January, 2024

Charged massless scalar fields in a charged $C$-metric black hole: Exact
solutions, Hawking radiation and scattering of scalar waves

Ming Chen, Gabriele Tartaglino-Mazzucchelli, Yao-Zhong Zhang

School of Mathematics and Physics, University of Queensland,
St Lucia, Brisbane, Queensland 4072, Australia
ming.chen@uq.edu.au,g.tartaglino-mazzucchelli<EMAIL_ADDRESS>

We study Hawking radiation and wave scattering of charged scalar fields in a
charged $C$-metric black hole background. The conformally invariant wave
equation for charged scalar fields can be separated into radial and angular
parts, each with five singularities. We show that the radial and angular
equations can be respectively transformed into the general Heun equation. We
explore exact solutions of the radial Heun equation in terms of the local Heun
functions and connection coefficients. Exact behaviours of the asymptotic wave
functions are determined without approximations. We apply the exact results to
derive Hawking radiation, quasi-normal modes and superradiance. Since
quasinormal modes are significant for black holes through gravitional waves,
we present the numeric results for quasinormal modes, and show the dependence
upon the $C$-metric parameters and the charge of scalar fields. The analytic
expressions of the solutions allow us to make fast numerical calculations of
high precision without restrictions on the model parameters.

###### Contents

  1. 1 Introduction
  2. 2 Heun’s equations
  3. 3 Asymptotic behaviors and Hawking radiation
    1. 3.1 Asymptotic behaviors
    2. 3.2 Hawking radiation
  4. 4 Boundary conditions for Quasinormal modes and superradiance at asymptotic limits
    1. 4.1 Coefficients of asymptotic behaviors in the tortoise coordinate
    2. 4.2 Applications to QNMs
    3. 4.3 Applications to superradiance
  5. 5 Exact GHE and applications to quasinormal modes and superradiance
  6. 6 Numerical results for QNMs
    1. 6.1 Acceleration modes
    2. 6.2 Near extremal modes
    3. 6.3 Photon sphere modes
  7. 7 Conclusions and Discussions
  8. A Derivation of the radial Heun equation

##  1 Introduction

The existence of black holes (BHs) is one of the most striking predictions in
Einstein’s theory of general relativity. BHs are exact solutions of the
Einstein field equations, characterized by elementary macroscopic quantities
such as mass, angular momentum and electric charge. Their physics has been an
immensely fruitful research ground, playing a vital role in examining
gravitational fields under extreme conditions. Research on BHs has recently
experienced substantial advancements, propelled by detecting gravitational
waves from the neutron stars and black hole collisions [1, 2]. Additionally,
there have been groundbreaking measurements of the central BHs in the
M87-galaxies [3]. These developments mark a significant leap forward in
understanding the dynamics and properties of BHs in our cosmos.

Black holes were predicted to enjoy remarkable phenomena, such as Hawking
radiation, when quantum effects come into play [4]. Under specific conditions,
they can amplify incident waves at the cost of the black hole spins [5] or
charges [6, 7] through, e.g., the Penrose process and superradiance [8].

Perhaps more significantly, the detected gravitational waves from a pair of
merging BHs [1] indicate that the interaction of two BHs can be conditionally
divided into four stages: the Newtonian stage, the merger of two BHs into a
single one, the ringdown phase, and the formation of a single final BH with
the consequent ringing [1, 2]. The ringdown phase or ringdown waveform is
originated from the distorted final resultant black hole and comprises the
superposition of quasinormal modes (QNMs). Each QNM possesses a complex
frequency, where the real part corresponds to the oscillation frequency and
the imaginary part is the inverse of the damping time. This damping time is
uniquely determined by the mass and angular momentum of the black hole. As the
frequencies and damping time of QNMs are directly related to the “No Hair”
theorem of BHs, a precise identification of QNMs serves as a conclusive
indicator for BHs and provides a crucial test for general relativity in the
context of strong gravitational fields [9].

Furthermore, it is well known that small perturbations of a black hole
background take the form of damped oscillations, which also lead to QNMs. By
causality, QNMs can be calculated when the perturbation is purely ingoing at
the exterior event horizon is purely outgoing at spatial infinity [10, 11, 12,
13]. Various types of fields have been used as the test fields to induce
perturbations in different black hole backgrounds, based on solutions of the
Teukolsky master equations [14, 15].

In most cases, the master wave equations can be separated into a set of
ordinary differential equations (ODEs). Various methods have been employed to
solve these separated ODEs, both approximately and analytically [16, 17, 18,
19, 20, 21] — at least for subclasses of gravitational backgrounds that allow
for that. In recent years, more researchers have been interested in solving
the ODEs analytically in terms of the Heun functions by transforming the ODEs
into Heun’s equations [22, 23, 24, 25, 26]. In [20, 27, 28], the exact
formulation for wave scatterings of BHs and Kerr-AdS5 type spacetimes was
presented based on the general Heun equation (GHE) and their exact solutions.
In [18, 29, 30], QNMs of the Kerr-dS BHs were studied in terms of the Heun
functions.

In this paper, we develop and employ the analytic approach to study solutions
and applications of the master equation in a charged $C$-metric background
[31]. The $C$-metric serves as a mathematical model for describing a pair of
black holes that are causally separated and moving apart from each other with
opposite accelerations. This metric is an extension of the Schwarzschild
solution and introduces an extra parameter associated with the acceleration of
the black hole, in addition to its mass parameter [32, 33]. In the charged
version of the $C$-metric, an electric charge parameter is included to account
for the influence of an electromagnetic field. The $C$-metric represents a
class of boost-symmetric black hole geometry and can be interpreted as a black
hole that has been accelerated under its interaction with a local cosmological
medium.

In [34, 35], the conformally invariant Klein-Gordon (KG) wave equation for a
massless neutral scalar field in the charged $C$-metric black hole was
derived. Then [31] further generalizes this coupling to the charged scalar
field case. The master equations in both cases can all be separated into
radial and angular parts (see Appendix A of [31]). Using the asymptotic
behaviors of the solutions of the separated ODEs and the Mathematica package
QNMSpectral developed in [10], QNMs and superradiance of the $C$-metric black
hole were investigated in [31]. These results indicated that the $C$-metric
remains stable under the perturbation of neutral or charged scalar fields.
This stability is reflected in the frequencies of QNM, which exhibit a
monotonic decay over time.

In this work, we study Hawking radiation and wave scattering of a charged
massless scalar field in the charged $C$-metric black hole background. The
corresponding conformally invariant KG wave equation separates into two ODEs
with five singularities. We show that the radial and angular ODEs can be
respectively transformed into the general Heun equation. Exact solutions of
the radial Heun equation are obtained in terms of the local Heun functions and
connection coefficients. This enables us to determine the exact behaviours of
the asymptotic wave functions. We apply the exact results to analyse Hawking
radiation, quasi-normal modes and superradiance. Numerical simulations for
quasinormal modes are also presented.

This paper is organized as follows. In Sec. 2, we derive the Heun equations
describing the radial and angular parts of the KG wave equation in the
$C$-metric background. In Sec. 3, we determine the asymptotic behaviors of the
radial wave function and compute the Hawking radiation. In Sec. 4, we derive
the exact coefficients of the asymptotic wave functions in the tortoise
coordinates and apply them to discuss the boundary conditions for QNMs and
superradiance. In Sec. 5, we present exact solutions of the Heun equations via
connection coefficients and obtain an analytic formulation of QNMs and
superradiance. Sec. 6 gives our numerical simulation results on QNMs and
provides a comparison of our results with those of other approaches. We
present the conclusions of the paper in Sec. 7. In Appendix A we present more
detail regarding the derivation of the radial Heun equation of sec. 3. Note
that, throughout the paper, we set the gravitational constant and the speed of
light equal to 1.

##  2 Heun’s equations

The line element of the charged $C$-metric in the Boyer-Lindquist coordinates
reads [31, 34, 35]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}\bigg{(}-f(r)dt^{2}+f^{-1}(r)dr^{2}+P(\theta)^{-1}r^{2}d\theta^{2}+P(\theta)r^{2}\sin^{2}\theta
d\varphi^{2}\bigg{)},$ (1a) where $\Lambda=1-\tilde{\alpha}r\cos\theta$ works
as a conformal factor, $\tilde{\alpha}$ is the acceleration parameter which is
positive real in the de Sitter spacetime, and the functions $f(r)$ and
$P(\theta)$ are given by $\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\bigg{(}1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\bigg{)}(1-\tilde{\alpha}^{2}r^{2}),$
(1b) $\displaystyle P(\theta)$ $\displaystyle=$ $\displaystyle
1-2\tilde{\alpha}M\cos\theta+\tilde{\alpha}^{2}Q^{2}\cos^{2}\theta,$ (1c)

with $\theta\in(0,\pi)$ and $M$ being the mass of the $C$-metric black hole.
$Q$ is the electric charge, and the electromagnetic potential associated with
the charged black hole source is $A_{\mu}=(-Q/r,0,0,0)$.

The conformally invariant wave equation for a massless charged scalar field
$\phi$ in the $C$-metric background is given by [31]

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
r}\bigg{(}r^{2}f(r)\frac{\partial\tilde{\phi}}{\partial
r}\bigg{)}-\frac{r^{4}}{r^{2}f(r)}\frac{\partial^{2}\tilde{\phi}}{\partial
t^{2}}-\frac{2iqQr}{r^{2}f(r)}\frac{\partial\tilde{\phi}}{\partial
t}+\frac{q^{2}Q^{2}}{r^{2}f(r)}\tilde{\phi}$ (2)
$\displaystyle+\frac{1}{6}\bigg{(}r^{2}f^{\prime\prime}(r)+4rf^{\prime}(r)+2f(r)\bigg{)}\tilde{\phi}+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\bigg{(}P(\theta)\sin\theta\frac{\partial}{\partial\theta}\tilde{\phi}\bigg{)}$
$\displaystyle+\frac{1}{P(\theta)\sin^{2}\theta}\frac{\partial^{2}\tilde{\phi}}{\partial\varphi^{2}}+\frac{1}{6}\bigg{(}P^{\prime\prime}(\theta)+3\cot\theta
P^{\prime}(\theta)-2P(\theta)\bigg{)}\tilde{\phi},$

where $\tilde{\phi}=\Lambda^{-1}\phi$ and $q$ is the charge of the scalar
field.

The conformally invariant wave equation (2) is separable [36]. Indeed, we
consider the Ansatz

$\tilde{\phi}(r,\theta,\varphi,t)=e^{-i\omega
t}R(r)e^{im\varphi}\,\Theta(\theta),$ (3)

where $\omega$ is the frequency (energy) of the scalar field or wave under BH
perturbations, and $m\in\mathbb{Z}$ is the azimuthal number. Moreover, $R(r)$
and $\Theta(\theta)$ are functions of the radial and angular variables $r$ and
$\theta$, respectively [37, 38]. Then, substituting (3) into (2), we find that
the wave equation is separated into the radial and angular ODEs,

$\displaystyle
R(r)^{\prime\prime}+\frac{\Delta^{\prime}(r)}{\Delta(r)}R(r)^{\prime}+\frac{1}{\Delta(r)}\bigg{[}\frac{\Omega(r)}{\Delta(r)}+\frac{1}{6}\Delta^{\prime\prime}(r)+\lambda\bigg{]}R(r)=0,$
(4a)
$\displaystyle\Theta^{\prime\prime}(\theta)+\left(\frac{P^{\prime}(\theta)}{P(\theta)}+\cot\theta\right)\Theta^{\prime}(\theta)+\frac{1}{P(\theta)}\bigg{[}\frac{-m^{2}}{P(\theta)\sin^{2}\theta}+2\tilde{\alpha}M\cos\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\tilde{\alpha}^{2}Q^{2}\cos^{2}\theta+\frac{\tilde{\alpha}^{2}Q^{2}-1}{3}+\lambda\bigg{]}\Theta(\theta)=0,$
(4b)

where the prime denotes the derivative with respect to the argument, $\lambda$
is the separation constant and

$\displaystyle\Delta(r)$ $\displaystyle=$ $\displaystyle
r^{2}f(r)=-\tilde{\alpha}^{2}(r^{2}-2Mr+Q^{2})\bigg{(}r^{2}-\frac{1}{\tilde{\alpha}^{2}}\bigg{)},~{}~{}~{}$
(5a) $\displaystyle\Omega(r)$ $\displaystyle=$
$\displaystyle\omega^{2}r^{4}-2qQ\omega r+q^{2}Q^{2}.$ (5b)

Before proceeding further, it is worth mentioning that when waves propagate
near or cross the horizon of a black hole, the extreme gravitational time
dilation leads to a substantial change in the observed frequencies $\omega$.
Typically, this effect leads to a discrete set of complex frequencies
representing exponential decay of the waves if the imaginary part is negative,
and dynamical instability of the waves if the imaginary part is positive [10].
It is thus reasonable to assume that $\omega$ is complex, i.e.
$\omega=\omega_{{}_{R}}+i\omega_{{}_{I}}$, where
$\omega_{{}_{R}}=\textrm{Re}[\omega]$, $\omega_{{}_{I}}=\textrm{Im}[\omega]$,
introduced above, are respectively the real and imaginary parts. In the
following we will use $J(r)$ to denote the square root of $\Omega(r)$,

$J(r)=\sqrt{\Omega(r)},$ (6)

which can be expressed as

$\displaystyle
J(r)=\bigg{[}\sqrt{\frac{\bar{r}+a}{2}}+i\;\textrm{sgn}(b)\sqrt{\frac{\bar{r}-a}{2}}\bigg{]},$
(7)

with $\bar{r}=\sqrt{a^{2}+b^{2}}$ being the modulus, with
$a=(\omega_{{}_{R}}^{2}-\omega_{{}_{I}}^{2})r^{4}+q^{2}Q^{2}-2qQ\omega_{{}_{R}}r$,
and $b=2(\omega_{{}_{R}}\omega_{{}_{I}}r^{4}-qQ\omega_{{}_{I}}r)$.

The separated ODEs (4a) and (4) have five regular singularities. We now show
that both equations can be transformed into the general Heun differential
equation via the change of variables.

We start with the radial equation. Setting

$\displaystyle R(r)=\Delta^{-\frac{1}{2}}(r)\Psi(r),$ (8)

we obtain from (4a) the ODE for $\Psi(r)$,

$\displaystyle\Psi^{\prime\prime}(r)+\bigg{[}\frac{1}{4}\big{(}\frac{\Delta^{\prime}(r)}{\Delta(r)}\big{)}^{2}-\frac{1}{3}\frac{\Delta^{\prime\prime}(r)}{\Delta(r)}+\frac{\Omega(r)}{\Delta^{2}(r)}+\frac{\lambda}{\Delta(r)}\bigg{]}\Psi(r)=0.~{}~{}~{}$
(9)

This equation has four finite regular singularities as well as one infinite
regular singularity at $r=\infty$. The finite singularities are determined by

$\displaystyle\Delta(r)=-\tilde{\alpha}^{2}(r-r_{+})(r-r_{-})(r-r^{\prime}_{+})(r-r^{\prime}_{-})=0,$
(10)

where $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ and
$r^{\prime}_{\pm}=\pm\frac{1}{\tilde{\alpha}}$. We set the ordering of the
four roots as $r^{\prime}_{-}<0\leq r_{-}<r_{+}<r^{\prime}_{+}$. Then
$r=r_{\pm},r_{+}^{\prime}$ are event, Cauchy and acceleration horizons,
respectively, while $r=r_{-}^{\prime}$ is not physical. We are interested in
the static region $r_{+}<r<r^{\prime}_{+}$, in which $\Delta(r)$ is positive
and the metric has fixed signature [34, 24]. This ordering requires that
$r_{+}\leq\frac{1}{\tilde{\alpha}}$, i.e.,
$\tilde{\alpha}(M+\sqrt{M^{2}-Q^{2}})\leq 1$. When the equality holds, the
black hole becomes extremal. This case is also known as the Nariai limit.
Another extremal limit occurs when $M=Q$. In this case, the event and Cauchy
horizons coincide. Note that in the static region $P(\theta)>0$ for all
$\theta\in(0,\pi)$ [39, 31].

We make the following variable change

$\displaystyle
z:=\frac{r^{\prime}_{+}-r_{-}}{r^{\prime}_{+}-r_{+}}\frac{r-r_{+}}{r-r_{-}}.$
(11)

Under this transformation,

$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r=r_{+}\rightarrow
z=0,\;\;\quad r=r^{\prime}_{+}\rightarrow z=1,$ (12a)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r=r^{\prime}_{-}\rightarrow
z=z_{r},\;\;\quad r=r_{-}\rightarrow z=\infty,$ (12b)

where $z_{r}=z_{\infty}\frac{r^{\prime}_{-}-r_{+}}{r^{\prime}_{-}-r_{-}}$ with
$z_{\infty}=\frac{r^{\prime}_{+}-r_{-}}{r^{\prime}_{+}-r_{+}}$. Thus this
transformation maps the region of interest, $r_{+}<r<r^{\prime}_{+}$, to the
one $0<z<1$. In terms of the new variable $z$, the radial equation (9) becomes

$\displaystyle\Psi^{\prime\prime}(z)+\frac{2}{z-z_{\infty}}\Psi^{\prime}(z)$
$\displaystyle~{}~{}~{}+\frac{(r_{+}-r_{-})^{2}z^{2}_{\infty}}{(z-z_{\infty})^{4}}\bigg{[}\frac{1}{4}\bigg{(}\frac{\Delta^{\prime}(r)}{\Delta(r)}\bigg{)}^{2}-\frac{1}{3}\frac{\Delta^{\prime\prime}(r)}{\Delta(r)}+\frac{\Omega(r)}{\Delta^{2}(r)}+\frac{\lambda}{\Delta(r)}\bigg{]}\Psi(z)=0.~{}~{}~{}$
(13)

Setting

$\displaystyle\Psi(z)=z^{C_{1}}(z-1)^{C_{2}}(z-z_{r})^{C_{3}}(z-z_{\infty})^{-1}\mathcal{Y}(z),$
(14)

after tedious computations (see Appendix A for details), we arrive at the
following radial ODE

$\displaystyle\mathcal{Y}^{\prime\prime}(z)+\bigg{(}\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-z_{r}}\bigg{)}\mathcal{Y}^{\prime}(z)+\frac{\alpha\beta
z-{q_{r}}}{z(z-1)(z-z_{r})}\mathcal{Y}(z)=0,~{}~{}~{}$ (15)

where

$\displaystyle\gamma=2C_{1},\;\;\quad\delta=2C_{2},\;\;\quad\epsilon=2C_{3},\;\;\quad\alpha=1\pm\sum_{j=1}^{4}\tilde{B}_{j},$
(16a)
$\displaystyle\beta=1\pm\sum_{j=1}^{3}\tilde{B}_{j}\mp\tilde{B}_{4},\;\;\quad{q_{r}}=2C_{1}C_{3}+2C_{1}C_{2}z_{r}-Az_{r},$
(16b)

with

$\displaystyle
C_{j}=\frac{1}{2}\pm\tilde{B}_{j},\;\;\quad\tilde{B}_{j}:=i\frac{J(r_{j})}{\Delta^{\prime}(r_{j})},$
(17a) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bigg{[}\frac{r^{\prime}_{+}-r_{+}}{r^{\prime}_{+}-r_{-}}+\frac{1}{2}\frac{r_{-}-r_{+}}{r_{-}-r^{\prime}_{+}}-\frac{1}{2}\frac{(r_{-}-r_{+})(r^{\prime}_{+}-r_{+})}{(r^{\prime}_{+}-r_{-})(r^{\prime}_{-}-r_{+})}\bigg{]}$
(17b)
$\displaystyle+\frac{1}{z^{2}_{r}}\bigg{(}2E+D+\frac{2E}{z_{r}}\bigg{)}-\frac{\lambda}{\alpha^{2}}\frac{1}{(r_{-}-r^{\prime}_{+})(r_{-}-r^{\prime}_{-})}\frac{z_{\infty}}{z_{r}},$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{J^{2}(r_{+})}{[\Delta^{\prime}(r_{-})]^{2}}z^{4}_{\infty},$
(17c) $\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{-4\omega^{2}r_{-}r^{3}_{+}+2qQ\omega(r_{-}+3r_{+})-4q^{2}Q^{2}}{[\Delta^{\prime}(r_{-})]^{2}}z^{3}_{\infty}.$
(17d)

Here and throughout, we use the identification

$r_{1},r_{2},r_{3},r_{4}\Longleftrightarrow
r_{+},r^{\prime}_{+},r^{\prime}_{-},r_{-},$ (18)

respectively. It can be easily checked that
$\gamma+\delta+\epsilon=\alpha+\beta+1$. So the radial equation (15) is the
general Heun equation with accessory parameter ${q_{r}}$.

We now consider the angular part. Making the variable change $x=\cos\theta$
and setting

$\displaystyle\Theta(x)=[(x^{2}-1)P(x)]^{\frac{1}{2}}\mathcal{X}(x),$ (19)

where $P(x)=1-2\tilde{\alpha}Mx+\tilde{\alpha}^{2}Q^{2}x^{2}$, then the
angular ODE (4) becomes

$\displaystyle\mathcal{X}^{\prime\prime}(x)+\bigg{[}\frac{1}{4}\bigg{(}\frac{\mathcal{P}^{\prime}(x)}{\mathcal{P}(x)}\bigg{)}^{2}-\frac{1}{3}\frac{\mathcal{P}^{\prime\prime}(x)}{\mathcal{P}(x)}+\frac{(im)^{2}}{\mathcal{P}^{2}(x)}+\frac{\lambda}{\mathcal{P}(x)}\bigg{]}\mathcal{X}(x)=0.$
(20)

Here

$\displaystyle\mathcal{P}(x)$ $\displaystyle=$ $\displaystyle(x^{2}-1)P(x)$
(21a) $\displaystyle\equiv$
$\displaystyle\tilde{\alpha}^{2}Q^{2}\,(x-x_{+})(x-x_{-})(x-x^{\prime}_{+})(x-x^{\prime}_{-}),$
(21b) $\displaystyle x_{+}$ $\displaystyle=$ $\displaystyle-1,\quad
x_{-}=\frac{1}{\tilde{\alpha}Q^{2}}(M+\sqrt{M^{2}-Q^{2}}),$ (21c)
$\displaystyle x^{\prime}_{+}$ $\displaystyle=$ $\displaystyle 1,\quad
x^{\prime}_{-}=\frac{1}{\tilde{\alpha}Q^{2}}(M-\sqrt{M^{2}-Q^{2}}).$ (21d)

Let $w^{s}_{\infty}=\frac{x^{\prime}_{+}-x_{-}}{x^{\prime}_{+}-x_{+}}$,
$w_{s}=z^{s}_{\infty}\,\frac{x^{\prime}_{-}-x_{+}}{x^{\prime}_{-}-x_{-}}$.
Similarly to the radial case, making the variable change

$\displaystyle
w:=\frac{x^{\prime}_{+}-x_{-}}{x^{\prime}_{+}-x_{+}}\frac{x-x_{+}}{x-x_{-}},$
(22)

and setting

$\displaystyle\mathcal{X}(w)=w^{{\cal C}_{1}}(w-1)^{{\cal
C}_{2}}(w-w_{s})^{{\cal C}_{3}}(w-w^{s}_{\infty})^{-1}\mathcal{Y}_{s}(w),$
(23)

we find after a long computation that the angular part is also transformed
into the general Heun equation,

$\displaystyle\mathcal{Y}_{s}^{\prime\prime}(w)+\bigg{(}\frac{\gamma_{s}}{w}+\frac{\delta_{s}}{w-1}+\frac{\epsilon_{s}}{w-w_{s}}\bigg{)}\mathcal{Y}_{s}^{\prime}(w)+\frac{\alpha_{s}\beta_{s}w-q_{s}}{w(w-1)(w-w_{s})}\mathcal{Y}_{s}(w)=0,~{}~{}~{}$
(24)

with

$\displaystyle\gamma_{s}=2{\cal C}_{1},\;\;\quad\delta_{s}=2{\cal
C}_{2},\;\;\quad\epsilon_{s}=2{\cal C}_{3},\;\;\quad\alpha_{s}=1,$ (25a)
$\displaystyle\beta_{s}=1\mp 2\tilde{\cal B}_{4},\;\;\quad q_{s}=2{\cal
C}_{1}{\cal C}_{3}+2{\cal C}_{1}{\cal C}_{2}w_{s}-{\cal A}w_{s},~{}~{}~{}$
(25b) and $\displaystyle{\cal C}_{j}=\frac{1}{2}\pm\tilde{\cal
B}_{j},\;\;\quad\tilde{\cal
B}_{j}:=\frac{m}{\mathcal{P}^{\prime}(x_{j})},\quad j=1,2,3,4,$ (25c)

with the following correspondence used here and throughout the rest of our
paper

$x_{1},x_{2},x_{3},x_{4}\Longleftrightarrow
x_{+},x^{\prime}_{+},x^{\prime}_{-},x_{-}.$ (26)

Here $\sum_{j=1}^{4}\tilde{\cal B}_{j}=0$ and ${\cal A}$ has the same form of
$A$ in (17b) with the following replacements: $r\mapsto x,z_{r}\mapsto
w_{s},z_{\infty}\mapsto w^{s}_{\infty},\tilde{\alpha}\mapsto
i\tilde{\alpha}Q,D\mapsto\frac{-4(im)^{2}}{[\mathcal{P}^{\prime}(x_{-})]^{2}}(w^{s}_{\infty})^{3}$
and
$E\mapsto\frac{(im)^{2}}{[\mathcal{P}^{\prime}(x_{-})]^{2}}(w^{s}_{\infty})^{4}$.
We remark that the parameters in (24) satisfy the condition
$\gamma_{s}+\delta_{s}+\epsilon_{s}=\alpha_{s}+\beta_{s}+1$, as required.

##  3 Asymptotic behaviors and Hawking radiation

In this section we determine the asymptotic behaviours of solutions of the
radial Heun equation (15) and apply them to analyse Hawking radiation.

###  3.1 Asymptotic behaviors

The radial GHE (15) has two linearly independent solutions in the vicinity of
$z=0$ [28, 40, 18],

$\displaystyle\ y_{01}(z)$ $\displaystyle=$
$\displaystyle\textrm{HeunG}[a,q_{r},\alpha,\beta,\gamma,z],$ (27a)
$\displaystyle\ y_{02}(z)$ $\displaystyle=$ $\displaystyle
z^{1-\gamma}\,\textrm{HeunG}[a,q_{r}+(1-\gamma)(\epsilon+a\delta),1+\beta-\gamma,1+\alpha-\gamma,2-\gamma,\delta,z],~{}~{}~{}~{}~{}~{}$
(27b)

where $a=z_{r}$ and $\textrm{HeunG}[\,\ldots,z]$ is the local Heun function

$\displaystyle\textrm{HeunG}[a,q,\alpha,\beta,\gamma,z]=\sum^{\infty}_{n=0}c_{n}z^{n},$
(28)

with coefficients $c_{n}$ defined by the three-term recurrence relation,

$\displaystyle-qc_{0}+a\gamma c_{1}=0,$ (29a) $\displaystyle
P_{n}c_{n-1}-(Q_{n}+q)c_{n}+R_{n}c_{n+1}=0,\;(n\geq 1),$ (29b)

where

$\displaystyle P_{n}$ $\displaystyle=$ $\displaystyle(n-1+\alpha)(n-1+\beta),$
(30a) $\displaystyle Q_{n}$ $\displaystyle=$ $\displaystyle
n[(n-1+\gamma)(a+1)+a\delta+\epsilon],$ (30b) $\displaystyle R_{n}$
$\displaystyle=$ $\displaystyle a(n+1)(n+\gamma).$ (30c)

The local Heun function (28) is normalized at $z=0$ as

$\displaystyle\textrm{HeunG}[a,q,\alpha,\beta,\gamma,0]=1.$ (31)

Asymptotically, when $z\rightarrow 0$, it holds

$\displaystyle\ y_{01}(z)$ $\displaystyle\sim$ $\displaystyle
1+\mathcal{O}(z),$ (32a) $\displaystyle\ y_{02}(z)$ $\displaystyle\sim$
$\displaystyle z^{1-\gamma}(1+\mathcal{O}(z)).$ (32b)

It then follows from (14) and (8) that the asymptotic radial wave function at
the exterior event horizon is given by

$\displaystyle R(r)\sim(r-r_{+})^{\tilde{B}_{1}}+(r-r_{+})^{-\tilde{B}_{1}}.$
(33)

Similarly, at $z=1$, there will be two linearly independent solutions,

$\displaystyle\ y_{11}(z)$ $\displaystyle=$
$\displaystyle\textrm{HeunG}[1-a,\alpha\beta-q,\alpha,\beta,\gamma,\delta,1-z],$
(34a) $\displaystyle\ y_{12}(z)$ $\displaystyle=$
$\displaystyle(1-z)^{1-\delta}\textrm{HeunG}[1-a,((1-a)\gamma+\epsilon)(1-\delta),\alpha\beta-q,1+\beta-\delta,$
(34b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1+\alpha-\delta,2-\delta,\gamma,1-z].~{}~{}~{}~{}~{}~{}$

Asymptotically, when $z\rightarrow 1$, it holds

$\displaystyle\ y_{11}(z)$ $\displaystyle\sim$ $\displaystyle
1+\mathcal{O}(1-z),$ (35a) $\displaystyle\ y_{12}(z)$ $\displaystyle\sim$
$\displaystyle(1-z)^{1-\delta}(1+\mathcal{O}(1-z)).$ (35b)

We thus obtain a similar asymptotic radial wave functions at the acceleration
horizon,

$\displaystyle
R(r)\sim(r-r^{\prime}_{+})^{\tilde{B}_{2}}+(r-r^{\prime}_{+})^{-\tilde{B}_{2}}.$
(36)

These asymptotic solutions are the same as those obtained via the Damour-
Ruffini-Sannan (DRS) method [47, 48, 49].

The tortoise coordinate and the surface gravity are defined as follow

$\displaystyle
dr_{*}:=\frac{dr}{\Delta(r)};\;\;\quad\kappa:=\frac{\Delta^{\prime}(r_{j})}{2r^{2}_{j}},\quad
j=1,2,3,4.$ (37)

This implies that the tortoise coordinate is

$\displaystyle
r_{*}=\sum^{4}_{j}\frac{\ln|r-r_{j}|}{\Delta^{\prime}(r)}=\frac{\ln|r-r_{+}|}{2\kappa(r_{+})r^{2}_{+}}+\frac{\ln|r-r_{-}|}{2\kappa(r_{-})r^{2}_{-}}+\frac{\ln|r-r^{\prime}_{+}|}{2\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}+\frac{\ln|r-r^{\prime}_{-}|}{2\kappa(r^{\prime}_{-})r^{\prime
2}_{-}}.~{}~{}~{}$ (38)

In terms of the tortoise coordinate, eq. (4a) is written as

$\displaystyle\frac{d^{2}R(r)}{dr^{2}_{*}}+\big{[}\Omega(r)-V_{\textrm{eff}}(r)\big{]}R(r)=0,$
(39)

where
$V_{\textrm{eff}}(r)=-\Delta(r)\big{(}\frac{1}{6}\Delta^{\prime\prime}(r)+\lambda\big{)}$.
In the asymptotic limit $r\rightarrow r_{j}$, we have

$\displaystyle\frac{d^{2}R(r)}{dr^{2}_{*}}+\Omega(r)R(r)=0.$ (40)

Its solutions can be expressed in terms of the parameters of the GHE,

$\displaystyle R(r)\sim(r-r_{j})^{\pm\tilde{B}_{j}},\;\;j=1,2,$ (41)

which are consistent with those from GHE results (33) and (36).

###  3.2 Hawking radiation

As the first application of the results based on the general Heun equations,
we consider Hawking radiation, which can be interpreted as a scattering
problem with wave modes crossing the BH event horizon or the acceleration
horizon [4].

It can be easily checked in the ordering $r^{\prime}_{-}<0\leq
r_{-}<r_{+}<r^{\prime}_{+}$ that $\kappa(r_{+})>0$ and
$\kappa(r^{\prime}_{+})<0$. Thus, from the results in the previous section,
the spatial-dependent ingoing and outgoing waves have the form,

$\displaystyle
R_{\textrm{in/out}}(r>r_{j})=(r-r_{j})^{\pm\frac{iJ(r_{j})}{2\kappa(r_{j})r^{2}_{j}}}.$
(42)

As per the different signs of $\textrm{Re}[J]$, there will be two branches.

For $\textrm{Re}[J]>0$, we have the following ingoing/outgoing solutions
around different horizons. Around the event horizon $(r-r_{+})\rightarrow
0^{+}$,

$\displaystyle\ R_{\textrm{in}}(r>r_{+})$ $\displaystyle=$
$\displaystyle(r-r_{+})^{-\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}},$ (43a)
$\displaystyle\ R_{\textrm{out}}(r>r_{+})$ $\displaystyle=$
$\displaystyle(r-r_{+})^{\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}},$ (43b)

and around the acceleration horizon $(r-r^{\prime}_{+})\rightarrow 0^{+}$,

$\displaystyle\ R_{\textrm{in}}(r>r^{\prime}_{+})$ $\displaystyle=$
$\displaystyle(r-r^{\prime}_{+})^{\frac{iJ(r^{\prime}_{+})}{2\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}},$ (44a) $\displaystyle\ R_{\textrm{out}}(r>r^{\prime}_{+})$
$\displaystyle=$
$\displaystyle(r-r^{\prime}_{+})^{-\frac{iJ(r^{\prime}_{+})}{2\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}}.$ (44b)

Considering the time-dependent damping properties, focusing on the outgoing
waves, we have

$\displaystyle\ R_{\textrm{out}}(t,r>r_{+})$ $\displaystyle=$ $\displaystyle
e^{-i\omega t}(r-r_{+})^{\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}},$ (45a)
$\displaystyle\ R_{\textrm{out}}(t,r>r^{\prime}_{+})$ $\displaystyle=$
$\displaystyle e^{-i\omega
t}(r-r^{\prime}_{+})^{-\frac{iJ(r^{\prime}_{+})}{2\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}}.$ (45b)

As a result, the outgoing wave from the event horizon is (45a).

This wave can be analytically continued inside the event horizon
($r_{-}<r<r_{+}$) by the lower half complex $r$-plane around the unit circle
$r=r_{+}-i0$: $r-r_{+}\rightarrow(r_{+}-r)e^{-i\pi}$ [21],

$\displaystyle R_{\textrm{out}}^{c}(t,r<r_{+})$ $\displaystyle=$
$\displaystyle e^{-i\omega
t}[(r_{+}-r)e^{-i\pi}]^{\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}}$ (46)
$\displaystyle=$ $\displaystyle e^{-i\omega
t}(r_{+}-r)^{\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}}(e^{-i\pi})^{\frac{i[\textrm{Re}[J(r_{+})]+i\textrm{Im}[J(r_{+})]}{2\kappa(r_{+})r^{2}_{+}}}$
$\displaystyle=$ $\displaystyle e^{-i\omega
t}\bigg{[}(r_{+}-r)^{\frac{iJ(r_{+})}{2\kappa(r_{+})r^{2}_{+}}}\bigg{]}\bigg{[}e^{\frac{i\pi\textrm{Im}[J(r_{+})]}{2\kappa(r_{+})r^{2}_{+}}}\bigg{]}\bigg{[}e^{\frac{\pi\textrm{Re}[J(r_{+})]}{2\kappa(r_{+})r^{2}_{+}}}\bigg{]},$

where $R_{\textrm{out}}^{c}(t,r<r_{+})$ denotes the analytically continued
outgoing wave.

After analytic continuation, we can get the scattering probability and
outgoing energy density for Hawking radiation:

$\displaystyle\Gamma(J)_{r_{+}}=\bigg{|}\frac{R_{\textrm{out}}(t,r>r_{+})}{R_{\textrm{out}}^{c}(t,r<r_{+})}\bigg{|}^{2}=e^{-\frac{\pi\textrm{Re}[J(r_{+})]}{\kappa(r_{+})r^{2}_{+}}},$
(47a) $\displaystyle
N(J)_{r_{+}}=\frac{\Gamma_{J}}{1-\Gamma_{J}}=\frac{1}{e^{\frac{\hbar\textrm{Re}[J(r_{+})]}{k_{B}T_{+}r^{2}_{+}}}-1},$
(47b)

where $k_{B}$ is the Boltzmann constant and $T_{+}$ the Hawking-Unruh
temperature at the event horizon: $k_{B}T_{+}=\frac{\hbar\kappa(r_{+})}{\pi}$.
When the $qQ$-dependent term can be ignored, and the frequency $\omega$ is
real, i.e. $\textrm{Re}J(r_{+})\rightarrow\omega r^{2}_{+}$, the results in
(47) reduce to those in [21, 19, 20] for a spherically symmetric background,

$\displaystyle\Gamma(J)_{r_{+}}\rightarrow
e^{\frac{-\pi\omega}{\kappa(r_{+})}};\;\;\;N(J)_{r_{+}}\rightarrow
N(\omega)_{r_{+}}=\frac{1}{e^{\frac{\hbar\omega^{2}}{k_{B}T_{+}}-1}}.$ (48)

Similarly, we can also get the Hawking radiation for the acceleration horizon

$\displaystyle\Gamma(J)_{r^{\prime}_{+}}$ $\displaystyle=$ $\displaystyle
e^{\frac{\pi\textrm{Re}[J(r^{\prime}_{+})]}{\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}},$ (49a) $\displaystyle N(J)_{r^{\prime}_{+}}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{-\frac{\hbar\textrm{Re}[J(r^{\prime}_{+})]}{k_{B}T^{\prime}_{+}r^{\prime
2}_{+}}}-1},$ (49b)

where $T^{\prime}_{+}$ is the Hawking-Unruh temperature at the acceleration
horizon: $k_{B}T^{\prime}_{+}=\frac{\hbar\kappa(r^{\prime}_{+})}{\pi}$.

For $\textrm{Re}[J]<0$, after the same procedure as above, for the event
horizon, we obtain

$\displaystyle\Gamma(J)_{r_{+}}=e^{\frac{\pi\textrm{Re}[J(r_{+})]}{\kappa(r_{+})r^{2}_{+}}},$
(50a) $\displaystyle
N(J)_{r_{+}}=\frac{\Gamma_{J}}{1-\Gamma_{J}}=\frac{1}{e^{-\frac{\hbar\textrm{Re}[J(r_{+})]}{k_{B}\tilde{T}_{+}r^{2}_{+}}}-1},$
(50b)

with $k_{B}\tilde{T}_{+}=-\frac{\hbar\kappa(r_{+})}{\pi}$. At the acceleration
horizon, it holds

$\displaystyle\Gamma(J)_{r^{\prime}_{+}}$ $\displaystyle=$ $\displaystyle
e^{-\frac{\pi\textrm{Re}[J(r^{\prime}_{+})]}{\kappa(r^{\prime}_{+})r^{\prime
2}_{+}}},$ (51a) $\displaystyle N(J)_{r^{\prime}_{+}}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\frac{\hbar\textrm{Re}[J(r^{\prime}_{+})]}{k_{B}\tilde{T}^{\prime}_{+}r^{\prime
2}_{+}}}-1},$ (51b)

with $k_{B}\tilde{T}^{\prime}_{+}=-\frac{\hbar\kappa(r^{\prime}_{+})}{\pi}$.

##  4 Boundary conditions for Quasinormal modes and superradiance at
asymptotic limits

Asymptotically, the boundary conditions of the wave functions can generally be
characterized by the tortoise coordinates $r_{*}\rightarrow\pm\infty$ and
$x_{*}\rightarrow\pm\infty$ ($x\rightarrow\mp 1$) for the radial and angular
parts, respectively [31, 13].

In the following, we discuss the asymptotic behaviors of the GHE solutions in
general without considering any specific boundary or regularity conditions. We
will then discuss the boundary conditions for two applications, QNMs and
superradiance.

###  4.1 Coefficients of asymptotic behaviors in the tortoise coordinate

We first determine the proportionality coefficients in the asymptotic wave
functions (33), (36) and (41).

From the definition of tortoise coordinate in eq. (38), asymptotically when
$r\rightarrow r_{j}$, we have

$\displaystyle
r_{*}\sim\sum^{4}_{j}\frac{\ln|r-r_{j}|}{\Delta^{\prime}(r_{j})}+d_{j},$ (52)

where $d_{j}$ are constant coefficients defined by

$d_{j}:=\sum_{k\neq j}\frac{\ln|r_{j}-r_{k}|}{\Delta^{\prime}(r_{k})},\;\quad
j,k=1,2,3,4.$ (53)

Solving (52) gives,

$\displaystyle r-r_{j}\sim
e^{-d_{j}\Delta^{\prime}(r_{j})}e^{\Delta^{\prime}(r_{j})r_{*}}.$ (54)

Then from (8), (14), (32) and (35), we can determine the asymptotic wave
solutions in terms of the tortoise coordinate $r_{*}$ as follows.

Around $z=0$, we have,

$\displaystyle\ R_{01}(r\rightarrow r_{+})\sim(r-r_{+})^{\pm\tilde{B}_{1}},$
(55a) $\displaystyle\ R_{02}(r\rightarrow
r_{+})\sim(r-r_{+})^{\mp\tilde{B}_{1}},$ (55b)

or in tortoise coordinate, $r_{*}\rightarrow-\infty$,

$\displaystyle\ R_{01}(r_{*}\rightarrow-\infty)$ $\displaystyle\sim$
$\displaystyle A_{01}^{(\mp)}e^{\pm iJ(r_{+})r_{*}},$ (56a) $\displaystyle\
R_{02}(r_{*}\rightarrow-\infty)$ $\displaystyle\sim$ $\displaystyle
A_{02}^{(\pm)}e^{\mp iJ(r_{+})r_{*}},$ (56b)

where the constant coefficients are

$\displaystyle\ A_{01}^{(\mp)}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{C_{2}}(-z_{r})^{C_{3}}(-z_{\infty})^{-1}}{\sqrt{-\tilde{\alpha}^{2}\prod_{k\neq
1}(r_{+}-r_{k})}}\,e^{\mp id_{1}J(r_{+})},$ (57a) $\displaystyle\
A_{02}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{C_{2}}(-z_{r})^{C_{3}}(-z_{\infty})^{-1}}{\sqrt{-\tilde{\alpha}^{2}\prod_{k\neq
1}(r_{+}-r_{k})}}\,e^{\pm id_{1}J(r_{+})}.$ (57b)

Around $z=1$, we get,

$\displaystyle\ R_{11}(r\rightarrow
r^{\prime}_{+})\sim(r-r^{\prime}_{+})^{\pm\tilde{B}_{2}},$ (58a)
$\displaystyle\ R_{12}(r\rightarrow
r^{\prime}_{+})\sim(r-r^{\prime}_{+})^{\mp\tilde{B}_{2}},$ (58b)

or in tortoise coordinate, or $r_{*}\rightarrow+\infty$,

$\displaystyle\ R_{11}(r_{*}\rightarrow+\infty)\sim A_{11}^{(\mp)}e^{\pm
iJ(r^{\prime}_{+})r_{*}},$ (59a) $\displaystyle\
R_{12}(r_{*}\rightarrow+\infty)\sim A_{12}^{(\pm)}e^{\mp
iJ(r^{\prime}_{+})r_{*}},$ (59b)

with the following constant coefficients,

$\displaystyle\ A_{11}^{(\mp)}$ $\displaystyle=$
$\displaystyle\frac{(1)^{C_{1}}(1-z_{r})^{C_{3}}(1-z_{\infty})^{-1}}{\sqrt{-\tilde{\alpha}^{2}\prod_{k\neq
2}(r^{\prime}_{+}-r_{k})}}\,e^{\mp id_{2}J(r^{\prime}_{+})},$ (60a)
$\displaystyle\ A_{12}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{(1)^{C_{1}}(1-z_{r})^{C_{3}}(1-z_{\infty})^{-1}}{\sqrt{-\tilde{\alpha}^{2}\prod_{k\neq
2}(r^{\prime}_{+}-r_{k})}}\,e^{\pm id_{2}J(r^{\prime}_{+})}.$ (60b)

Similarly, we can determine the coefficients for the angular solutions from
(19), (23), (32) and (35) as follows.

From (4), after making a similar variable change, one obtain an equation
similar to (4a),

$\displaystyle\mathcal{X}^{\prime\prime}(x)+\frac{\mathcal{P}^{\prime}(x)}{\mathcal{P}(x)}\mathcal{X}^{\prime}(x)+\frac{1}{\mathcal{P}(x)}\bigg{[}\frac{-m^{2}}{\mathcal{P}(x)}+\frac{1}{6}\mathcal{P}^{\prime\prime}(x)+\lambda\bigg{]}\mathcal{X}(x)=0,$
(61)

Let $w_{*}$ denote the tortoise coordinate for the angular equation, which is
defined as,

$\displaystyle
dw_{*}:=\frac{dx}{\mathcal{P}(x)};\;\;w_{*}\sim\sum^{4}_{j}\frac{\ln|x-x_{j}|}{\mathcal{P}^{\prime}(x)}+d^{s}_{j},$
(62)

where $d^{s}_{j}$ are constant coefficients defined by

$d^{s}_{j}:=\sum_{k\neq
j}\frac{\ln|x_{j}-x_{k}|}{\mathcal{P}^{\prime}(x_{k})},\;\quad j,k=1,2,3,4.$
(63)

Solving (62) gives,

$\displaystyle x-x_{j}\sim
e^{-d^{s}_{j}\mathcal{P}^{\prime}(x_{j})}e^{\mathcal{P}^{\prime}(x_{j})w_{*}}.$
(64)

In terms of $w_{*}$, equation (61) takes a form analogue to (39),

$\displaystyle\frac{d^{2}\mathcal{X}(x)}{dw^{2}_{*}}+\big{[}-m^{2}-V_{\textrm{eff}}(x)\big{]}\mathcal{X}(x)=0,$
(65)

where
$V_{\textrm{eff}}(x)=-\mathcal{P}(x)\big{(}\frac{1}{6}\mathcal{P}^{\prime\prime}(x)+\lambda\big{)}$.
In the asymptotic limit $x\rightarrow x_{j}$, we have

$\displaystyle\frac{d^{2}\mathcal{X}(x)}{dw^{2}_{*}}-m^{2}\mathcal{X}(x)=0.$
(66)

Its boundary conditions are

$\displaystyle\mathcal{X}(x)\sim e^{\pm mw_{*}},$ (67)

In terms of the parameters of the GHE, this become

$\displaystyle\mathcal{X}(x)\sim(x-x_{j})^{\frac{1}{2}+{\cal
C}_{j}},(x-x_{j})^{\frac{3}{2}-{\cal C}_{j}}\;\;j=1,2.$ (68)

In more detail, around $w=0$ ($x\rightarrow x_{+}$), it holds

$\displaystyle\ \mathcal{X}_{01}(x\rightarrow x_{+})$ $\displaystyle\sim$
$\displaystyle(x-x_{+})^{(1\pm\tilde{B}^{s}_{1})},$ (69a) $\displaystyle\
\mathcal{X}_{02}(x\rightarrow x_{+})$ $\displaystyle\sim$
$\displaystyle(x-x_{+})^{(1\mp\tilde{B}^{s}_{1})}.$ (69b)

In terms of $w_{*}$, we have

$\displaystyle\ \mathcal{X}_{01}(w_{*}\rightarrow+\infty)$ $\displaystyle\sim$
$\displaystyle A^{(\pm)s}_{01}e^{(\mathcal{P}^{\prime}(x_{+})\pm m)w_{*}},$
(70a) $\displaystyle\ \mathcal{X}_{02}(w_{*}\rightarrow+\infty)$
$\displaystyle\sim$ $\displaystyle
A^{(\mp)s}_{02}e^{(\mathcal{P}^{\prime}(x_{+})\mp m)w_{*}},$ (70b)

with
$\mathcal{P}^{\prime}(x_{+})=-2(1+2\tilde{\alpha}M+\tilde{\alpha}^{2}Q^{2})$
and

$\displaystyle\ A^{(\pm)s}_{01}$ $\displaystyle=$
$\displaystyle(-1)^{C^{s}_{2}}(-z_{s})^{C^{s}_{3}}(-z^{s}_{\infty})^{-1}\sqrt{\tilde{\alpha}^{2}Q^{2}\prod_{k\neq
1}(x_{+}-x_{k})}~{}e^{-d^{s}_{1}\mathcal{P}^{\prime}(x_{+})(1\pm\tilde{B}^{s}_{1})},$
(71a) $\displaystyle\ A^{(\mp)s}_{02}$ $\displaystyle=$
$\displaystyle(-1)^{C^{s}_{2}}(-z_{s})^{C^{s}_{3}}(-z^{s}_{\infty})^{-1}\sqrt{\tilde{\alpha}^{2}Q^{2}\prod_{k\neq
1}(x_{+}-x_{k})}~{}e^{-d^{s}_{1}\mathcal{P}^{\prime}(x_{+})(1\mp\tilde{B}^{s}_{1})}.$
(71b)

Around $w=1$ ($x\rightarrow x^{\prime}_{+}$), it holds

$\displaystyle\ \mathcal{X}_{11}(x\rightarrow x^{\prime}_{+})$
$\displaystyle\sim$
$\displaystyle(x-x^{\prime}_{+})^{(1\pm\tilde{B}^{s}_{2})},$ (72a)
$\displaystyle\ \mathcal{X}_{12}(x\rightarrow x^{\prime}_{+})$
$\displaystyle\sim$
$\displaystyle(x-x^{\prime}_{+})^{(1\mp\tilde{B}^{s}_{2})}.$ (72b)

In terms of $w_{*}$, they are

$\displaystyle\ \mathcal{X}_{11}(w_{*}\rightarrow-\infty)$ $\displaystyle\sim$
$\displaystyle A^{(\pm)s}_{11}e^{(\mathcal{P}^{\prime}(x^{\prime}_{+})\pm
m)w_{*}},$ (73a) $\displaystyle\ \mathcal{X}_{12}(w_{*}\rightarrow-\infty)$
$\displaystyle\sim$ $\displaystyle
A^{(\mp)s}_{12}e^{(\mathcal{P}^{\prime}(x^{\prime}_{+})\mp m)w_{*}},$ (73b)

with
$\mathcal{P}^{\prime}(x^{\prime}_{+})=2(1-2\tilde{\alpha}M+\tilde{\alpha}^{2}Q^{2})$
and

$\displaystyle\ A^{(\pm)s}_{11}$ $\displaystyle=$
$\displaystyle(1)^{C^{s}_{1}}(1-z_{s})^{C^{s}_{3}}(1-z^{s}_{\infty})^{-1}\sqrt{\tilde{\alpha}^{2}Q^{2}\prod_{k\neq
2}(x^{\prime}_{+}-x_{k})}~{}e^{-d^{s}_{2}\mathcal{P}^{\prime}(x^{\prime}_{+})(1\pm\tilde{B}^{s}_{2})},$
(74a) $\displaystyle\ A^{(\mp)s}_{12}$ $\displaystyle=$
$\displaystyle(1)^{C^{s}_{1}}(1-z_{s})^{C^{s}_{3}}(1-z^{s}_{\infty})^{-1}\sqrt{\tilde{\alpha}^{2}Q^{2}\prod_{k\neq
2}(x^{\prime}_{+}-x_{k})}~{}e^{-d^{s}_{2}\mathcal{P}^{\prime}(x^{\prime}_{+})(1\mp\tilde{B}^{s}_{2})}.$
(74b)

Comparing these equations with (70), (73) and (67), we see that the exact
results are different from the asymptotic solutions. As can be seen later, the
same is true for our numeric results.

###  4.2 Applications to QNMs

When a wave propagates near or crosses the horizon of a black hole, the
extreme gravitational time dilation causes a significant alteration in the
observed frequency. This effect usually manifests as a discrete set of complex
frequencies, the so-called QNMs, which characterize the decay or growth of the
wave. The real part of the QNM describes the damped oscillation, and the
imaginary part the exponential decay, provided it is negative, while a
positive imaginary part indicates a dynamical instability [10].

Quasi normal modes can generally be understood as an eigenvalue problem of the
wave equation in the black hole background under the following boundary
conditions, characterized by the frequencies $\omega$ [13, 41, 42, 43]. More
precisely, there are only ingoing waves at the event horizon, while the
ingoing modes are discarded at the acceleration horizon because they are
unobservable beyond this horizon.

In view of (55), the outgoing and ingoing wave modes at the event horizon are
(provided that Re$J(r_{+})>0$),

$\displaystyle\ \textrm{outgoing}:\;\;R_{01}(r\rightarrow
r_{+})\sim(r-r_{+})^{-\tilde{B}_{1}},$ (75a) $\displaystyle\
\textrm{ingoing}:\;\;R_{02}(r\rightarrow
r_{+})\sim(r-r_{+})^{+\tilde{B}_{1}},$ (75b)

or from (56),

$\displaystyle\ \textrm{outgoing}:\;\;R_{01}(r_{*}\rightarrow-\infty)$
$\displaystyle\sim$ $\displaystyle A_{01}^{(+)}e^{-iJ(r_{+})r_{*}},$ (76a)
$\displaystyle\ \textrm{ingoing}:\;\;R_{02}(r_{*}\rightarrow-\infty)$
$\displaystyle\sim$ $\displaystyle A_{02}^{(-)}e^{+iJ(r_{+})r_{*}},$ (76b)

since $r_{*}\rightarrow-\infty$ at $r\rightarrow r_{+}$.

Similarly, from equation (58), we have outgoing and ingoing wave modes
(provided that Re$J(r^{\prime}_{+})<0$),

$\displaystyle\ \textrm{outgoing}:\;\;R_{11}(r\rightarrow
r^{\prime}_{+})\sim(r-r^{\prime}_{+})^{-\tilde{B}_{2}},$ (77a) $\displaystyle\
\textrm{ingoing}:\;\;R_{12}(r\rightarrow
r^{\prime}_{+})\sim(r-r^{\prime}_{+})^{+\tilde{B}_{2}},$ (77b)

or from (59),

$\displaystyle\ \textrm{outgoing}:\;\;R_{11}(r_{*}\rightarrow+\infty)$
$\displaystyle\sim$ $\displaystyle A_{11}^{(+)}e^{-iJ(r^{\prime}_{+})r_{*}},$
(78a) $\displaystyle\ \textrm{ingoing}:\;\;R_{12}(r_{*}\rightarrow+\infty)$
$\displaystyle\sim$ $\displaystyle A_{12}^{(-)}e^{+iJ(r^{\prime}_{+})r_{*}},$
(78b)

since $r_{*}\rightarrow+\infty$ at $r\rightarrow r^{\prime}_{+}$. It follows
that the wave functions under the boundary conditions to determine the QNMs
are

$\displaystyle R(r)\sim\left\\{\begin{array}[]{ll}\ R_{02}(r),\quad
r\rightarrow r_{+},\\\ \\\ \ R_{11}(r),\quad r\rightarrow
r^{\prime}_{+}.\end{array}\right.$ (82)

Based on (82), we can use the standard matching procedure to derive the QNMs
[34, 31, 35]. The logic of this method is similar to that for studying
quasibound modes (QBMs): Solving the radial GHE in two different asymptotic
regions and matching the solutions in their overlapped region (see, e.g. [18,
40] and references therein).

To apply this procedure, we should determine the solutions of the angular GHE
and their regularity conditions. It can be shown that, to ensure regularity,
we need to impose $|m|<2$. From (69) at $x\rightarrow x_{+}$
($w_{*}\rightarrow+\infty$), we have

$\displaystyle\ \mathcal{X}_{01}(x\rightarrow x_{+})\sim
A^{(-)s}_{01}(x-x_{+})^{1-\tilde{\cal B}_{1}},~{}~{}~{}|m|<2,$ (83a)
$\displaystyle\ \mathcal{X}_{02}(x\rightarrow x_{+})\sim
A^{(-)s}_{02}(x-x_{+})^{1+\tilde{\cal B}_{1}},~{}~{}~{}|m|<2,$ (83b)

Similarly from (72) at $x\rightarrow x^{\prime}_{+}$
($w_{*}\rightarrow-\infty$), we get

$\displaystyle\ \mathcal{X}_{11}(x\rightarrow x^{\prime}_{+})\sim
A^{(-)s}_{11}(x-x^{\prime}_{+})^{1-\tilde{\cal B}_{2}},~{}~{}~{}|m|<2,$ (84a)
$\displaystyle\ \mathcal{X}_{12}(x\rightarrow x^{\prime}_{+})\sim
A^{(-)s}_{12}(x-x^{\prime}_{+})^{1+\tilde{\cal B}_{2}},~{}~{}~{}|m|<2.$ (84b)

###  4.3 Applications to superradiance

Superradiance denotes the interaction between the wave functions and the black
hole’s rotations. An incident wave from the accelerating horizon, scattering
off the BH, will be partially reflected backward the acceleration horizon and
partially transmitted through the potential barrier and into the event
horizon.

The boundary conditions for superradiance contain the ingoing wave modes at
the event horizon and the transmitting wave modes (outgoing and ingoing)
across the acceleration horizon [31]. From (76) and (78),

$\displaystyle R(r)\sim\left\\{\begin{array}[]{ll}\ R_{02}(r),\quad
r\rightarrow r_{+},\\\ \\\ \ R_{11}(r)+R_{12}(r),\quad r\rightarrow
r^{\prime}_{+}.\end{array}\right.$ (88)

To obtain the reflection and transmission amplitudes, we write

$\displaystyle R(r)\sim\left\\{\begin{array}[]{ll}\
\mathcal{T}e^{+iJ(r_{+})r_{*}},\quad r\rightarrow r_{+},\\\ \\\ \
\mathcal{I}e^{+iJ(r^{\prime}_{+})r_{*}}+\mathcal{R}e^{-iJ(r^{\prime}_{+})r_{*}},\quad
r\rightarrow r^{\prime}_{+},\end{array}\right.$ (92)

where $\mathcal{T}$ is the transmission wave into the event horizon,
$\mathcal{I}$ the incident wave into the acceleration horizon and
$\mathcal{R}$ the reflection wave away from the acceleration horizon. Then,
the Wronskians for these waves and their linearly independent solutions are

$\displaystyle
W[\mathcal{T}e^{+iJ(r_{+})r_{*}},\mathcal{T}^{*}e^{-iJ(r_{+})r_{*}}]$
$\displaystyle=$ $\displaystyle|\mathcal{T}|^{2}[-2iJ(r_{+})],$ (93a)
$\displaystyle
W[\mathcal{I}e^{+iJ(r^{\prime}_{+})r_{*}},\mathcal{I}^{*}e^{-iJ(r^{\prime}_{+})r_{*}}]$
$\displaystyle=$ $\displaystyle|\mathcal{I}|^{2}[-2iJ(r^{\prime}_{+})],$ (93b)
$\displaystyle
W[\mathcal{R}e^{-iJ(r^{\prime}_{+})r_{*}},\mathcal{R}^{*}e^{+iJ(r^{\prime}_{+})r_{*}}]$
$\displaystyle=$
$\displaystyle|\mathcal{R}|^{2}[+2iJ(r^{\prime}_{+})].~{}~{}~{}~{}~{}~{}$
(93c)

They must coincide at both boundaries, i.e.

$\displaystyle|\mathcal{T}|^{2}[-2iJ(r_{+})]=|\mathcal{I}|^{2}[-2iJ(r^{\prime}_{+})]+|\mathcal{R}|^{2}[2iJ(r^{\prime}_{+})].$
(94)

This yields

$\displaystyle|\mathcal{R}|^{2}=|\mathcal{I}|^{2}-\frac{J(r_{+})}{J(r^{\prime}_{+})}|\mathcal{T}|^{2}.$
(95)

The previous relation implies that $\frac{J(r_{+})}{J(r^{\prime}_{+})}<0$.
Hence, the reflection amplitude will be larger than the incident amplitude,
which is the indication that there is superradiance. This agrees with the
result in [31]. However, we still need to settle the parameters to calculate
the exact superradiant amplifications. This can be achieved using the exact
solutions.

##  5 Exact GHE and applications to quasinormal modes and superradiance

In the previous section (Sec.4.2), we studied the asymptotic behaviours of the
radial wave functions in the respective convergence region around $z=0$ and
$z=1$ and in their overlapped convergence region by matching the solutions.

In this section, we explore the exact connection-coefficient formalism
proposed in [28], which provides an effective approach to many of the
scattering-related problems. Specifically, the connection coefficients enable
us to establish an overlapped convergence disk, in which we can determine the
exact behaviors of the radial solutions at an arbitrary point in the disk
without analytic continuation.

Let $C_{11},C_{12},C_{21},C_{22}$ denote the connection coefficients
connecting the two exact solutions in (27) (of convergence at $z=0$) and (34)
(which is convergent at $z=1$). Then, by definition, in the overlapped
convergence region, the two solutions are related by

$\displaystyle\ y_{01}(z)$ $\displaystyle=$ $\displaystyle
C_{11}y_{11}(z)+C_{12}y_{12}(z),$ (96a) $\displaystyle\ y_{02}(z)$
$\displaystyle=$ $\displaystyle C_{21}y_{11}(z)+C_{22}y_{12}(z).$ (96b)

Then, we can obtain the connection coefficients around $z=0$,

$\displaystyle C_{11}=\frac{W[y_{01},y_{12}]}{W[y_{11},y_{12}]}$ ,
$\displaystyle\qquad C_{12}=\frac{W[y_{01},y_{11}]}{W[y_{12},y_{11}]},$ (97a)
$\displaystyle C_{21}=\frac{W[y_{02},y_{12}]}{W[y_{11},y_{12}]}$ ,
$\displaystyle\qquad C_{22}=\frac{W[y_{02},y_{11}]}{W[y_{12},y_{11}]},$ (97b)

where $W$ is the Wronskian,

$W[f_{1}(z),f_{2}(z)]=\bigg{|}\begin{array}[]{cc}f_{1}(z)&f_{2}(z)\\\
f^{\prime}_{1}(z)&f^{\prime}_{2}(z)\\\ \end{array}\bigg{|}.$ (98)

Around $z=1$ [28, 29], they are given by

$\displaystyle D_{11}=\frac{W[y_{11},y_{02}]}{W[y_{01},y_{02}]}$ ,
$\displaystyle\qquad D_{12}=\frac{W[y_{11},y_{01}]}{W[y_{02},y_{01}]},$ (99a)
$\displaystyle D_{21}=\frac{W[y_{12},y_{02}]}{W[y_{01},y_{02}]}$ ,
$\displaystyle\qquad D_{22}=\frac{W[y_{12},y_{01}]}{W[y_{02},y_{01}]}.$ (99b)

Consider the asymptotic behaviors of the wave functions in terms of the
connection coefficients. Within the context of conformal diagrams [44], we can
distinguish the wave functions in the four parts with different boundary
conditions as follows [28, 45].

  * •

$\mathcal{R}_{\textrm{in}}(r)$: around $z=0$ or $r=r_{+}$, with a reference
horizon between $z=0$ and $z=1$, no outgoing waves from $z=0$;

Corresponding to $r_{*}\rightarrow-\infty$, only ingoing modes should be
present for the event horizon.

  * •

$\mathcal{R}_{\textrm{up}}(r)$: around $z=1$ or $r=r^{\prime}_{+}$, with a
reference horizon (or potential barrier) between $z=0$ and $z=1$, no ingoing
waves from $z=1$;

Corresponding to $r_{*}\rightarrow+\infty$, only outgoing modes should be
present for the acceleration horizon.

  * •

$\mathcal{R}_{\textrm{down}}(r)=\mathcal{R}^{*}_{\textrm{up}}(r)$.

  * •

$\mathcal{R}_{\textrm{out}}(r)=\mathcal{R}^{*}_{\textrm{in}}(r)$.

Their relations with the GHE solutions are [28, 45, 29]

$\displaystyle\mathcal{R}_{\textrm{in}}(r)=\left\\{\begin{array}[]{ll}\
R_{02}(r),\;\;r\rightarrow r_{+};\\\ \\\ \
C_{21}R_{11}(r)+C_{22}R_{12}(r),\;\;r\rightarrow
r^{\prime}_{+},\end{array}\right.$ (103)
$\displaystyle\mathcal{R}_{\textrm{up}}(r)=\left\\{\begin{array}[]{ll}\
D_{11}R_{01}(r)+D_{12}R_{02}(r),\;\;r\rightarrow r_{+};\\\ \\\ \
R_{11}(r),\;\;r\rightarrow r^{\prime}_{+},\end{array}\right.$ (107)

where

$R_{Ii}=\Delta^{-\frac{1}{2}}(r)z^{\frac{\gamma}{2}}(z-1)^{\frac{\delta}{2}}(z-z_{r})^{\frac{\epsilon}{2}}(z-z_{\infty})^{-1}y_{Ii}(z),$
(108)

with $I=0,1$ and $i=1,2$. Similar relations can be found for the angular wave
solutions $\mathcal{X}^{s}_{Ii}$.

Comparing (103) with (82), it can be seen that in terms of the connection
coefficients, the QNMs are obtained from (103) by setting $C_{22}=0$, or
equivalently from (107) by requiring $D_{11}=0$. For the superradiance, in
view of (88), we find that in terms of the connection coefficient formalism,
one should look at equation (103). Comparing with (95), we obtain

$\displaystyle|C_{21}|^{2}|A_{11}|^{2}=|C_{22}|^{2}|A_{12}|^{2}-\frac{J(r_{+})}{J(r^{\prime}_{+})}|A_{02}|^{2}.$
(109)

One defines the amplification factor as [8, 31]

$\displaystyle Z_{0m}=\frac{|\mathcal{R}|^{2}}{|\mathcal{I}|^{2}}-1,$ (110)

where “0” denotes the scalar field with spin weight $s=0$. Then, the
amplification factor for the superradiance is [28, 29]

$\displaystyle
Z_{0m}=-\frac{J(r_{+})}{J(r^{\prime}_{+})}\bigg{|}\frac{\prod_{k\neq
2}(r^{\prime}_{+}-r_{k})}{\prod_{k\neq
1}(r_{+}-r_{k})]}\bigg{|}\bigg{(}\frac{1-z_{\infty}}{z_{\infty}}\bigg{)}^{2}\frac{1}{|C_{22}|^{2}},$
(111)

which can be simplified to

$\displaystyle
Z_{0m}=-\frac{\tilde{B}_{1}}{\tilde{B}_{2}}\bigg{(}\frac{1-z_{\infty}}{z_{\infty}}\bigg{)}^{2}\frac{1}{|C_{22}|^{2}}.$
(112)

Here we have used $\prod_{k\neq
j}(r^{\prime}_{j}-r_{k})=\Delta^{\prime}(r_{j})$ and
$\tilde{B}_{j}:=i\frac{J(r_{j})}{\Delta^{\prime}(r_{j})}$, with $j,k=1,2,3,4$.
We can then see that superradiance exists in (111), provided that
$\frac{J(r_{+})}{J(r^{\prime}_{+})}<0$. This agrees with the result in [28].

##  6 Numerical results for QNMs

In this section, we carry out numerical computations for QNMs. In [34, 35,
31], the program package QNMspectral developed in [10] was applied to do
numerical simulations (see also [13]). We will use the Mathematica package
designed to solve Heun equations based on the procedure used in [30]. Since
the angular Heun equation is independent of $\omega$, we are able to first
determine $\lambda$ and then use the obtained $\lambda$ values to determine
$\omega$ subsequently.

By the regularity consideration in Section 4.2, there are four cases while
analysing the solutions of the angular Heun equation for the computation of
$\lambda$:

  * •

$1-\tilde{\cal B}_{1}\geq 0$ and $1-\tilde{\cal B}_{2}\geq 0$, which
corresponds to

$\displaystyle W[y^{s}_{01},y^{s}_{11}]=0.$ (113)

  * •

$1-\tilde{\cal B}_{1}\geq 0$ and $1+\tilde{\cal B}_{2}\geq 0$, which
corresponds to

$\displaystyle W[y^{s}_{01},y^{s}_{12}]=0.$ (114)

  * •

$1+\tilde{\cal B}_{1}\geq 0$ and $1-\tilde{\cal B}_{2}\geq 0$, which
corresponds to

$\displaystyle W[y^{s}_{02},y^{s}_{11}]=0.$ (115)

  * •

$1+\tilde{\cal B}_{1}\geq 0$ and $1+\tilde{\cal B}_{2}\geq 0$, which
corresponds to

$\displaystyle W[y^{s}_{02},y^{s}_{12}]=0.$ (116)

Every cases can lead to $\lambda$. In this work, we choose the first branch of
solution to discuss numerical results.

For the determination of $\omega$ from the radial part, we choose the
connection coefficient $C_{22}=0$, that is

$\displaystyle\frac{W[y_{02},y_{11}]}{W[y_{12},y_{11}]}=0.$ (117)

These conditions enable us to compute the separation constant $\lambda$ and,
subsequently the frequency $\omega$.

In the following numeric outputs, we show the results of $\lambda$ and
$\omega$ which lead to a convergent local Heun functions HeunG$[\,\ldots,z]$
in the simulation.

###  6.1 Acceleration modes

Acceleration modes are purely imaginary modes when $qQ=0$ and $\tilde{\alpha}$
is not a too big value. When $qQ=0$, for $z=\frac{5}{10}$, $m=0$,
$(M,Q,\tilde{\alpha},0)=(1,\frac{1}{2},\tilde{\alpha},0)$ with
$\tilde{\alpha}=0.1,0.2,0.3$, we find a series of outputs for $\lambda$.
Corresponding to each $\lambda$ value, we find there are a series of outputs
for $\omega$ (up to $\mathcal{O}(10^{-5}$)). The results are shown in Table 1.

$\lambda$ | $\omega$
---|---
$\lambda=13.09782$ | $\omega=\pm 0.03505i;\,\pm 0.12773i;\,\pm 0.27955i$
$\lambda=92.00843$ | $\omega=\pm 0.19681i;\,\pm 0.30231i;\,\pm 0.40124i$
$\lambda=249.78348$ | $\omega=\pm 0.06997i;\,\pm 0.16049i;\,\pm 0.34988i$
$\vdots$ | $\vdots$
Table 1: $\omega$ at $\tilde{\alpha}=0.1$ for different $\lambda$

Similarly, for $\tilde{\alpha}=0.2$ and $0.3$, we find a series of numerical
values for $\lambda$, respectively; see Table 2 for these results.

$\tilde{\alpha}$ | $\lambda$
---|---
$0.2$ | $\lambda\sim 6;\,44;\,120;\,\ldots$
Table 2: $\lambda$ outputs for $\tilde{\alpha}=0.2$.

In Tables 3 and 4, we present the $\omega$ values corresponding to the
specific $\lambda$ values obtained.

$\lambda$ | $\omega$
---|---
$\lambda=6.18154$ | $\omega=\pm 0.08599i;\,\pm 0.25704i;\,\pm 0.71084i$
Table 3: Outputs of $\omega$ at $\tilde{\alpha}=0.2$ for specific $\lambda$ $\lambda$ | $\omega$
---|---
$\lambda=3.67468$ | $\omega=\pm 0.12674i;\,\pm 0.32354i;\,\pm 0.73738i$
Table 4: Outputs of $\omega$ at $\tilde{\alpha}=0.3$ for specific $\lambda$

From the results above, we see that there are indeed $\omega$ values whose
norms are roughly the same as those of $\tilde{\alpha}$. See Table 5 below.

$\tilde{\alpha}$ | $\omega$
---|---
$0.1$ | $\omega=\pm 0.12773i$
$0.2$ | $\omega=\pm 0.25704i$
$0.3$ | $\omega=\pm 0.32354i$
Table 5: Outputs of $\omega$ for different $\tilde{\alpha}$

However, there are various other values of $\omega$ for each $\tilde{\alpha}$.
Moreover, we have no reason to rule out unstable modes.

We now investigate the effect of $qQ$ on $\omega$.

$q$ | $\omega$
---|---
$0$ | $\omega=-2.45640\times 10^{-26}-0.12773i$
$0.001$ | $\omega=5\times 10^{-7}-0.14147i$
$0.01$ | $\omega=5\times 10^{-6}-0.14368i$
$0.1$ | $\omega=5\times 10^{-5}-0.14465i$
$0.2$ | $\omega=1\times 10^{-4}-0.14508i$
$0.3$ | $\omega=1.5\times 10^{-4}-0.14523i$
$\vdots$ | $\vdots$
$2.0$ | $2.00975\times 10^{-3}-0.15511i$
Table 6: Varying $q$ for negative branch of $\tilde{\alpha}=0.1$

As can be seen from table 6, there are real parts in the acceleration modes.
These real parts are proportional to the values of $qQ$ but subdominant
compared with the imaginary parts. In other words, the increase of $qQ$ leads
to the increase of the decay rate or the decrease of the lifetime, as seen
from Fig.1.

Figure 1: The green dots denote the modes varying with the increase of $q$
from 0 to 2 (i.e. $qQ$ from 0 to 1). The orange dots denote the other two
acceleration modes.

However, we did not observe any turning points for the real parts, in contrast
to Fig.2 in [31]. For the $m=1,2,-1,-2$ cases, we find the results given in
Table 7222In our tables, the symbol “$--$” means that we did not find a
converging solution to the angular/radial Heun equation..

$m$, $\lambda$ | $\omega$
---|---
$m=2,\lambda=--$ | $--$
$m=1,\lambda=3.20225$ | $\omega=\pm 0.04197i;\pm 0.14467i;\pm 0.32099i$
$m=0,\lambda=13.09782$ | $\omega=\pm 0.03505i;\pm 0.12773i;\pm 0.27955i$
$m=-1,\lambda=20.24490$ | $\omega=\pm 0.09563i;\pm 0.14926i;\pm 0.36783i$
$m=-2,\lambda=26.01624$ | $\omega=\pm 0.25469i;\pm 0.34846i$
Table 7: Outputs of $\omega$ at $\tilde{\alpha}=0.1$ and $q=0$ for varying $m$

###  6.2 Near extremal modes

Near extremal (NE) modes are also pure imaginary in the limit $Q\rightarrow
M$. In the following, we will take $\frac{Q}{M}=0.999$ to study this limit.
For $z=\frac{5}{10}$, $(l,m)=(0,0)$,
$(M,Q,\tilde{\alpha},0)=(1,\frac{999}{1000},\tilde{\alpha},0)$ with
$\tilde{\alpha}=0.1,0.3,0.5$, we find the results given in Table 8.

$\tilde{\alpha}$ | $\omega$ | $\lambda$
---|---|---
$0.1$ | $--$ | $\lambda=3.30654$
$0.3$ | $\omega=-0.14694i;\,-2.05724i;-\,3.08586$ | $\lambda=1.01289$
$0.5$ | $\omega=-0.12475i;\,-0.37425i;\pm 1.49700i;\,\pm 2.49500i$ | $\lambda=0.50044$
Table 8: $\omega$ at $Q=0.999$ and $q=0$ for varying $\tilde{\alpha}$

For $\tilde{\alpha}\leq 0.1$, we did not find any $\omega$ that makes the Heun
function converge. That is, for a small acceleration parameter, there is no NE
mode.

Regarding the effect of $qQ$ on $\omega$, it turns out that there is no
standard pattern like that in Fig.3 of [31]. The simulation is too sensitive
to the $qQ$ parameter at the NE limit to yield convergent solutions.

###  6.3 Photon sphere modes

Photon sphere (PS) modes are oscillatory QNMs. It is clear from (5b) that when
$qQ=0$, the PS modes are symmetric and this symmetry is broken when $qQ\neq
0$.

We first investigate the $qQ=0$ case. For $z=\frac{5}{10}$, $(l,m)=(0,0)$,
$(M,Q,\tilde{\alpha},0)=(1,\frac{1}{2},\tilde{\alpha},0)$ with
$\tilde{\alpha}=0.1,0.2,0.3$, we have the results given in Table 9.

$\tilde{\alpha}$ | $\omega$
---|---
$0.1$ | $\omega=5.19700-9.05388i;-5.40843-8.53097i$
$0.2$ | $--;\;\;\;\;\;\;\;\;\omega=-0.35652-1.20282i$
$0.3$ | $\omega=-0.12674i$
Table 9: Outputs of $\omega$ at $\tilde{\alpha}=0.1,0.2,0.3$

With the increase of $\tilde{\alpha}$, one observe that the oscillation
frequencies decrease while the lifetime increases distinctively. This is
generally consistent with the trend of the r.h.s of Fig.1 in [31].

For the effect of $qQ$ on $\Omega$, we show the numeric results for
$\tilde{\alpha}=0.1$ in Table 10.

$q$ | $\omega$
---|---
$0$ | $\omega=5.19700-9.05388i;-5.40843-8.53097i$
$0.1$ | $\omega=5.18795-9.01970i;-5.40469-8.53595i$
$0.2$ | $\omega=5.15238-9.08646i;-5.40127-8.54160i$
$0.3$ | $\omega=5.21390-9.02358i;-5.39789-8.54748i$
$0.4$ | $\omega=5.20503-9.03020i;-5.39450-8.55351i$
$0.5$ | $\omega=5.21119-9.00279i;-5.39110-8.55962i$
$0.6$ | $\omega=5.16298-9.00165i;-5.38769-8.56580i$
$0.7$ | $\omega=5.19550-9.01807i;-5.38427-8.57203i$
$0.8$ | $\omega=5.16185-9.01152i;-5.38083-8.57831i$
$0.9$ | $\omega=5.19494-9.04580i;-5.37738-8.58463i$
$1.0$ | $\omega=5.31502-9.01849i;-5.37391-8.59098i$
$1.1$ | $\omega=5.23809-9.05574i;-5.37043-8.59735i$
$1.2$ | $\omega=5.20695-9.02102i;-5.36694-8.60376i$
$1.3$ | $\omega=5.20970-9.02110i;-5.36343-8.61019i$
$1.4$ | $\omega=5.20632-9.00793i;-5.35991-8.61665i$
$1.5$ | $\omega=5.20163-9.03112i;-5.35638-8.62313i$
$1.6$ | $\omega=5.20886-9.04105i;-5.35283-8.62963i$
$1.7$ | $\omega=5.23168-8.97363i;-5.34927-8.63615i$
$1.8$ | $\omega=5.29659-9.12748i;-5.34570-8.64269i$
$1.9$ | $\omega=5.51560-8.98556i;-5.34211-8.64924i$
$2.0$ | $\omega=5.17890-9.02047i;-5.33851-8.65582i$
Table 10: Varying $qQ\in(0,1)$ for PS modes at $\tilde{\alpha}=0.1$

Our numeric results demonstrate that there are acceleration modes, NE modes,
as well as PS modes. However, besides the stable modes, we do find unstable
QNMs, and there is no reason to rule them out. Thus, there is a possibility
that the $C$-metric BHs are unstable against linear massless charged scalar
perturbations. We leave a more detailed analysis of this question to the
future.

##  7 Conclusions and Discussions

We have shown that the radial and angular ODEs of the conformally invariant
wave equation for the massless charged KG equation in the charged $C$-metric
background can respectively be transformed into the general Heun equations. We
have investigated the asymptotic behaviors of the radial wave functions from
both the Heun solutions and the DRS approximations. The two approaches provide
the same wave functions asymptotically. We have applied these wave functions
to explore Hawking radiation, obtaining the scattering probability and energy
density. Furthermore, we have also provided the exact proportionality
coefficients in these wave functions. The wave solutions were then further
analysed under different physical boundary conditions, such as those for QNMs
and superradiance. We have also obtained the exact wave solutions in terms of
local Heun functions and connection coefficients. These enabled us to
determine QNMs and superradiance.

We have used a code in Mathematica to solve for $\omega$ and $\lambda$ in the
radial and angular equations, arriving at numerical results for QNMs. We
underline that our approach is based on the Heun equations and proves to be
efficient in solving the wave functions and determining the QNMs. It would be
interesting to develop this approach further for computing QNMs, superradiance
and other physical quantities. For backgrounds that can be treated exactly
thanks to, e.g., the reduction to Heun equation, our analysis provides an
effective alternative compared to other well-established numerical techniques,
such as the recurrence method and the DRS approximation. The methods presented
in this paper can be applied to study exact solutions and wave scattering of
test fields in other backgrounds. It is also of particular interest to
consider Dirac fields in a $C$-metric black hole. We hope to examine some of
the above problems and report their results elsewhere.

## Acknowledgments

M.Chen has been supported by the UQ Research Training Scholarship.
G.Tartaglino-Mazzucchelli has been supported by the Australian Research
Council (ARC) Future Fellowship FT180100353, ARC Discovery Project
DP240101409, and the Capacity Building Package of the University of
Queensland. Y.Z.Zhang has been partially supported by Australian Research
Council Discovery Project DP190101529.

##  Appendix A Derivation of the radial Heun equation

In this appendix, we provide the computation details for deriving the radial
Heun equation of section 2.

Setting $\Psi(z)=(z-z_{\infty})^{-1}\Phi(z)$, we transform (2) to the form

$\displaystyle\Phi^{\prime\prime}(z)+\frac{(r_{+}-r_{-})^{2}z^{2}_{\infty}}{(z-z_{\infty})^{4}}\bigg{[}\frac{1}{4}\big{(}\frac{\Delta^{\prime}(r)}{\Delta(r)}\big{)}^{2}-\frac{1}{3}\frac{\Delta^{\prime\prime}(r)}{\Delta(r)}+\frac{\Omega(r)}{\Delta^{2}(r)}+\frac{\lambda}{\Delta(r)}\bigg{]}\Phi(z)=0.$
(118)

After re-expressing all the functions of $r$ in terms of the new variable $z$
defined in (11), we arrive at the following ODE with the singularity
$z=z_{\infty}$ removed,

$\displaystyle\Phi^{\prime\prime}(z)+\bigg{\\{}\bigg{[}\frac{A_{1}}{z}+\frac{A_{2}}{z-1}+\frac{A_{3}}{z-z_{r}}\bigg{]}+\bigg{[}\frac{B_{1}}{z^{2}}+\frac{B_{2}}{(z-1)^{2}}+\frac{B_{3}}{(z-z_{r})^{2}}\bigg{]}\bigg{\\}}\Phi(z)=0.$
(119)

Here, we have introduced the various constants

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bigg{[}\frac{r^{\prime}_{+}-r_{+}}{r^{\prime}_{+}-r_{-}}+\frac{1}{2}\frac{r_{-}-r_{+}}{r_{-}-r^{\prime}_{+}}-\frac{1}{2}\frac{(r_{-}-r_{+})(r^{\prime}_{+}-r_{+})}{(r^{\prime}_{+}-r_{-})(r^{\prime}_{-}-r_{+})}\bigg{]}$
(120a)
$\displaystyle+\frac{1}{z^{2}_{r}}\bigg{(}2E+D+\frac{2E}{z_{r}}\bigg{)}-\frac{\lambda}{\alpha^{2}}\frac{1}{(r_{-}-r^{\prime}_{+})(r_{-}-r^{\prime}_{-})}\frac{z_{\infty}}{z_{r}},$
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bigg{[}\frac{r^{\prime}_{+}-r_{+}}{r_{+}-r_{-}}-\frac{1}{2}\frac{r_{-}-r^{\prime}_{+}}{r_{-}-r_{+}}-\frac{1}{2}\frac{(r_{-}-r^{\prime}_{+})(r^{\prime}_{+}-r_{+})}{(r^{\prime}_{+}-r^{\prime}_{-})(r_{-}-r_{+})}\bigg{]}$
(120b)
$\displaystyle+\frac{1}{(z_{r}-1)^{2}}\bigg{[}2\frac{A+B+C+D+E}{z_{r}-1}+(2A+B)-(2E+D)\bigg{]}$
$\displaystyle+\frac{\lambda}{\alpha^{2}}\frac{1}{(r_{-}-r^{\prime}_{+})(r_{-}-r^{\prime}_{-})}\frac{z_{\infty}}{z_{r}-1},$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\frac{(r^{\prime}_{+}-r_{+})(r^{\prime}_{-}-r_{-})}{(r^{\prime}_{+}-r_{-})(r_{+}-r_{-})}+\frac{1}{6}\frac{(r_{-}-r^{\prime}_{-})^{2}(r^{\prime}_{+}-r_{+})}{(r_{-}-r_{+})(r^{\prime}_{+}-r_{-})(r^{\prime}_{-}-r_{+})}$
(120c)
$\displaystyle+\frac{1}{6}\frac{(r_{-}-r^{\prime}_{-})^{2}(r^{\prime}_{+}-r_{+})}{(r_{-}-r^{\prime}_{+})(r^{\prime}_{+}-r^{\prime}_{-})(r_{-}-r_{+})}-\frac{1}{z^{2}_{r}}\bigg{[}(2E+D)+\frac{2E}{z_{r}}\bigg{]}$
$\displaystyle-\frac{1}{(z_{r}-1)^{2}}\bigg{[}\;2\frac{A+B+C+D+E}{z_{r}-1}+(2A+B)-(2E+D)\bigg{]}$
$\displaystyle-\frac{\lambda}{\alpha^{2}}\frac{1}{(r_{-}-r^{\prime}_{+})(r_{-}-r^{\prime}_{-})}\frac{z_{\infty}}{z_{r}}\frac{1}{z_{r}-1},$

with

$\displaystyle A=\frac{\Omega(r_{-})}{[\Delta^{\prime}(r_{-})]^{2}},$ (121a)
$\displaystyle
B=\frac{-4\omega^{2}r_{-}^{3}r_{+}+2qQ\omega(3r_{-}+r_{+})-4q^{2}Q^{2}}{[\Delta^{\prime}(r_{-})]^{2}}z_{\infty},$
(121b) $\displaystyle
C=\frac{6[\omega^{2}r_{-}^{2}r^{2}_{+}-qQ\omega(r_{-}+r_{+})+q^{2}Q^{2}]}{[\Delta^{\prime}(r_{-})]^{2}}z^{2}_{\infty},$
(121c) $\displaystyle
D=\frac{-4\omega^{2}r_{-}r^{3}_{+}+2qQ\omega(r_{-}+3r_{+})-4q^{2}Q^{2}}{[\Delta^{\prime}(r_{-})]^{2}}z^{3}_{\infty},$
(121d) $\displaystyle
E=\frac{\Omega(r_{+})}{[\Delta^{\prime}(r_{-})]^{2}}z^{4}_{\infty},$ (121e)
$\displaystyle\left(\Delta^{\prime}(r_{-})\right)^{2}=\alpha^{4}(r_{-}-r_{+})^{2}(r_{-}-r^{\prime}_{+})^{2}(r_{-}-r^{\prime}_{-})^{2},$
(121f)

and

$\displaystyle B_{1}-\frac{1}{4}$ $\displaystyle=$
$\displaystyle\frac{\Omega(r_{+})}{[\Delta^{\prime}(r_{+})]^{2}},$ (122a)
$\displaystyle B_{2}-\frac{1}{4}$ $\displaystyle=$
$\displaystyle\frac{\Omega(r^{\prime}_{+})}{[\Delta^{\prime}(r^{\prime}_{+})]^{2}},$
(122b) $\displaystyle B_{3}-\frac{1}{4}$ $\displaystyle=$
$\displaystyle\frac{\Omega(r^{\prime}_{-})}{[\Delta^{\prime}(r^{\prime}_{-})]^{2}}.$
(122c)

For convenience, we introduce the quantity $B_{4}$ which is defined from $A$
above via the relation

$\displaystyle
B_{4}-\frac{1}{4}=A=\frac{\Omega(r_{-})}{[\Delta^{\prime}(r_{-})]^{2}}.$ (123)

Then, it can be verified that the parameters $A_{i}$ and $B_{i}$ satisfy the
following relations:

$\displaystyle A_{1}+A_{2}+A_{3}=0,$ (124a)
$\displaystyle(A_{1}+A_{3})+(A_{1}+A_{2})z_{r}=B_{1}+B_{2}+B_{3}-B_{4}.$
(124b)

We now make the substitution

$\displaystyle\Phi(z)=z^{C_{1}}(z-1)^{C_{2}}(z-z_{r})^{C_{3}}\mathcal{Y}(z).$
(125)

This substitution transforms (119) into the form

$\displaystyle\mathcal{Y}^{\prime\prime}(z)+\bigg{\\{}\frac{2C_{1}}{z}+\frac{2C_{2}}{z-1}+\frac{2C_{3}}{z-z_{r}}\bigg{\\}}\mathcal{Y}^{\prime}(z)$
$\displaystyle~{}~{}~{}+\bigg{\\{}\frac{A_{1}}{z}+\frac{A_{2}}{z-1}+\frac{A_{3}}{z-z_{r}}+\frac{2C_{1}C_{2}}{z(z-1)}+\frac{2C_{1}C_{3}}{z(z-z_{r})}+\frac{2C_{2}C_{3}}{(z-1)(z-z_{r})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{B_{1}+C^{2}_{1}-C_{1}}{z^{2}}+\frac{B_{2}+C^{2}_{2}-C_{2}}{(z-1)^{2}}+\frac{B_{3}+C^{2}_{3}-C_{3}}{(z-z_{r})^{2}}\bigg{\\}}\mathcal{Y}(z)=0.$
(126)

Choosing $C_{j},i=1,2,3$ such that

$\displaystyle C^{2}_{j}-C_{j}+B_{j}=0,$ (127a)
$\displaystyle\Longrightarrow\;\;C_{j}=\frac{1}{2}\pm
i\sqrt{B_{j}-\frac{1}{4}},$ (127b)

we can eliminate the $1/z^{2},1/(z-1)^{2},1/(z-z_{r})^{2}$ terms in (A). By
means of the relation (124a), we have

$\displaystyle\mathcal{Y}^{\prime\prime}(z)+\bigg{\\{}\frac{2C_{1}}{z}+\frac{2C_{2}}{z-1}+\frac{2C_{3}}{z-z_{r}}\bigg{\\}}\mathcal{Y}^{\prime}(z)$
$\displaystyle~{}~{}~{}+\bigg{\\{}\frac{[-(A_{1}+A_{3})-(A_{1}+A_{2})z_{r}+2C_{1}C_{2}+2C_{1}C_{3}+2C_{2}C_{3}]z-q}{z(z-1)(z-z_{r})}\bigg{\\}}\mathcal{Y}(z)=0,~{}~{}~{}~{}~{}~{}~{}$
(128)

where $q=2C_{1}C_{3}+2C_{1}C_{2}z_{r}-A_{1}z_{r}$. This equation can be
written in the form of the general Heun equation (15), i.e.

$\displaystyle\mathcal{Y}^{\prime\prime}(z)+\bigg{(}\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-z_{r}}\bigg{)}\mathcal{Y}^{\prime}(z)+\frac{\alpha\beta
z-q}{z(z-1)(z-z_{r})}\mathcal{Y}(z)=0,~{}~{}~{}~{}$ (129)

where $\gamma,\delta,\epsilon,q$ are given in (16), and $\alpha$ and $\beta$
are

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\bigg{(}C_{1}+C_{2}+C_{3}-\frac{1}{2}\bigg{)}$ (130a)
$\displaystyle\pm\sqrt{(C_{1}^{2}-C_{1})+(C_{2}^{2}-C_{2})+(C_{3}^{2}-C_{3})+\frac{1}{4}+(A_{1}+A_{3})+(A_{1}+A_{2})z_{r}},~{}~{}~{}~{}~{}~{}$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\bigg{(}C_{1}+C_{2}+C_{3}-\frac{1}{2}\bigg{)}$ (130b)
$\displaystyle\mp\sqrt{(C_{1}^{2}-C_{1})+(C_{2}^{2}-C_{2})+(C_{3}^{2}-C_{3})+\frac{1}{4}+(A_{1}+A_{3})+(A_{1}+A_{2})z_{r}}.~{}~{}~{}~{}~{}~{}$

It can be easily seen that these parameters satisfy the required condition
$\gamma+\delta+\epsilon=\alpha+\beta+1$ for the GHE.

We now show that the $\alpha$ and $\beta$ expressions above are exactly the
same as those in (16). Using the identity (124b) and the relation (127a), the
expressions for $\alpha$ and $\beta$ can be simplified to the following

$\displaystyle\alpha=\bigg{(}C_{1}+C_{2}+C_{3}-\frac{1}{2}\bigg{)}\pm
i\sqrt{B_{4}-\frac{1}{4}},$ (131a)
$\displaystyle\beta=\bigg{(}C_{1}+C_{2}+C_{3}-\frac{1}{2}\bigg{)}\mp
i\sqrt{B_{4}-\frac{1}{4}}.$ (131b)

Introducing $\tilde{B}_{j}=\sqrt{B_{j}-\frac{1}{4}},j=1,2,3,4$, then (127b)
can be written as

$\displaystyle
C_{j}=\frac{1}{2}\pm\tilde{B}_{j},\qquad\tilde{B}_{j}:=i\frac{J(r_{j})}{\Delta^{\prime}(r_{j})},$
(132)

with the same identification as before:
$r_{1},r_{2},r_{3},r_{4}\Longleftrightarrow
r_{+},r^{\prime}_{+},r^{\prime}_{-},r_{-}$. In terms of $\tilde{B}_{j}$,
$\alpha$ and $\beta$ become

$\displaystyle\alpha=1\pm\sum_{j=1}^{4}\tilde{B}_{j},\;\;\beta=1\pm\sum_{j=1}^{3}\tilde{B}_{j}\mp\tilde{B}_{4},$
(133)

which are nothing but the ones in (16)-(17a). This completes our derivation of
(15).

The derivation of the angular Heun equation is completely analogue to the
derivation above.

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