
quantum number $\ell$ can be read from the behavior only around $\theta=0$ as
the degree of vanishing as

$\displaystyle
S^{(a=0)}_{\ell}(\theta)=\theta^{\ell}\left(1+O(\theta,n^{-1})\right).$ (C.11)

Although it is not still unclear why we can obtain the quantum number $\ell$
only by the behavior around $\theta=0$, not by the behavior both at $\theta=0$
and $\theta=\pi/2$, such behavior is useful for the quasinormal mode analysis
of the Myers-Perry black hole.

Next we consider the contributions by $a\neq 0$ to the spheroidal harmonics.
Let us assume that the spheroidal harmonics $S(\theta)$ is modified by the
contributions from $a$ and $\omega$ as

$\displaystyle S(\theta)=S^{(a=0)}_{\ell}(\theta)\left(1+\delta
S(\theta)\right).$ (C.12)

Substituting this into eq. (C.5) and linearizing the equation with respect to
$\delta S(\theta)$, we can obtain the equation for $\delta S(\theta)$. The
obtained equation, however, does not have $a$ and $\omega$ dependences in its
equation at the leading order of the large $D$ limit. This is because the
contributions of $a$ and $\omega$ are not leading order effects in eq. (C.5)
in the $1/D$ expansion. Thus $\delta S(\theta)$ can be set to zero also for
the spheroidal harmonics $S(\theta)$ at the leading order in the $1/D$
expansion. Hence the spheroidal harmonics has the following large $D$ form

$\displaystyle S(\theta)$ $\displaystyle=$ $\displaystyle
S^{(a=0)}_{\ell}(\theta)+O(1/n)$ (C.13) $\displaystyle=$
$\displaystyle\sin^{\ell-j}{\theta}\cos{{}^{j}}{\theta}+O(1/n),$

and the separation constant also becomes

$\displaystyle\Lambda=\ell n\left(1+O(n^{-1})\right),$ (C.14)

where $\ell$ is non-negative integer. The quantum number along $\theta$
direction, $\ell$, can be obtained again only by the behavior around
$\theta=0$. If one considers the $1/D$ corrections to the spheroidal harmonics
we can obtain the $1/D$ correction to the separation constant. In the analysis
of this paper we consider only the leading order results of the quasinormal
modes, so we do not pursue this analysis in more detail here.

##  Appendix D Detail analysis on the vector type perturbation

In this appendix we show the detail analysis of the vector type perturbation
of the Myers-Perry- black hole. The vector type perturbation on the Myers-
Perry black hole is decomposed to

$\displaystyle h^{(V)}_{\mu\nu}dx^{\mu}dx^{\nu}=e^{-i\omega
t}e^{im\phi}\Bigl{[}f^{(V)\text{Sch}}_{I}\mathbb{Y}^{(V)j}_{A}e^{(I)}e^{(A)}+H^{(V)}_{L}\mathbb{Y}^{(V)j}_{AB}e^{(A)}e^{(B)}\Bigr{]},$
(D.1)

where the vielbein and the vector harmonics were defined in section 3. The
decoupling perturbation variables are defined by

$\displaystyle
F^{(V)}_{0}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{0}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{2},$
(D.2) $\displaystyle
F^{(V)}_{2}=\frac{\cosh{\rho}}{\sqrt{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{2}-\sinh{\rho}\sqrt{\frac{1-r_{0}(z)^{2}}{1+r_{0}(z)^{2}\sinh^{2}{\rho}}}f^{(V)}_{0},$
(D.3)

and

$\displaystyle~{}F^{(V)}_{1}=f^{(V)}_{1},~{}~{}F^{(V)}_{3}=f^{(V)}_{3},~{}~{}F^{(V)}_{L}=H^{(V)}_{L}.$
(D.4)

The decoupling perturbation variables are expanded by

$\displaystyle F^{(V)}_{I}=\sum_{k\geq
0}\frac{{}^{(k)}F^{(V)}_{I}}{n^{k}},~{}F^{(V)}_{T,L}=\sum_{k\geq
0}\frac{{}^{(k)}F_{L}^{(V)}}{n^{k}}.$ (D.5)

Changing coordinate by $\mathsf{R}=\cosh^{2}{\rho}$, the perturbation equation
at each order becomes

$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)}_{0}-\frac{{}^{(k)}F^{(V)}_{0}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(V)}_{0},$
(D.6)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)}_{1}-\frac{{}^{(k)}F^{(V)}_{1}}{4\mathsf{R}(\mathsf{R}-1)}={}^{(k)}\mathcal{S}^{(V)}_{1},$
(D.7)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)}_{2}={}^{(k)}\mathcal{S}^{(V)}_{2},$
(D.8)
$\displaystyle\frac{\partial}{\partial\mathsf{R}}\mathsf{R}(\mathsf{R}-1)\frac{\partial}{\partial\mathsf{R}}{}^{(k)}F^{(V)}_{3}={}^{(k)}\mathcal{S}^{(V)}_{3}.$
(D.9)

The boundary condition is the same with one of the scalar type perturbation,
the ingoing boundary condition on the horizon (4.4) and decoupling mode
condition (4.5) at asymptotic infinity of near zone.

##### Leading order $(k=0)$

The regular solution at the leading order is

$\displaystyle{}^{(0)}F^{V}_{0}={}^{(0)}F^{V}_{1}=\frac{W_{(0)}}{\sqrt{\mathsf{R}(\mathsf{R}-1)}},~{}~{}{}^{(0)}F^{V}_{2}={}^{(0)}F^{V}_{3}=0.$
(D.10)

The leading order solution has an undetermined function $W_{(0)}$ by the
boundary condition.

##### Next-to leading order $(k=1)$

The regular solution at the next to leading order is

$\displaystyle{}^{(1)}F^{V}_{0}=\frac{W_{(1)}}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+{}^{(1)}\hat{F}^{V}_{0},~{}{}^{(1)}F^{V}_{1}=\frac{W_{(1)}}{\sqrt{\mathsf{R}(\mathsf{R}-1)}}+{}^{(1)}\hat{F}^{V}_{1},~{}$
(D.11)

and

$\displaystyle{}^{(0)}F^{V}_{2}={}^{(1)}\hat{F}^{V}_{2},~{}{}^{(0)}F^{V}_{3}={}^{(1)}\hat{F}^{V}_{3},$
(D.12)

where terms with the hat come from the integrations of source terms. The next
to leading order solution has an undetermined function $W_{(1)}$ by the
boundary condition again. At this order we obtain the non-trivial condition
for $W_{(0)}$ as

$\displaystyle\cos{\theta}\sin^{2}{\theta}(1+a^{2}\cos^{2}{\theta})W^{\prime}_{(0)}(\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\Bigl{[}-j(1+a^{2}\cos^{2}{\theta})+i\cos^{2}{\theta}(a^{4}\cos^{2}{\theta}(\omega-i)-a^{3}m\cos^{2}{\theta}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+a^{2}(\cos^{2}{\theta}(\omega+i)+\omega-2i)+\omega-
am)\Bigr{]}W_{0}(\theta)=0.$ (D.13)

This condition can be solved explicitly as

$\displaystyle
W_{(0)}(\theta)=\frac{\cos^{j}{\theta}\sin^{{}^{(0)}\delta_{\omega}^{V}}{\theta}}{1+a^{2}\cos^{2}{\theta}},$
(D.14)

where

$\displaystyle{}^{(0)}\delta^{V}_{\omega}=-(1+a^{2})(j-1)-iam+i(1+a^{2})\omega.$
(D.15)

The harmonics condition on $W_{0}(\theta)$ is eq. (4.18) as

$\displaystyle{}^{(0)}\delta^{V}_{\omega}=\ell-j,$ (D.16)

and $\ell$ is parameterized by the non-negative integer $k_{V}$ by

$\displaystyle\ell=j+|m|+2k_{V}.$ (D.17)

This harmonics condition gives the leading order QNM frequency for the vector
type perturbation by

$\displaystyle\omega^{V}_{\text{LO}}=\frac{am}{1+a^{2}}-i\left(j-1+\frac{\ell-j}{1+a^{2}}\right).$
(D.18)

##### Next-to-next-to leading order

At this order we can obtain the solution and undetermined function in the same
manner. Furthermore we find the non-trivial condition on $W_{(1)}(\theta)$.
Although we omit the explicit form of it since it is not so illuminative, the
solution of the non-trivial condition gives the following behavior around
$\theta=0$ on the horizon $\mathsf{R}=1$

$\displaystyle
F^{V}_{(0)}=\frac{\theta^{\delta^{V}_{\omega}}}{\sqrt{\mathsf{R}-1}}(1+O(\theta,1/n^{2}))$
(D.19)

where

$\displaystyle\delta^{V}_{\omega}={}^{(0)}\delta^{V}_{\omega}+\frac{{}^{(1)}\delta^{V}_{\omega}}{n}.$
(D.20)

${}^{(0)}\delta^{V}_{\omega}$ is given in eq. (D.15).
${}^{(1)}\delta^{V}_{\omega}$ is

$\displaystyle{}^{(1)}\delta^{V}_{\omega}=a^{2}(-a^{2}(j-1)^{2}+2ia(j-1)m+j^{2}-2j-m^{2}+1)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-a(2a(j-1)+im\log{(1+a^{2})})+\omega(i(a^{2}-1)\log{(1+a^{2})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+2ia^{2}(a^{2}(j-1)+iam+j-1))+(1+a^{2})^{2}\omega^{2}.$
(D.21)

Then the harmonics condition gives

$\displaystyle{}^{(0)}\delta^{V}_{\omega}+\frac{{}^{(1)}\delta^{V}_{\omega}}{n}=\ell-j.$
(D.22)

This condition is solved by

$\displaystyle\omega^{V}=\omega^{V}_{\text{LO}}+\frac{\omega^{V}_{\text{NLO}}}{n}+O(1/n^{2}),$
(D.23)

where

$\displaystyle\omega^{V}_{\text{NLO}}=\frac{2am}{1+a^{2}}\Bigl{(}\ell-1+a^{2}(j-1)+\log{(1+a^{2})}\Bigr{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-i\Biggl{(}(j-1)^{2}+\frac{(1+a^{2})(\ell-j)(\ell+j-2)-2a^{2}m^{2}}{(1+a^{2})^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left(j-1-\frac{\ell-j}{1+a^{2}}+\frac{2(\ell-j)}{(1+a^{2})^{2}}\right)\log{(1+a^{2})}\Biggr{)}.$
(D.24)

##  Appendix E Explicit forms of $c_{i}(\theta)$ and $d_{i}(\theta)$

The scalar type perturbation can satisfy the boundary condition on the horizon
at the next to next to the leading order if the integration functions are
solutions of following equations:

$\displaystyle
c_{1}(\theta)C_{(1)}^{\prime\prime}(\theta)+c_{2}(\theta)C_{(1)}^{\prime}(\theta)+c_{3}(\theta)C_{(1)}(\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c_{4}(\theta)D_{(0)}^{\prime\prime}(\theta)+c_{5}(\theta)D_{(0)}^{\prime}(\theta)+c_{6}(\theta)D_{(0)}(\theta)=0,$
(E.1)

and

$\displaystyle
d_{1}(\theta)D_{(0)}^{\prime\prime\prime}(\theta)+d_{2}(\theta)D_{(0)}^{\prime\prime}(\theta)+d_{3}(\theta)D_{(0)}^{\prime}(\theta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+d_{4}(\theta)D_{(0)}(\theta)+d_{5}(\theta)C_{(1)}^{\prime}(\theta)+d_{6}(\theta)C_{(1)}(\theta)=0.$
(E.2)

Each coefficients are following:

$\displaystyle
c_{1}(\theta)=\frac{\cos^{2}{\theta}\sin^{3}{\theta}\left(1+a^{2}\cos{\theta}^{2}\right)}{2\left(1+a^{2}\right)^{5/2}},$
(E.3) $\displaystyle
c_{2}(\theta)=\frac{\cos{\theta}\sin^{2}{\theta}}{2\left(1+a^{2}\right)^{5/2}}\Bigl{[}2a^{2}\cos^{4}{\theta}\left(a^{2}(j-i\omega)+iam-i\omega+2\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\cos^{2}{\theta}\left(a^{2}(4j-2i\omega-5)+2iam-2i\omega-2\right)+2j+1\Bigr{]},$
(E.4) $\displaystyle
c_{3}(\theta)=\frac{\sin^{2}{\theta}}{2\left(1+a^{2}\right)^{5/2}}\Bigl{[}a^{2}\cos^{6}{\theta}\left(a^{2}(j-i\omega)+iam-i\omega+2\right)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\cos^{2}{\theta}\left(a^{2}\left(3j^{2}+j(-4-2i\omega)-1\right)+2iajm+j(-4-2i\omega)+4\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\cos^{4}{\theta}\Bigl{(}a^{4}\left(3j^{2}+j(-6-4i\omega)-\omega^{2}+6i\omega+4\right)+2a^{3}m(2ij+\omega-3i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-a^{2}\left(m^{2}+2\left(2i(j-2)\omega+\omega^{2}-1\right)\right)+2am(\omega-i)-\omega(\omega-2i)\Bigr{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+j^{2}+2j-3\Bigr{]},$ (E.5) $\displaystyle
c_{4}(\theta)=-\frac{\sin^{2}{\theta}}{\left(1+a^{2}\right)^{4}\cos{\theta}}\Bigl{[}\cos^{2}{\theta}\left(a^{2}(-5j+3i\omega+2)-3iam+j+3i\omega+5\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\cos^{4}{\theta}\left(a^{4}(-2j+3i\omega+3)-3ia^{3}m+a^{2}(j+2i\omega+1)-i\omega\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-ia^{2}\left(1+a^{2}\right)\omega\cos^{6}{\theta}-3j-5\Bigr{]},$
(E.6) $\displaystyle
c_{5}(\theta)=\frac{\sin{\theta}}{\left(1+a^{2}\right)^{4}\cos^{2}{\theta}\left(1+a^{2}\cos^{2}{\theta}\right)}\Bigl{[}\cos^{2}{\theta}\Bigl{(}a^{2}\left(10j^{2}+2j(8-3i\omega)-7i\omega-3\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+ia(6j+7)m-2j^{2}+j(-19-6i\omega)-7i\omega-9\Bigr{)}$
$\displaystyle~{}~{}~{}~{}+a^{4}\left(a^{2}+1\right)\omega\cos^{10}{\theta}\left(a^{2}(2ij+2\omega-i)-2am+2\omega+3i\right)$
$\displaystyle~{}~{}~{}~{}+\cos^{4}{\theta}\Bigl{(}a^{4}\left(12j^{2}+j(-6-16i\omega)-3\omega^{2}-i\omega-9\right)+a^{3}m(16ij+6\omega+i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-a^{2}\left(6j^{2}+4j(7+3i\omega)+3m^{2}+6\omega^{2}-9i\omega+2\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+am(-2ij+6\omega-9i)+j(5+4i\omega)-3\omega^{2}+10i\omega+4\Bigr{)}$
$\displaystyle~{}~{}~{}~{}+a^{2}\cos^{8}{\theta}\Bigl{(}a^{6}\left(j^{2}+j(-3-4i\omega)-3\omega^{2}+4i\omega+2\right)+2a^{5}m(2ij+3\omega-2i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-a^{4}\left(2j^{2}+j(-2-4i\omega)+3m^{2}+2\omega(\omega+i)\right)+a^{3}m(-2ij+2\omega+i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+a^{2}\left(8ij\omega+j+5\omega^{2}-5i\omega-2\right)+am(-4\omega+i)+\omega(4\omega+i)\Bigr{)}$
$\displaystyle~{}~{}~{}~{}+\cos^{6}{\theta}\Bigl{(}a^{6}\left(6j^{2}+j(-12-14i\omega)-6\omega^{2}+10i\omega+5\right)+2a^{5}m(7ij+6\omega-5i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-a^{4}\left(6j^{2}+j(7+4i\omega)+6m^{2}+10\omega^{2}-15i\omega-15\right)+2a^{3}m(-2ij+5\omega-4i)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+a^{2}\left(2j(3+5i\omega)-2\omega^{2}+3i\omega+6\right)+am(-2\omega+i)+2\omega(\omega-i)\Bigr{)}$
$\displaystyle~{}~{}~{}~{}+3j^{2}+13j+5\Bigr{]},$ (E.7) $\displaystyle
d_{1}(\theta)=-\frac{\cos{\theta}\sin^{3}{\theta}}{\left(1+a^{2}\right)^{2}\left(1+a^{2}\cos^{2}{\theta}\right)},$
(E.8) $\displaystyle
d_{2}(\theta)=-\frac{\sin^{2}{\theta}}{\left(1+a^{2}\right)^{2}\left(1+a^{2}\cos^{2}{\theta}\right)^{2}}\Bigl{[}3j+1$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\cos^{2}{\theta}\left(a^{2}(6j-3i\omega-4)+3iam-3i\omega-1\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+a^{2}\cos^{4}{\theta}\left(3a^{2}(j-i\omega-1)+3iam-3i\omega+1\right)\Bigr{]},$
(E.9) $\displaystyle
d_{3}(\theta)=-\frac{\sin{\theta}}{\left(1+a^{2}\right)^{2}\cos{\theta}\left(1+a^{2}\cos^{2}{\theta}\right)^{3}}\Bigl{[}\cos^{2}{\theta}\Bigl{(}a^{2}\left(12j^{2}-6ij\omega+j+i\omega+3\right)$
$\displaystyle~{}~{}+ia(6j-1)m-6ij\omega-6j+i\omega-1\Bigr{)}+a^{4}\cos^{8}{\theta}\Bigl{(}a^{4}\left(3j^{2}-6ij\omega-4j-3\omega^{2}+5i\omega+2\right)$
$\displaystyle~{}~{}+a^{3}m(6ij+6\omega-5i)+a^{2}\left(-6ij\omega+4j-3m^{2}-6\omega^{2}+2i\omega-2\right)+3am(2\omega+i)$
$\displaystyle~{}~{}-3\omega(\omega+i)\Bigr{)}+a^{2}\cos^{6}{\theta}\Bigl{(}a^{4}\left(12j^{2}-18ij\omega-17j-6\omega^{2}+15i\omega+7\right)+3a^{3}m(6ij+4\omega-5i)$
$\displaystyle~{}~{}+a^{2}\left(-18ij\omega+2j-6m^{2}-12\omega^{2}+13i\omega+5\right)+2am(6\omega+i)-6\omega^{2}-2i\omega+8\Bigr{)}$
$\displaystyle~{}~{}+\cos^{4}{\theta}\Bigl{(}a^{4}\left(18j^{2}+j(-17-18i\omega)-3\omega^{2}+11i\omega-1\right)+a^{3}m(18ij+6\omega-11i)$
$\displaystyle~{}~{}-a^{2}\left(2j(4+9i\omega)+3m^{2}+6\omega^{2}-12i\omega+10\right)+am(6\omega-i)+(-3\omega+i)\omega\Bigr{)}$
$\displaystyle~{}~{}+3j^{2}+5j+1\Bigr{]},$ (E.10) $\displaystyle
d_{5}(\theta)=\frac{\sqrt{1+a^{2}}\cos{\theta}^{2}\sin^{2}{\theta}}{\left(1+a^{2}\right)\left(1+a^{2}\cos^{2}{\theta}\right)},$
(E.11) $\displaystyle
d_{6}(\theta)=\frac{\cos{\theta}\sin{\theta}\left(\cos^{2}{\theta}\left(a^{2}(j-i\omega-2)+iam-i\omega\right)+j-1\right)}{\sqrt{1+a^{2}}\left(1+a^{2}\cos^{2}{\theta}\right)}.$
(E.12)

The coefficients $c_{6}(\theta)$ and $d_{4}(\theta)$ have very lengthy forms.
They are written as

$\displaystyle
c_{6}(\theta)=\frac{1}{\left(a^{2}+1\right)^{4}\cos^{3}{\theta}\left(1+a^{2}\cos^{2}{\theta}\right)^{2}}\sum_{k=0}^{k=7}\hat{c}_{6}^{(k)}\cos^{2k}{\theta},$
(E.13)

and

$\displaystyle
d_{4}(\theta)=\frac{1}{\left(1+a^{2}\right)^{2}\cos^{2}{\theta}\left(1+a^{2}\cos^{2}{\theta}\right)^{4}}\sum_{k=0}^{k=6}\hat{d}_{4}^{k}\cos^{2k}{\theta},$
(E.14)

where

$\displaystyle\hat{c}_{6}^{(0)}=j\left(j^{2}+8j+12\right),$ (E.15)
$\displaystyle\hat{c}_{6}^{(1)}=j\Bigl{(}a^{2}\left(5j^{2}-3ij\omega+23j-10i\omega+16\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+ia(3j+10)m-j^{2}-3ij(\omega-5i)-10i\omega-28\Bigr{)},$
(E.16)
$\displaystyle\hat{c}_{6}^{(2)}=a^{4}\left(10j^{3}+j^{2}(18-13i\omega)-j\left(3\omega^{2}+19i\omega+8\right)-2\left(\omega^{2}-2i\omega+4\right)\right)$
$\displaystyle~{}~{}+a^{3}m\left(13ij^{2}+6j\omega+19ij+4\omega-4i\right)-a^{2}\Bigl{(}5j^{3}+5j^{2}(9+2i\omega)$
$\displaystyle~{}~{}+j\left(3m^{2}+6\omega^{2}+44\right)+2\left(m^{2}+2\omega^{2}-2i\omega+4\right)\Bigr{)}+am\left(-2ij^{2}+j(6\omega-17i)+4\omega\right)$
$\displaystyle~{}~{}+j^{2}(7+3i\omega)+j\left(-3\omega^{2}+19i\omega+20\right)-2\omega^{2},$
(E.17)
$\displaystyle\hat{c}_{6}^{(3)}=a^{6}\left(10j^{3}+j^{2}(-4-22i\omega)-j\left(11\omega^{2}+14\right)+i\left(\omega^{3}+8\omega+8i\right)\right)$
$\displaystyle~{}~{}~{}+a^{5}m\left(22ij^{2}+22j\omega-i\left(3\omega^{2}+8\right)\right)+a^{4}\Bigl{(}-10j^{3}+j^{2}(-42-9i\omega)$
$\displaystyle~{}~{}~{}+j\left(-11m^{2}+(-19\omega+39i)\omega\right)+i\left(3m^{2}+4\right)\omega+3i\omega^{3}+4\omega^{2}+8\Bigr{)}$
$\displaystyle~{}~{}~{}-ia^{3}m\left(8j^{2}+2j(19+9i\omega)+m^{2}+6\omega^{2}-8i\omega-4\right)+a^{2}\Bigl{(}j^{2}(22+13i\omega)$
$\displaystyle~{}~{}~{}+j\left(m^{2}-5\omega^{2}+30i\omega+34\right)+m^{2}(4+3i\omega)+3i\omega^{3}+8\omega^{2}-4i\omega+16\Bigr{)}$
$\displaystyle~{}~{}~{}-am(j(4\omega-6i)+(8+3i\omega)\omega)+3j\omega^{2}-9ij\omega-4j+i\omega^{3}+4\omega^{2},$
(E.18)
$\displaystyle\hat{c}_{6}^{(4)}=a^{8}\left(5j^{3}+j^{2}(-10-18i\omega)-3j\omega(5\omega-6i)+3i\omega^{3}+7\omega^{2}+8\right)$
$\displaystyle~{}~{}~{}+a^{7}m\left(18ij^{2}+6j(5\omega-3i)+(-14-9i\omega)\omega\right)+a^{6}\Bigl{(}-10j^{3}+j^{2}(-6+4i\omega)$
$\displaystyle~{}~{}~{}+j\left(-15m^{2}-19\omega^{2}+28i\omega+18\right)+m^{2}(7+9i\omega)+8i\omega^{3}+15\omega^{2}-10i\omega+32\Bigr{)}$
$\displaystyle~{}~{}~{}-ia^{5}m\left(12j^{2}+4j(5+4i\omega)+3m^{2}+16\omega^{2}-21i\omega-12\right)+a^{4}\Bigl{(}j^{2}(24+22i\omega)$
$\displaystyle~{}~{}~{}+j\left(3m^{2}+7\omega^{2}-10i\omega+8\right)+m^{2}(6+8i\omega)+6i\omega^{3}+7\omega^{2}-12i\omega+16\Bigr{)}$
$\displaystyle~{}~{}~{}-a^{3}m(14j(\omega-i)+(4+5i\omega)\omega)+a^{2}\Bigl{(}j\left(11\omega^{2}-20i\omega-6\right)+m^{2}(-1-i\omega)$
$\displaystyle~{}~{}~{}-3\omega^{2}-2i\omega-8\Bigr{)}+am(3+2i\omega)\omega+(-2-i\omega)\omega^{2},$
(E.19)
$\displaystyle\hat{c}_{6}^{(5)}=a^{2}\Biggl{(}a^{8}\left(j^{3}+j^{2}(-3-7i\omega)+j\left(-9\omega^{2}+10i\omega+2\right)+\omega\left(3i\omega^{2}+6\omega-4i\right)\right)$
$\displaystyle~{}~{}~{}+a^{7}m\left(7ij^{2}+2j(9\omega-5i)-i(-3\omega+2i)^{2}\right)+a^{6}\Bigl{(}-5j^{3}+j^{2}(9+11i\omega)$
$\displaystyle~{}~{}~{}-3j\left(3m^{2}+\omega(\omega+4i)\right)+m^{2}(6+9i\omega)+6i\omega^{3}+\omega^{2}-4i\omega-8\Bigr{)}$
$\displaystyle~{}~{}~{}-ia^{5}m\left(8j^{2}-6j+3m^{2}+12\omega^{2}-i\omega-4\right)+a^{4}\Bigl{(}2j^{2}(5+9i\omega)$
$\displaystyle~{}~{}~{}+j\left(3m^{2}+21\omega^{2}-32i\omega-6\right)+2i\left(3m^{2}-1\right)\omega-17\omega^{2}-24\Bigr{)}$
$\displaystyle~{}~{}~{}+a^{3}m\left(-2j(9\omega-5i)+3i\omega^{2}+14\omega-4i\right)+a^{2}\Bigl{(}5j(3\omega-2i)\omega+m^{2}(-2-3i\omega)$
$\displaystyle~{}~{}~{}-6i\omega^{3}-13\omega^{2}-16\Bigr{)}+3am(1+2i\omega)\omega+\omega\left(-3i\omega^{2}-\omega+2i\right)\Biggr{)},$
(E.20)
$\displaystyle\hat{c}_{6}^{(6)}=a^{4}\Biggl{(}a^{8}\omega\left(-ij^{2}-2j\omega+ij+i\omega^{2}+\omega\right)+a^{7}m\left(ij^{2}+j(4\omega-i)+(-2-3i\omega)\omega\right)$
$\displaystyle~{}~{}~{}+a^{6}\left(-j^{3}+j^{2}(3+6i\omega)-j\left(2m^{2}-5\omega^{2}+12i\omega+2\right)+m^{2}(1+3i\omega)+(-7\omega+6i)\omega\right)$
$\displaystyle~{}~{}~{}-ia^{5}m\left(2j^{2}+j(-5-6i\omega)+m^{2}+9i\omega+4\right)+a^{4}\Bigl{(}j^{2}(1+7i\omega)+j\left(m^{2}+\omega(16\omega-9i)\right)$
$\displaystyle~{}~{}~{}-2m^{2}+\omega\left(-6i\omega^{2}-13\omega+4i\right)\Bigr{)}+a^{3}m\left(j(-10\omega+2i)+9i\omega^{2}+8\omega-4i\right)$
$\displaystyle~{}~{}~{}+a^{2}\left(j\left(9\omega^{2}+4i\omega+2\right)+m^{2}(-1-3i\omega)+\omega\left(-8i\omega^{2}-\omega+2i\right)\right)$
$\displaystyle~{}~{}~{}+3iam\omega(2\omega+i)+\omega\left(-3i\omega^{2}+4\omega+4i\right)\Biggr{)},$
(E.21)
$\displaystyle\hat{c}_{6}^{(7)}=a^{6}\left(a^{2}+1\right)\omega\Bigl{(}a^{4}\left(ij^{2}+j(2\omega-i)+(-1-i\omega)\omega\right)+a^{3}m(-2j+2i\omega+1)$
$\displaystyle~{}~{}~{}+a^{2}\left(j(2\omega+3i)-i\left(m^{2}+2\omega^{2}+2i\omega+2\right)\right)+am(-3+2i\omega)$
$\displaystyle~{}~{}~{}~{}-i\omega^{2}+3\omega+2i\Bigr{)},$ (E.22)
$\displaystyle\hat{d}_{4}^{(0)}=-j(j+2)^{2},$ (E.23)
$\displaystyle\hat{d}_{4}^{(1)}=j\Bigl{(}a^{2}\left(-6j^{2}+j(-11+3i\omega)+2i\omega-8\right)-ia(3j+2)m$
$\displaystyle~{}~{}~{}~{}~{}+j(5+3i\omega)+2i\omega+8\Bigr{)},$ (E.24)
$\displaystyle\hat{d}_{4}^{(2)}=a^{4}\left(-15j^{3}+j^{2}(-3+15i\omega)+j\left(3\omega^{2}-5i\omega+2\right)-2\left(\omega^{2}-2i\omega+4\right)\right)$
$\displaystyle~{}~{}~{}~{}+a^{3}m\left(-15ij^{2}-6j\omega+5ij+4\omega-4i\right)+a^{2}\Bigl{(}j^{2}(17+15i\omega)+j\left(3m^{2}+6\omega^{2}-10i\omega+26\right)$
$\displaystyle~{}~{}~{}~{}-2\left(m^{2}+2\omega^{2}-2i\omega+4\right)\Bigr{)}+am(4\omega+j(-6\omega+5i))+j\left(3\omega^{2}-5i\omega-4\right)-2\omega^{2},$
(E.25)
$\displaystyle\hat{d}_{4}^{(3)}=a^{6}\left(-20j^{3}+6j^{2}(3+5i\omega)+2j\left(6\omega^{2}-15i\omega+5\right)-i\omega\left(\omega^{2}-10i\omega-4\right)\right)$
$\displaystyle~{}~{}~{}~{}+a^{5}m\left(-30ij^{2}-6j(4\omega-5i)+3i\omega^{2}+20\omega-4i\right)+a^{4}\Bigl{(}6j^{2}(3+5i\omega)$
$\displaystyle~{}~{}~{}~{}+4j\left(3m^{2}+6\omega^{2}-10i\omega+6\right)+m^{2}(-10-3i\omega)-3i\omega^{3}-20\omega^{2}-8i\omega+16\Bigr{)}$
$\displaystyle~{}~{}~{}~{}+a^{3}m\left(j(-24\omega+10i)+i\left(m^{2}+6\omega^{2}-20i\omega+12\right)\right)+a^{2}\Bigl{(}2j\left(6\omega^{2}-5i\omega-9\right)$
$\displaystyle~{}~{}~{}~{}-3i\left(m^{2}+4\right)\omega-3i\omega^{3}-10\omega^{2}+16\Bigr{)}+3iam\omega^{2}-i\omega^{3},$
(E.26)
$\displaystyle\hat{d}_{4}^{(4)}=-a^{2}\Biggl{(}a^{6}\left(15j^{3}+j^{2}(-22-30i\omega)-2j\left(9\omega^{2}-20i\omega+1\right)+3i\omega^{3}+16\omega^{2}-4i\omega+8\right)$
$\displaystyle~{}~{}~{}+a^{5}m\left(30ij^{2}+4j(9\omega-10i)-9i\omega^{2}-32\omega+4i\right)+a^{4}\Bigl{(}j^{2}(-2-30i\omega)$
$\displaystyle~{}~{}~{}-2j\left(9m^{2}+18\omega^{2}-20i\omega+2\right)+m^{2}(16+9i\omega)+9i\omega^{3}+30\omega^{2}+8i\omega+16\Bigr{)}$
$\displaystyle~{}~{}~{}-a^{3}m\left(-36j\omega+3im^{2}+18i\omega^{2}+28\omega+12i\right)+a^{2}\Bigl{(}j\left(26-18\omega^{2}\right)+m^{2}(-2+9i\omega)$
$\displaystyle~{}~{}~{}+9i\omega^{3}+12\omega^{2}+4i\omega+16\Bigr{)}+am\left(-9i\omega^{2}+4\omega+8i\right)+3i\omega^{3}-2\omega^{2}-8i\omega+8\Biggr{)},$
(E.27)
$\displaystyle\hat{d}_{4}^{(5)}=a^{4}\Biggl{(}a^{6}\left(-6j^{3}+3j^{2}(3+5i\omega)+2j\left(6\omega^{2}-10i\omega-1\right)+\omega\left(-3i\omega^{2}-10\omega+4i\right)\right)$
$\displaystyle~{}~{}~{}+a^{5}m\left(-15ij^{2}-4j(6\omega-5i)+9i\omega^{2}+20\omega-4i\right)+a^{4}\Bigl{(}j^{2}(-7+15i\omega)$
$\displaystyle~{}~{}~{}+2j\left(6m^{2}+\omega(12\omega-5i)\right)+m^{2}(-10-9i\omega)-9i\omega^{3}-16\omega^{2}+8\Bigr{)}$
$\displaystyle~{}~{}~{}+a^{3}m\left(-2j(12\omega+5i)+3im^{2}+18i\omega^{2}+12\omega+4i\right)+a^{2}\Bigl{(}2j\left(6\omega^{2}+5i\omega-7\right)$
$\displaystyle~{}~{}~{}+m^{2}(4-9i\omega)-9i\omega^{3}-2\omega^{2}+4i\omega+16\Bigr{)}+iam\left(9\omega^{2}+8i\omega-8\right)$
$\displaystyle~{}~{}~{}-3i\omega^{3}+4\omega^{2}+8i\omega+8\Biggr{)},$ (E.28)
$\displaystyle\hat{d}_{4}^{(6)}=-a^{6}\Biggl{(}a^{6}\left(j^{3}+j^{2}(-1-3i\omega)-3j\omega(\omega-i)+(2+i\omega)\omega^{2}\right)$
$\displaystyle~{}+a^{5}m\left(3ij^{2}+j(6\omega-3i)+(-4-3i\omega)\omega\right)+a^{4}\Bigl{(}j^{2}(3-3i\omega)$
$\displaystyle~{}-j\left(3m^{2}+6\omega^{2}+2i\omega+2\right)+m^{2}(2+3i\omega)+\omega\left(3i\omega^{2}+2\omega+4i\right)\Bigr{)}$
$\displaystyle~{}-ia^{3}m\left(j(-5+6i\omega)+m^{2}+6\omega^{2}+4\right)+a^{2}\Bigl{(}j\left(-3\omega^{2}-5i\omega+2\right)$
$\displaystyle~{}+i\left(m^{2}(3\omega+2i)+\omega\left(3\omega^{2}+2i\omega+4\right)\right)\Bigr{)}+am(4-3i\omega)\omega+i\omega^{2}(\omega+2i)\Biggr{)}.$
(E.29)

##  Appendix F Explicit form of $\delta^{S}_{\omega}$

The QNM condition for the scalar type perturbation of the Myers-Perry black
hole is

$\displaystyle\delta^{S}_{\omega}=\ell-j.$ (F.1)

This condition can be rewritten as

$\displaystyle\delta^{S}_{\omega}-\ell+j=0\Leftrightarrow\frac{F(\omega)}{G(\omega)}=0,$
(F.2)

where

$\displaystyle
G(\omega)=\left(1+a^{2}\right)^{3}\Bigl{[}a^{4}(j-i\omega)^{2}+2a^{3}m(\omega+ij)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-a^{2}\left(j(-2\ell+2i\omega+4)+2i\ell\omega+m^{2}+2\omega^{2}-4i\omega-4\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2am(i\ell+\omega-2i)+(\ell-i\omega-2)^{2}\Bigr{]},$
(F.3)

and

$\displaystyle
F(\omega)=f_{5}\omega^{5}+f_{4}\omega^{4}+f_{3}\omega^{3}+f_{2}\omega^{2}+f_{1}\omega+f_{0}.$
(F.4)

Each coefficient is

$\displaystyle f_{5}=i\left(1+a^{2}\right)^{5},$ (F.5) $\displaystyle
f_{4}=-\left(1+a^{2}\right)^{4}\left(a^{2}(5j-2)+5iam+5\ell-6\right),$ (F.6)
$\displaystyle
f_{3}=-i\left(1+a^{2}\right)^{3}\Bigl{(}a^{4}j(10j-7)+4ia^{3}(5j-2)m+a^{2}\bigl{(}j(20\ell-21)$
$\displaystyle~{}~{}-9\ell-10m^{2}+12\bigr{)}+2ia(10\ell-11)m+10\ell^{2}-23\ell+12\Bigr{)},$
(F.7) $\displaystyle
f_{2}=\left(1+a^{2}\right)^{2}\Bigl{(}a^{6}j^{2}(10j-9)+3ia^{5}j(10j-7)m+a^{4}\bigl{(}3j^{2}(10\ell-9)$
$\displaystyle~{}~{}-6j\left(4\ell+5m^{2}-5\right)+2\left(\ell+6m^{2}-4\right)\bigr{)}-ia^{3}m\bigl{(}j(57-60\ell)+27\ell$
$\displaystyle~{}~{}+10m^{2}-30\bigr{)}+a^{2}\left(30j(\ell-1)^{2}-15\ell^{2}-30\ell
m^{2}+34\ell+30m^{2}-16\right)$
$\displaystyle~{}~{}+3ia\left(10\ell^{2}-21\ell+10\right)m+10\ell^{3}-33\ell^{2}+32\ell-8\Bigr{)},$
(F.8) $\displaystyle
f_{1}=i\left(1+a^{2}\right)\Bigl{(}5a^{8}(j-1)j^{3}+2ia^{7}j^{2}(10j-9)m+a^{6}j\bigl{(}5j^{2}(4\ell-3)$
$\displaystyle~{}~{}-3j\left(7\ell+10m^{2}-8\right)+4\ell+21m^{2}-12\bigr{)}-4ia^{5}m\bigl{(}-3j^{2}(5\ell-4)$
$\displaystyle~{}~{}+j\left(12\ell+5m^{2}-12\right)-\ell-2m^{2}+3\bigr{)}+a^{4}\bigl{(}3j^{2}\left(10\ell^{2}-17\ell+8\right)$
$\displaystyle~{}~{}+j\left(-27\ell^{2}+\ell\left(56-60m^{2}\right)+51m^{2}-24\right)+4\ell^{2}+3\ell\left(9m^{2}-4\right)+m^{2}\left(5m^{2}-24\right)\bigr{)}$
$\displaystyle~{}~{}-2ia^{3}m\left(-6j\left(5\ell^{2}-9\ell+4\right)+15\ell^{2}+2\ell\left(5m^{2}-14\right)-9m^{2}+12\right)$
$\displaystyle~{}~{}+a^{2}\left(j\left(20\ell^{3}-57\ell^{2}+52\ell-12\right)-11\ell^{3}+\ell^{2}\left(32-30m^{2}\right)+3\ell\left(19m^{2}-8\right)-24m^{2}\right)$
$\displaystyle~{}~{}+4ia\left(5\ell^{3}-15\ell^{2}+13\ell-3\right)m+\ell\left(5\ell^{3}-21\ell^{2}+28\ell-12\right)\Bigr{)},$
(F.9)

and

$\displaystyle
f_{0}=-\left(a^{2}j+iam+\ell\right)^{2}\Bigl{(}a^{6}(j-1)j^{2}+3ia^{5}(j-1)jm$
$\displaystyle~{}~{}+a^{4}\left(3j^{2}(\ell-1)+j\left(-4\ell-3m^{2}+6\right)+2\left(\ell+m^{2}-2\right)\right)$
$\displaystyle~{}~{}-ia^{3}m\left(j(7-6\ell)+5\ell+m^{2}-6\right)+a^{2}\bigl{(}j\left(3\ell^{2}-8\ell+6\right)-3\ell^{2}$
$\displaystyle~{}~{}-3\ell
m^{2}+10\ell+4m^{2}-8\bigr{)}+3ia\left(\ell^{2}-3\ell+2\right)m+(\ell-2)^{2}(\ell-1)\Bigr{)}.$
(F.10)

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