# Superradiantly tability analysis on dyonic stringly black hole

Jia-Hui Huang<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of
Quantum Engineering and Quantum Materials, School of Physics and
Telecommunication Engineering, South China Normal University, Guangzhou
510006, China  Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center, South China Normal University,
Guangzhou 510006, China  Guangdong Provincial Key Laboratory of Nuclear
Science, Institute of quantum matter, South China Normal University, Guangzhou
510006, China  Mu-Zi Zhang  Tian-Tian Cao  Yi-Feng Zou  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China  Guangdong Provincial
Key Laboratory of Nuclear Science, Institute of quantum matter, South China
Normal University, Guangzhou 510006, China  Zhan-Feng Mai
<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department of
Physics, School of Science, Tianjin University, Yaguan Road 135, Jinnan
District, 300350 Tianjin, P. R. China

###### Abstract

The black hole superradiance phenomena states that when a scalar wave
perturbation is properly scattering off a charged or rotating black hole, the
energy of the reflected scalar wave may be amplified. If this amplification
process can occur back and forth through certain confining mechanism, it will
lead to strong instability of the black hole, which is so-called ”black hole
bomb”. In this paper, the superradiant stability is investigated for dyonic
black holes in string theory. Although the electric charged black hole in
string theory has been proved superrdiantly stable, it is found that the
dyonic black hole is more unstable than a RN black hole due to the magnetic
charge. Furthermore, by our analysis on the effective potential outside the
event horizon, we give the parameters region associated with the superradiance
stability on dyonic stringly black hole.

##  I Introduction

The (in)stability problem of black holes is an interesting topic in black hole
physics. Regge and Wheelerwheeler1957  proved that the spherically symmetric
Schwarzschild black hole is stable under perturbations. The stability problems
of rotating or charged black holes are more complicated due to the significant
effect of superradiance. Superradiance effect can occur in both classical and
quantum scattering processes Manogue1988 ; Greiner1985 ; Cardoso2004 ;
Brito:2015oca ; Brito:2014wla . When a charged bosonic wave is impinging upon
a charged rotating black hole, the wave reflected by the event horizon is
amplified if the wave frequency $\omega$ satisfied the following superradiant
regime

$\omega<m\Omega+e\Phi,$ (1)

where $e$ is the charge of the bosonic wave, $m$ is the azimuthal number,
$\Omega$ is the angular velocity of black hole horizon and $\Phi$ is the
electromagnetic potential of the black hole P1969 ; Ch1970 ; M1972 ; Ya1971 ;
Bardeen1972 ; Bekenstein1973 ; Ya1972 . This means that when the incoming wave
is scattered, the wave extracts rotational energy from rotating black holes
and electromagnetic energy from charged black holes. The Eq. (1) then is
called superradiance condition. According to the black hole bomb mechanism
proposed by Press and TeukolskyPTbomb , if there is a mirror between the black
hole horizon and space infinity, the amplified wave has been trapped as well
as can be reflected back and forth between the mirror and the black hole,
growing exponentially, resulting in the superradiant instability of the black
hole.

The superradiant properties of different kinds of black holes have been
extensively studied in the literature, such as Hawking radiation (for a
review, see Brito:2015oca ). It has been illustrated that the angular momentum
of black hole will trigger the superradiant instability. A rotating black hole
can be superradiantly unstable caused by a massive scalar field when the
parameters of the black hole and the scalar filed are in certain parameter
spaces. In this case, the mass of the scalar field acts as a natural mirror. A
lot of work has been done to identify these parameter spaces, e.g.
Strafuss:2004qc ; Konoplya:2006br ; Cardoso:2011xi ; Dolan:2012yt ;
Hod:2012zza ; Hod:2014pza ; Aliev:2014aba ; Hod:2016iri ; Degollado:2018ypf ;
Huang:2019xbu . A rapidly spinning black hole that is impinged upon by a
complex and massive vector field is discussed in East:2017ovw ; East:2017mrj .
The superradiant instability of rotating black holes in curved space is also
reported Cardoso:2004hs ; Cardoso:2013pza ; Zhang:2014kna ; Delice:2015zga ;
Aliev:2015wla ; Wang:2015fgp ; Ferreira:2017tnc ; Konoplya:2013rxa ;
Kokkotas:2015uma .

It is well known that superradiant instability can also occur for charged
black holes. When there is a mirror or a cavity outside a Reissner-Nordstrom
(RN) black hole horizon, this black hole is superradiantly unstable in certain
parameter spaces Herdeiro:2013pia ; Li:2014gfg ; Degollado:2013bha ; Sanchis-
Gual:2015lje ; Fierro:2017fky ; Gonzalez:2017gwa . However, in flat
background, RN black holes have been proved to be superradiantly stable
against charged massive scalar perturbationHod:2013eea ; Huang:2015jza ;
Hod:2015hza ; DiMenza:2014vpa . It is pointed that when the parameters of a RN
black hole and a charged massive scalar field satisfy superradiant conditions,
there is no effective trapping potential outside the black hole horizon, which
acts as a mirror to reflect the superradiant modes Huang:2015jza . When
charged black holes are in curved backgrounds, such as (anti-)de Sitter space,
these backgrounds provide natural reflecting boundary conditions and support
the existence of superradiant instability Wang:2014eha ; Bosch:2016vcp ;
Huang:2016zoz ; Gonzalez:2017shu ; Zhu:2014sya . The discussion of black hole
superradiant property has also been extended to the analogue of RN black hole
in string theory and it has been shown that stringy RN black hole is
superradiantly stable against charged massive scalar perturbationLi:2013jna ;
Konoplya:2011qq . But when a mirror is introduced, superradiant modes exist
and the stringy RN black hole becomes unstable Li:2014xxa ; Li:2014fna ;
Li:2015mqa .

In addition, magnetic field can trigger superradiant instability for a black
hole. In Konoplya:2007yy ; Konoplya:2008hj , the authors use an approximation
method to show that when a scalar field is propagating on the Ernst
background, the magnetic field will induce an effective mass $\mu_{eff}\propto
B$($B$ is the magnetic field strength) for the scalar field, leading to the
superradiant instability. In a further work Brito:2014nja , the authors
considered Kerr-Newman black holes immersed in a uniform magnetic field and
showed that the magnetic field can confine scalar perturbations leading to
long-lived modes, which trigger superradiant instabilities.

Based on our illustration above, it is thus worth asking a question whether a
series of asymptotic flat dyonic (with both electric charge and magnetic
charge) stringly black hole is superradiantly stable against a massive charged
perturbative scalar field. In our previous work Huang:2015jza , it was
demonstrated that the effective potential outside the outer horizon have not
potential well when both the superradiance condition and bound state condition
were imposed, we thus showed that electrically charged RN black holes are
superradiantly stable under massive charged scalar perturbation. However, in
this paper, we will consider the superradiant behavior of the dyonic black
holes under a charged massive scalar perturbation. For our case, the metric of
the dyonic black hole is similar to a electrically charged RN black hole,but
pthe magnetic field also leads to the superradiant instability of the dyonic
black holes and scalar perturbation system.

##  II The Superradiance Condition of Dyonic Black Hole

As our demonstration in Huang:2015jza , the electric charged of RN black hole
in asymptotic flat will not trigger the superradiance instability under the
superradiance condition $\omega<\omega_{c}=\frac{qQ_{e}}{r_{+}}$ in which $Qe$
and $r_{+}$ denote the electric charge and event horizon of the RN black hole
respectively. A worth-answering question is whether the magnetic charge
associated with black hole will change the superradiance condition. From the
view of classical electromagnetic dynamic, the Lorentz force associated to the
static magnetic field does not do work as well as increase(decrease) the
kinetic energy carried by a given charged particle. Therefore, we suppose a
conjecture that the magnetic charged carried by the dyonic black hole will not
chage the superradiance condition. In following, we shall prove this
conjecture.

At first, we give a brief introduction on the scalar perturbation related to
the stringly dyonic black hole. The action reduced from low energy effective
theory of string theory in four dimension, written in the Einstein-frame, is
given (here we adopt the natural unit as $G=c=\hbar=k=1$)

$S=\frac{1}{16\pi}\int_{M}\text{d}^{4}x\sqrt{-g}\left(R-\frac{1}{2}\left(\partial\phi\right)^{2}-\text{e}^{-2\phi}F^{\mu\nu}F_{\mu\nu}\right),$
(2)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $\phi$
denote the strength of Maxwell field and dilaton field respectively. By
performing variation on the action, we will obtain the equation of motion

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi$
(4)
$\displaystyle+2\text{e}^{-2\phi}F^{\mu}_{\rho}F_{\nu}{}^{\rho}-\frac{1}{2}g_{\mu\nu}\left(\frac{1}{2}(\partial\phi)^{2}-\text{e}^{-2\phi}F^{\mu\nu}F_{\mu\nu}\right)\,,$
(6) $\displaystyle\nabla_{\nu}\left(\text{e}^{-2\phi}F^{\mu\nu}\right)=0\,,$
(8) $\displaystyle\square\phi+2\text{e}^{-2\phi}F^{\mu\nu}F_{\mu\nu}\phi=0.$

The theory Eq. (2) admits a series of stringly dyonic black hole solutions
with the ADM mass $M$, electric charge $Q_{e}$ and magnetic charge
$Q_{m}$Horowitz:1992jp ; Sen:1992fr . Specifically, the metric of this class
of stringly dyonic black hole can be written as follow,

$\displaystyle\text{d}{s^{2}}=-\frac{\Delta}{\rho(r)}\text{d}{t^{2}}+\frac{\rho(r)}{\Delta}\text{d}{r^{2}}+\rho(r)(\text{d}\theta^{2}+\sin^{2}\theta\text{d}\varphi^{2})\,,$
(9) (11) $\displaystyle
A_{\mu}=-\frac{Q_{e}}{r}\left(\text{d}t\right)_{\mu}-Q_{m}\cos\theta\left(\text{d}\varphi\right)_{\mu}\,,$
(13) $\displaystyle\text{e}^{2\phi}=\frac{r-r_{0}}{r+r_{0}}\,,$

where

$\displaystyle\Delta=(r-r_{+})(r-r_{-})\,,\rho(r)=r^{2}-r_{0}^{2}\,$ (14)
$\displaystyle r_{0}=\frac{Q^{2}_{m}-Q^{2}_{e}}{2M}\,,$ (15) $\displaystyle
r_{\pm}=M\pm\sqrt{M^{2}+r^{2}_{0}-Q^{2}_{m}-Q^{2}_{e}}\,.$ (16)

To proceed, we then derive the master equation associated with the
perturbative charged scalar field $\Psi$ under the dyonic black hole
background. Due to the static and spherical dyonic black hole background, the
equation of motion associated with $\Psi$ can reduce to a radial equation in 1
dimension in frequency domain. The dynamics of the charged massive scalar
field is described by the Klein-Gordon (KG) equation

$\left(({\nabla^{\nu}}-iq{A^{\nu}})({\nabla_{\nu}}-iq{A_{\nu}})-{\mu^{2}}\right)\Psi=0,$
(17)

where $q$ and $\mu$ are the charge and rest mass of the charged scalar field.
The nonzero vector field components are $A_{t}=-Q_{e}/r$ and
$A_{\varphi}=-Q_{m}\cos\theta$, describing the electric and magnetic charges
related to the black holeSen:1992fr .

Applying to the variables separation method in spherical black hole
background, we take the anstanz of $\Psi$ as

$\Psi(t,r,\theta,\varphi)=R(r)Y(\theta)\text{e}^{im\varphi}\text{e}^{-i\omega
t}\,,$ (18)

where $m$ is azimuthal harmonic index and $\omega$ is the frequency of the
scalar perturbation. Based on the periodic condition related to $\varphi$,

$\varphi\to\varphi+2\pi\,,$ (19)

the invariance of exponential complex function with respect to $\varphi$,
$\text{e}^{im\varphi}=\text{e}^{im\varphi}\text{e}^{i2m\pi}$, requires the
discrete spectrum of the azimuthal harmonic index, namely $m=0,\pm 1\,\pm
2\cdots$. We then focus on investigating the angular function $Y(\theta)$,
Plugging the anstanz Eq. (18) into the equation of motion Eq. (17), the
angular equation with respect to $Y(\theta)$ gives

$\displaystyle\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}Y)$
(20)
$\displaystyle+(\lambda_{lm}-\frac{m^{2}+q^{2}Q_{m}^{2}+2mqQ_{m}\cos\theta}{\sin^{2}\theta})Y=0\,,$
(21)

where $\lambda_{lm}$ is so-called separation constant, as well as usually set
to $\lambda=l(l+1)$. By Performing variable transformation as
$\eta=\cos\theta$ for convenience, we arrive that the angular equation Eq.
(20) will deform to a class of the Fuchs equation

$\displaystyle\frac{\text{d}^{2}Y}{\text{d}\eta^{2}}+(\frac{1}{\eta-1}+\frac{1}{\eta+1})\frac{\text{d}Y}{\text{d}\eta}+\left(-\frac{(m+qQ_{m})^{2}}{2(\eta-1)}\right.$
(22) (24)
$\displaystyle+\left.\frac{(m-qQ_{m})^{2}}{2(\eta+1)}-\lambda\right)\frac{Y}{(\eta-1)(\eta+1)}=0,$

In general, the physical boundary condition requires that the angular function
$Y(\theta)$ should be regular at $\theta=0,\pi$, corresponding to $\eta=\pm
1$, we then choose one branch of solution associated with Fuchs equation (22),

$\displaystyle Y(\eta)=$ $\displaystyle
C(1-\eta)^{\frac{1}{2}(m+qQ_{m})}(1+\eta)^{\frac{1}{2}(m-qQ_{m})}$ (25) (27)
$\displaystyle{}_{2}F_{1}\left(\alpha,\beta;\gamma,\frac{1-\eta}{2}\right)\,,$

where $C$ is a normalization constant. The parameters $\alpha,\beta$ and
$\gamma$ are given by

$\left\\{\begin{array}[]{l}{\alpha=m+\frac{1}{2}(1+\sqrt{1+4\lambda})}\\\
{}\cr{\beta=m+\frac{1}{2}(1-\sqrt{1+4\lambda})}\\\
{}\cr{\gamma=1+m-qQ_{m}}\end{array}\right.$ (28)

The regular condition of $Y(\eta)$ in the region $-1\leqslant\eta\leqslant 1$
also requires that

  * •

$\beta$ is a negative integer, corresponding to the hypergeometric function
reducing to hypergeometric polynomial which must be regular in $\eta=\pm 1$,
giving that the separation constant $\lambda=l(l+1)$ and $l>|m|$.

  * •

In addition to regularity of requirement associated with the hypergeometric
polynomial, the regularity of the factor
$(1-\eta)^{\frac{1}{2}(m+qQ_{m})}(1+\eta)^{\frac{1}{2}(m-qQ_{m})}$ in (25)
implies the following two inequalities,

$m>qQ_{m}\,,\quad l(l+1)>q^{2}Q_{m}^{2}\,.$ (29)

It is remarkable to note that different from the cases of RN black hole and
the electrically charged black hole in string theory, the solution of angular
equation $Y(\eta)$ will be no longer hypergeometric function due to the
interaction between the charged and magnetic field of the black hole. Through
the regular analysis on the hypergeometric function, the regularity associated
with the angular function $Y(\theta)$ gives the lower bounds of the angular
quantum numbers, namely $l(l+1)>q^{2}Q_{m}^{2}$. It means that due to the
interaction between the charged scalar and the magnetic field, $l$ will not
vanish, implying that the perturbative particle must spin around the black
hole under the magnetic field semi-classically.

Finally, we arrive the radial equation of the perturbative charged scalar
field in frequency domain,

$\Delta\frac{\text{d}}{\text{d}r}\left(\Delta\frac{\text{d}R(r)}{\text{d}r}\right)+U(r)R(r)=0\,,$
(30)

where

$U(r)=\left(\omega-\frac{qQ_{e}}{r}\right)^{2}\rho(r)^{2}-\Delta\left(l\left(l+1\right)-q^{2}Q_{m}^{2}-\mu^{2}\rho(r)\right)$
(31)

In order to study whether the superradiance condition of the dyonic black hole
against the massive charged perturbative scalar field will be changed by the
static magnetic charge, one need to impose both the ingoing and outgoing wave
boundary condition at the infinity, as well as the ingoing wave boundary
condition near the horizon region. That indicates that a beam of charged
scalar field propagate from spatial infinity to the black hole horizon and
then is scattered by the black hole and a part of this beam is reflected to
the infinity while the other part was absorbed by the black hole. To achieve
that, it is convenient to define the tortoise coordinate $y$ as

$\frac{\text{d}y}{\text{d}r}=\frac{r^{2}}{\Delta},$ (32)

and a new radial function as ${\tilde{R}}=rR$, the radial wave equation (30)
can be written in following standard wave function form

$\frac{\text{d}^{2}\tilde{R}}{\text{d}y^{2}}+\tilde{U}\tilde{R}=0,$ (33)

where the potential

$\tilde{U}=\frac{U}{r^{4}}+\frac{2\Delta^{2}}{r^{6}}-\frac{\Delta}{r^{5}}\frac{d\Delta}{dr}.$
(34)

To observe the behavior of the radial function $\tilde{R}$, one can
asymptotically perform Taylor expansion at both near horizon and the spatial
infinity, specifically have

$\lim_{r\to{r_{+}}}\tilde{U}=\left(\omega-\frac{qQ_{e}}{r_{+}}\right)^{2}\left(r_{+}^{2}-r_{0}^{2}\right)^{2},~{}~{}\lim_{r\to+\infty}\tilde{U}=\omega^{2}-{\mu^{2}}\,.$
(35)

The radial function $\tilde{R}$ then can be solved as

$\tilde{R}\varpropto\left\\{\begin{aligned} &{\cal
T}\text{e}^{ik_{+}y}\,,\quad&y\to-\infty\,,\cr&{\cal
L}\text{e}^{i\sqrt{\omega^{2}-\mu^{2}}y}+{\cal
R}{\text{e}^{-i\sqrt{\omega^{2}-\mu^{2}}y}}\,,\quad&y\to+\infty\,,\end{aligned}\right.$
(36)

where the wave number in the near horizon region $y\to-\infty$ is

$k_{+}=\pm\frac{1}{r_{+}}\left(\omega-\frac{qQ_{e}}{r_{+}}\right)\left(1-\left(\frac{r_{0}}{r_{+}}\right)^{2}\right)\,,$
(37)

and also the complex number ${\cal L}$, ${\cal R}$ and ${\cal T}$ are denoted
as the incident amplitude, reflection amplitude and the transmission amplitude
respectively. Remarkably, the sign of $\omega$ and $k_{+}$ is decided by the
requirement associated with the propagating direction denoted by the group
velocity. In general, we require the wave propagate from the spatial infinity
$y\to\infty$ to the black hole horizon $y\to-\infty$, implying that
$\frac{\partial\omega}{\partial k_{+}}<0$. We thus choose the negative sign.
On the other hand, the ${\cal R}$ and ${\cal T}$ are decided by the explicit
shape of the potential $\tilde{U}$. To obtain the superradiance condition, we
evaluate the conserved Wronskain associated with the solution $\tilde{R}$ and
its complex conjugate $\tilde{R}^{*}$ at both boundary, corresponding to the
conserved current density, then have

$|{\cal R}|^{2}=\left|{\cal L}\right|^{2}-\frac{\left(\omega-
qQ_{e}/r_{+}\right)\left(r_{+}^{2}-r_{0}^{2}p\right)}{\omega^{2}-\mu^{2}}\left|{\cal
T}\right|^{2}\,.$ (38)

One can easily observe that when

$0<\omega<\frac{qQ_{e}}{r_{+}}\,$ (39)

$|{\cal R}|^{2}>|{\cal L}|^{2}$, namely the reflection amplitude is larger
than the incident one, indicating that the superradiance phenomena occurs. On
the other hand, the above illustration also give the superradiance condition
Eq. (39). One can easily see that the superradiance condition of dyonic black
hole is the same as the case of stringly RN black holeLi:2013jna , even the RN
black hole in Einstein theoryDiMenza:2014vpa . Therefore, we have proved our
conjecture that the static magnetic field in dyonic black hole in string
theory will not effect the superradiance condition. It is remarkable to note
that the superradiance condition Eq. (39) which is important for us to
investigate the superradiance instability of the stringly dyonic black hole.

##  III Superradiant Stability analysis of dyonic black hole

In the last section, we proved that the magnetic field will not effect the
superradiance condition $\omega<qQ_{e}/r_{+}$ in which $q$, $Q_{e}$ and
$r_{+}$ are denotes the charge of the perturbative scalar field, the electric
charge and the location of the event horizon of the dyonic black hole
respectively. In other words, if one inject a beam of perturbative charged
scalar field whose the frequency satisfying the superradiance condition
$0<\omega<qQ_{e}/r_{+}$, only the electric energy of the dyonic black hole,
rather than the magnetic energy, will be extracted by the charged scalar field
due to the superrdiance phenomena. The next question we need to answer is
whether the dyonic black hole in string theory is superradiantly stable once
the superradiance mode of the perturbative charged scalar field has been
triggered on. In Ref. Li:2013jna ; DiMenza:2014vpa , it has been proved that
the pure electric charged black hole either in the Einstein gravity theory or
string theory is superradiantly stable against a perturbative charged scalar
field. Therefore, it is very interesting to investigate whether the magnetic
charge will trigger the superradiant instability in the stringly dyonic black
hole. In addition to the superradiance condition, another necessary condition
to trigger the superradiance instability of a given black hole is that there
exist a trapping well associated with the effective potential outside the
event horizon to trap the superradiance mode. If there does exist a trapping
well of the effective potential, a bound state of the perturbative scalar
field will arise and the superradiance amplification of such a bound state
will lead to instability of the black hole, which is so-called the ”black hole
bomb” mechanic.

To achieve to investigate whether there exist a trapping well triggering the
superradiant instability of the stringly dyonic black hole, we at first
consider the case of $Q_{e}=Q_{m}$. In this case, the derivative function of
effective potential, of which the numbers of roots indicates the existence of
the trapping well, owe a more simpler polynomial form to analyze. Therefore,
we adopt analytical method to study the associated superradiant instability.
However, when taking $Q_{e}\not=Q_{m}$ into account, one will find that the
number of roots of the derivative function related to the effective potential
cannot be given efficiently by analytical analysis. Therefore, we turn to
investigate the existence of the trapping well of the effective potential
outside the horizon by numerical method.

###  III.1 The Bound State Condition

From our illustration above, we study the superradiant stability on dyonic
black hole through analysing the existence of trapping well of the potential
in term of the radial equation outside the horizon to trap the bound state of
$\Psi$. Therefore, we firstly need to give a bound state condition.
Practically, To obtain the appropriate effective potential as well as deform
the radial equation Eq. (30) to a Schrödinger-like equation, it is convenient
to define a new radial function $\psi$,

$\psi=\Delta^{\frac{1}{2}}R\,.$ (40)

The radial equation Eq. (30) can be written as the following Schrödinger-like
equation

$\frac{\text{d}^{2}\psi}{\text{d}r^{2}}+(\omega^{2}-V)\psi=0\,,$ (41)

where the effective potential is

$V=\omega^{2}+\frac{1}{\Delta^{2}}\left(\Delta-U-(r-M)^{2}\right)\,.$ (42)

Remarkably, in general the radial equation as well as its effective potential
is investigated in tortoise coordinate. However, the effective potential given
by the radial equation in tortoise coordinate, namely $\tilde{U}$ in Eq. (34)
is not appropriate for our investigation on superradiant instability on dyonic
black hole since the effective potential in tortoise coordinate $\tilde{U}$ is
effectively approach to constants, rather than zero, in $y\to\pm\infty$
region. The first radial function $\tilde{R}$ then is free propagator in the
near horizon region $y\to\infty$. Nevertheless, from the definition of the
second radial function $\psi$, namely Eq. (40), one can find that $\psi$ is
effectively reduce to zero at the event horizon since $\Delta|_{r=r_{+}}=0$,
corresponding to the bound state condition. To match the bound state condition
at the spatial infinity region, we need to analyze the asymptotic behavior of
the potential $V$. We have

$V\to-\infty\,,\quad r\to r_{\pm}$ (43)

as well as

$V\to\mu^{2}+\frac{-4M\omega^{2}+2qQ\omega+2M\mu^{2}}{r}+{\cal
O}\left(\frac{1}{r^{2}}\right),\quad r\to+\infty.$ (44)

The leading order of $V$ implies that the radial equation in terms of $\psi$
will reduce to

$\frac{\text{d}^{2}\psi}{\text{d}r^{2}}+(\omega^{2}-\mu^{2})\psi=0\,,\quad
r\to\infty$ (45)

Interpreted $\psi$ as a wave function in bound state after superradiance
amplification locally, the bound state condition at spatial infinity requires
that

$\omega<\mu\,,$ (46)

corresponding the exponential decay mode associated with $\psi$ as
$\psi|_{r\to\infty}\varpropto\text{e}^{-\sqrt{\mu^{2}-\omega^{2}}r}$ at the
spatial infinity. Therefore, we finally obtain two necessary condition for us
to investigate the existence associated with the trapping well to trap the
bound state mode for the potential $V$ outside the horizon, namely the
superradiance condition as well as the bound state condition,

$\omega<\frac{qQ_{e}}{r_{+}}\,,\quad\omega<\mu.$ (47)

###  III.2 Analytical Analysis: The case of $Q_{e}=Q_{m}$

To proceed, we turn to study the general shape of the potential $V$, which can
generally depend on six parameters $\\{M,Q_{e},Q_{m},\omega,\mu,q,l\\}$ in
which the first three parameters are given by the dyonic black hole and the
last four parameters denote the property of the perturbative charged scalar
field $\Psi$. Naively we consider the case of $Q_{e}=Q_{m}=Q$. Recall the
subleading order of $V$ given in Eq. (44), it is easy to prove that

$\displaystyle-4M\omega^{2}+2qQ\omega+2M\mu^{2}$ (48) (49) $\displaystyle>$
$\displaystyle-4M\omega^{2}+2qQ\omega+2M\omega^{2}\quad$
$\displaystyle(\omega<\mu)$ (50) (51) $\displaystyle=$ $\displaystyle
2M\omega\left(\frac{qQ}{M}-\omega\right)\quad$ $\displaystyle(r_{+}>M)$ (52)
(53) $\displaystyle>$ $\displaystyle
2M\omega\left(\frac{qQ}{r_{+}}-\omega\right)>0\quad$
$\displaystyle(\omega<qQ/r_{+})\,,$ (54)

The asymptotic behavior of $V$, Eq. (44) and Eq. (48), implies that the
potential $V$ has at least one maximum in the region $r_{-}<r<r_{+}$ and one
maximum outside the event horizon ($r>r_{+}$).

In practice, if there was only one maximum outside the event horizon for $V$,
there will be no trapping well associated with $V$ to trap the bound state of
$\Psi$, indicating that the dyonic black hole will be superradiantly stable.
In following we will give a class of region related to the parameter space to
illustrate the superradiant stability for the dyonic black hole by analysing
the number of root associated with the derivative function of $V$. Since it
can be concluded that the dyonic black hole is superradiantly stable when the
potential $V$ has only one maximum outside the horizon. According to some
basic conclusion related to calculus, it indicates that one can find the
regions of parameters to illustrate the superradiant stability on dyonic black
hole under some constrains that the derivative function of $V$ has only one
root outside the horizon.

Through the analysis on the asymptotic behavior of $V$, it has been concluded
that there at least exist one maximum of $V$ when $r_{-}<r_{+}$ and $r>r_{+}$
respectively. For convenience, we define a new radial variable $z=r-r_{-}$.
The derivative function with respect to the $V$ then be given

$\frac{\text{d}V}{\text{d}z}=\frac{-2}{\Delta^{3}}\left(\left(\Delta-U-\left(z+r_{-}-M\right)^{2}\right)\frac{\text{d}\Delta}{\text{d}z}+\frac{1}{2}\Delta\frac{\text{d}U}{\text{d}z}\right).$
(55)

It is also easy to conclude that there exist at least two roots of the
numerator of $\frac{\text{d}V}{\text{d}z}$ when $z>0$ since the translation of
$r$ will not change the number of root associated with the numerator of
$V^{\prime}(z)$. Practically, the numerator of $V^{\prime}(z)$ can be written
in a polynomial form in terms of $z$

$f(z)=az^{4}+bz^{3}+cz^{2}+dz+e\,,$ (56)

where

$\displaystyle a$ $\displaystyle=$
$\displaystyle-2M\mu^{2}+4M\omega^{2}-2qQ\omega\,,$ (57) (58) $\displaystyle
b$ $\displaystyle=$ $\displaystyle
4\left(8M^{2}-6Mr_{+}+r_{+}^{2}\right)\omega^{2}-4qQ\left(5M-2r_{+}\right)\omega$
(59) $\displaystyle-$ $\displaystyle
2\left(l+l^{2}-2q^{2}Q^{2}+\left(6M^{2}-6Mr_{+}+r_{+}^{2}\right)\mu^{2}\right)\,,$
(60) (61) $\displaystyle c$ $\displaystyle=$ $\displaystyle
12\left(2M-r_{+}\right)^{3}\omega^{2}-18qQ\left(2M-r_{+}\right)^{2}\omega$
(62) $\displaystyle-$ $\displaystyle
6\left(\left(M-r_{+}\right)l\left(l+1\right)-3Mq^{2}Q^{2}\right.$ (63)
$\displaystyle+$
$\displaystyle\left.2r_{+}q^{2}Q^{2}+\mu^{2}\left(M-r_{+}\right)\left(2M-r_{+}\right)^{2}\right)\,,$
(64) (65) $\displaystyle d$ $\displaystyle=$ $\displaystyle
4\left(4M-3r_{+}\right)\left(2M-r_{+}\right)^{3}\omega^{2}$ (66)
$\displaystyle-$ $\displaystyle
4qQ\left(7M-5r_{+}\right)\left(2M-r_{+}\right)^{2}\omega$ (67)
$\displaystyle-$ $\displaystyle
4\left(2M^{2}-3Mr_{+}+r_{+}^{2}\right)^{2}\mu^{2}$ (68) $\displaystyle+$
$\displaystyle 4\left(7M^{2}-9Mr_{+}+3r_{+}^{2}\right)q^{2}Q^{2}$ (69)
$\displaystyle-$ $\displaystyle
4\left(l\left(l+1\right)-1\right)\left(M-r_{+}\right)^{2}\,,$ (70) (71)
$\displaystyle e$ $\displaystyle=$ $\displaystyle
2\left(r_{-}-r_{+}\right)r^{2}_{-}\left(qQ-\omega
r_{-}\right)^{2}+\frac{1}{2}\left(r_{-}-r_{+}\right)^{3}\,.$ (72)

The four roots of $f(z)$ , denoted by $z_{1},z_{2},z_{3},z_{4}$, should
satisfy the following two identities,

$\displaystyle z_{1}\times z_{2}\times z_{3}\times z_{4}=\frac{e}{a}\,,$ (73)
(76) $\displaystyle z_{1}\times z_{2}+z_{1}\times z_{3}+z_{1}\times z_{4}+$
$\displaystyle z_{2}\times z_{3}+z_{2}\times z_{4}+z_{3}\times
z_{4}=\frac{c}{a}\,.$

Based on some general algebraic quaternary equation properties, in principle,
these four roots are in complex plane. According to our discussion on Eq.
(48), There already have been two real roots. It is easy to see that
$V^{\prime}(z)$ will only have two real roots since if there is complex root
among $z_{1},z_{2},z_{3},z_{4}$, its complex conjugate is also a root.
Therefore in the following discussion, we naively suppose a worse case that
all four roots are real. Recall the superradiant conditions $\omega<qQ/r_{+}$
as well as the bound state condition $\omega<\mu$, one can easily prove that
$a<0$ and $e<0$. As we illustrate above, the asymptotic behaviors of the
effective potential indicates that there are at least two maxima for the
effective potential $V(z)$ when $z>0$, corresponding to two positive roots
$z_{1},z_{2}$ of $V^{\prime}(z)=0$. The equation Eq. (73) implies the another
two roots $z_{3},z_{4}$ are both positive or negative. If $z_{3},z_{4}$ are
both negative, then the $V^{\prime}(z)$ will only have two real roots outside
the inner horizon, implying that there is no trapping well outside the event
horizon and the black hole is thus superradiantly stable. From equation (76),
we conclude that _$c/a <0$ (i.e. $c>0$) is a sufficient condition for negative
$z_{3},z_{4}$ and therefore a sufficient condition for superradiantly stable
dyonic black holes_.

In following, we are focusing the parameter regions of dyonic black hole and
scalar field where $c>0$. Coefficient $c$ can be treated as a quadratic
polynomial of $\omega$, $c=c(\omega)$ and the coefficient of the quadratic
term is positive. The intercept of $c(\omega)$ is

$\displaystyle c(0)=-6\left((M-r_{+})l(l+1)-3Mq^{2}Q^{2}\right.$ (77) (79)
$\displaystyle\left.+2r_{+}q^{2}Q^{2}+\mu^{2}(M-r_{+})(2M-r_{+})^{2}\right).$

The discriminant of the quadratic equation with respect to $\omega$, namely
$c(\omega)=0$, is

$\displaystyle\Delta_{c}=$ $\displaystyle
36(2M-r_{+})^{3}\left(8l^{2}(M-r_{+})+8l(M-r_{+})+32\mu^{2}M^{3}\right.$ (80)
(82) $\displaystyle-64\mu^{2}M^{2}r_{+}-6Mq^{2}Q^{2}+40\mu^{2}Mr_{+}^{2}$ (84)
$\displaystyle\left.+7q^{2}Q^{2}r_{+}-8\mu^{2}r_{+}^{3}\right).$

In the following, we shall discuss more detail in two cases for $c>0$, finding
out a class of relevant parameter regions to give the superradiantly stable
dyonic black hole. For convenience, we define the following dimensionless
quality

$x=\frac{r_{+}}{M}\,.$ (85)

  * •

Case I: $\Delta_{c}<0$

In the case of $\Delta_{c}<0$, it is easy to prove that $c>0$ is always
holding. Recall the fact that $2M>r_{+}>M$ as well as $\Delta_{c}<0$, it is
equivalent to

$\displaystyle
8l^{2}\left(M-r_{+}\right)+8l\left(M-r_{+}\right)+32\mu^{2}M^{3}-64\mu^{2}M^{2}r_{+}$
(86) (88)
$\displaystyle-6Mq^{2}Q^{2}+40\mu^{2}Mr_{+}^{2}+7q^{2}Q^{2}r_{+}-8\mu^{2}r_{+}^{3}<0\,,$

which can also be rewritten as

$\mu^{2}>\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(7x-6)}{8(x-2)^{2}(x-1)}-\frac{l(l+1)}{r_{-}^{2}}\,.$
(89)

Recall definition of $r_{+}$, we have $1<x<2$. When the parameters of black
hole and dyonic black hole satisfy Eq. (89), the dyonic black hole is
superradiantly stable against the scalar perturbation. Two examples of the
effective potential in this case are shown in Fig. 1

Figure 1: Two examples of the effective potential in case I. The black hole
mass are chosen as $r_{+}=11414$ and $r_{+}=14000$ for the orange curve and
blue curve respectively. The other parameters are chosen as
$l=\omega=5,\mu=20,q=15,Q=7000.$

  * •

Case II-I: $\Delta_{c}>0$, Requiring $0<\omega<\omega_{-}$ then $c>0$.

In this case, denoting the two roots of $c(\omega)=0$ as $\omega_{\pm}$, we
firstly consider the case that $0<\omega<\omega_{-}$, indicating that $c>0$.
Explicitly, we have

$\omega_{\pm}=\frac{18qQr_{-}^{2}\pm\sqrt{\Delta_{c}}}{24r_{-}^{3}}.$ (90)

The condition $\Delta_{c}>0$ requires that $\mu$ satisfy

$\mu^{2}<\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(7x-6)}{8(x-2)^{2}(x-1)}-\frac{l(l+1)}{r_{-}^{2}}\,.$
(91)

Recall the superradiance condition, namely $\omega<qQ/r_{+}$, one can find
that $0<\omega<\omega_{-}\Rightarrow c>0$ must satisfy if requiring that
$qQ/r_{+}<\omega_{-}$,. Therefore, we follow the illustration above and
consider the constraint $qQ/r_{+}<\omega_{-}$, which explicitly is

$\displaystyle
qQ/r_{+}<\frac{18qQr_{-}^{2}-\sqrt{\Delta_{c}}}{24r_{-}^{3}}=\frac{3qQ}{4r_{-}}-\frac{\sqrt{\Delta_{c}}}{24r_{-}^{3}}$
(92) (94)
$\displaystyle\Leftrightarrow\boxed{\frac{\sqrt{\Delta_{c}}}{24r_{-}^{3}}<\frac{3qQ}{4r_{-}}-qQ/r_{+}}\,.$

Then the above inequality is equivalent to

$\displaystyle\frac{3qQ}{4r_{-}}-qQ/r_{+}>0~{}(r_{+}/r_{-}>4/3)\,,$ (95) (97)
$\displaystyle 18qQr_{-}^{2}r_{+}-24r_{-}^{3}qQ>r_{+}\sqrt{\Delta_{c}}\,.$

The second inequality will give that

$\displaystyle r_{+}^{2}(2M-r_{+})^{2}\mu^{2}-16M^{2}q^{2}Q^{2}$ (98) (100)
$\displaystyle+20Mq^{2}Q^{2}r_{+}+(l(l+1)-7q^{2}Q^{2})r_{+}^{2}>0,$

which can be simplified as

$\displaystyle\mu^{2}$ $\displaystyle>$
$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7r_{+}^{2}-20Mr_{+}+16M^{2}}{r_{-}^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
(101) (102) $\displaystyle=$
$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}\,.$
(103)

One can check that the above condition is consistent with Eq. (91),

$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
(104)
$\displaystyle<\mu^{2}<\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(7x-6)}{8(x-2)^{2}(x-1)}-\frac{l(l+1)}{r_{-}^{2}}\,.$
(105)

Therefore, together with the constraints on mass and charge ratio of dyonic
black hole, we then obtain a following superradiantly stable parameter region
of the scalar and dyonic black hole system in case II-I. We show the plot
associated with the effective potential in Fig. 2.

Figure 2: Two examples of the effective potential in case II-I. The black hole
mass are chosen as $r_{+}=20553$ and $r_{+}=20591$ for the orange curve and
blue curve respectively. The other parameters are chosen as
$l=\omega=5,\mu=20,q=15,Q=7000.$

$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
(106) (108)
$\displaystyle<\mu^{2}<\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(7x-6)}{8(x-2)^{2}(x-1)}-\frac{l(l+1)}{r_{-}^{2}}\,.$

with

$\frac{r_{+}}{r_{-}}>\frac{4}{3}\quad\text{or}\quad\left(\frac{Q}{M}\right)<\frac{2\sqrt{6}}{7}\,.$
(109)

  * •

Case II-II: Under $\Delta_{c}>0$, Requiring $\omega>\omega_{+}$ then $c>0$

Recall the superradiance condition, $\omega<qQ/r_{+}$. the requirement of
$c>0$ will give $\omega_{+}<\omega<qQ/r_{+}$. Therefore, in following we
consider the constraint $qQ/r_{+}>\omega_{+}$. Explicitly, we have

$\displaystyle
qQ/r_{+}>\frac{18qQr_{-}^{2}+\sqrt{\Delta_{c}}}{24r_{-}^{3}}=\frac{3qQ}{4r_{-}}+\frac{\sqrt{\Delta_{c}}}{24r_{-}^{3}}$
(110)
$\displaystyle\Leftrightarrow\boxed{\frac{\sqrt{\Delta_{c}}}{24r_{-}^{3}}<\frac{-3qQ}{4r_{-}}+qQ/r_{+}}\,,$
(111)

Then the above inequality is equivalent to

$\displaystyle\frac{3qQ}{4r_{-}}-qQ/r_{+}<0~{}\left(\frac{r_{+}}{r_{-}}<\frac{4}{3}\right)\,,$
(112) (114)
$\displaystyle-\left(18qQr_{-}^{2}r_{+}-24r_{-}^{3}qQ\right)>r_{+}\sqrt{\Delta_{c}}\,.$

Then we have

$\displaystyle r_{+}^{2}(2M-r_{+})^{2}\mu^{2}-16M^{2}q^{2}Q^{2}$ (115) (117)
$\displaystyle+20Mq^{2}Q^{2}r_{+}+(l(l+1)-7q^{2}Q^{2})r_{+}^{2}>0\,,$

which can be simplified as

$\mu^{2}>\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}.$
(118)

One can check that the above condition is consistent with Eq. (91),

$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
(119) (121)
$\displaystyle<\mu^{2}<\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(-6+7x)}{8(-2+x)^{2}(-1+x)}-\frac{l(l+1)}{r_{-}^{2}}.$

Therefore, it is concluded that in case II-II, the superradiantly stable
parameter space of the scalar and dyonic black hole system is

$\displaystyle\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
(122) (124)
$\displaystyle<\mu^{2}<\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(-6+7x)}{8(-2+x)^{2}(-1+x)}-\frac{l(l+1)}{r_{-}^{2}}.$

with

$\frac{r_{+}}{r_{-}}<\frac{4}{3}\left(\frac{Q}{M}>\frac{2\sqrt{6}}{7}\right)\,,\quad\omega_{+}<\omega<qQ/r_{+}.$
(125)

Furthermore, we also show the plot of the effective potential in Fig. 3

Figure 3: Two examples of the effective potential in case II-II. The black
hole mass are chosen as $r_{+}=14648$ and $r_{+}=14482$ for the orange curve
and blue curve respectively. The other parameters are chosen as
$l=\omega=5,\mu=20,q=15,Q=7000.$

##  IV Summary

In this paper, we study the superradiant stability of the dyonic black holes
in string theory against a charged massive bosonic perturbation. Although the
metric of a dyonic black hole is similar to that of a RN black hole and the
electric charged black hole in string theory has been proved superradiantly
stable, we find that the magnetic field of the dyonic black hole makes the
scalar and black hole system more unstable than RN black hole, giving that the
system is not superradiantly stale in whole parameter space. Therefore, we
discuss three cases for the superradiantly stable region.

For a general dyonic black hole (with the equal electric and magnetic charges)
the superradiantly stable parameter region of the system is
$\mu^{2}>\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(7x-6)}{8(x-2)^{2}(x-1)}-\frac{l(l+1)}{r_{-}^{2}}(x=r_{+}/M)$.
If the ratio between the black hole charge and the black hole mass satisfies
$\frac{Q}{M}<\frac{2\sqrt{6}}{7}$, the superradiantly stable parameter region
of the system becomes larger, which is
$\mu^{2}>\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$.
For the case $\frac{Q}{M}>\frac{2\sqrt{6}}{7}$, the superradiantly stable
parameter is
$\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{x^{2}(-6+7x)}{8(-2+x)^{2}(-1+x)}-\frac{l(l+1)}{r_{-}^{2}}>\mu^{2}>\frac{q^{2}Q^{2}}{r_{+}^{2}}\frac{7x^{2}-20x+16}{(x-2)^{2}}-\frac{l(l+1)}{r_{-}^{2}}$
and $\omega_{+}<\omega<qQ/r_{+}$. For each case, we present a picture to show
the superradiantly stable effective potential.

Recently, it has been pointed out that the magnetically charged black holes
have some interesting featuresMaldacena:2020skw . They can have long-lived
life even their masses are not large. They may have strong magnetic fields,
which results in restoring the electroweak symmetry in some regions around
them. It is worth noting that a relevant interesting feature is the Hawking
radiation effects are enhanced by the magnetic fields for near extremal black
holes. In the near extremal limit, the Hawking emission modes are in the
superradiant region. In this sense, the magnetic fields enhanced the emission
of superradiant modes. This is consistent with our result that magnetic fields
make black hole system more unstable. It will be interesting to study further
about magnetically charged black hole systems.

###### Acknowledgements.

Z.F.M thanks Run-Qiu Yang, H. Lü and Shou-Long Li for useful discussion and
proofreading. J.H.H. is supported by the Natural Science Foundation of
Guangdong Province (No.2016A030313444). This work is also partially supported
by Guangdong Major Project of Basic and Applied Basic Research (No.
2020B0301030008), Science and Technology Program of Guangzhou (No. 2019050001)
and Natural Science Foundation of Guangdong Province (No. 2020A1515010388).

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§ ANALYTIC SOLUTIONS TO THE WAVE EQUATION

In this section we solve for the acoustic signal due to arbitrary single particles in its most general form using perturbation theory. We first develop a perturbative approximation scheme for the non-homogeneous strongly damped wave equation Eq. (<ref>) based on viscosity. Then, we calculate Green's functions for the retarded propagator on each order, and finally, we provide an explicit solution for the first order in viscosity.

Throughout the section we will focus on the following form of Eq. (<ref>) where the source term has been replaced by an arbitrary function $f : \mathbb{R}^4 \to \mathbb{R}$ such that

\begin{equation}
\label{eq:wave-general}
\Box p(\bm{x}) + \lambda \Delta \partial_t p(\bm{x}) = - f(\bm{x}),
\end{equation}

where $\Box$ is the d'Alembert Operator, $\lambda = 1/\omega_0$ is the viscosity coefficient, $p$ is the pressure, and $f$ represents the source and is some function with compact support. Later we will impose more restrictions on $f$ to better represent the energy distribution of a moving particle. However, for now, we consider a general distribution to come up with Green's Functions for the problem.

§.§ Approximations Using Perturbation Theory

The solution for the pressure wave $p$ in Eq. (<ref>) can be found more easily by treating the damping term $\lambda \Delta \partial_t p(\bm{x})$ as a perturbation for small $\lambda$. Specifically, we can write the solution as a function of $\lambda$ so that

\begin{equation}
p(\bm{x}) = p_0(\bm{x}) + \lambda p_1(\bm{x}) + \cdots = \sum_{n=0}^{\infty} \lambda^n p_n(\bm{x}).
\label{eq:perturbed-pressure}
\end{equation}

Using Eq. (<ref>) we can rewrite Eq. (<ref>) as

\begin{equation}
\label{eq:wave-perturbed}
\Box p_0(\bm{x}) + \sum_{n=1}^{\infty}\lambda^n \left[\Box p_n(\bm{x}) + \Delta \partial_t p_{n-1}(\bm{x}) \right] = -f(\bm{x}).
\end{equation}

For all values of $\lambda$ we can derive the following recursive formula for the $n$th order correction in the pressure

\begin{align}
\label{eq:wave-perturb-recursive-1}
\Box p_0(\bm{x}) &= - f(\bm{x})\\
\label{eq:wave-perturb-recursive-2}
\Box p_n(\bm{x}) &= - \Delta \partial_t p_{n-1} (\bm{x}).
\end{align}

Using these expressions we can write the following partial differential equation

\begin{equation}
\label{eq:wave-perturb}
\Box^{n+1} p_n(\bm{x}) = - \left(- \Delta \partial_t\right)^n f(\bm{x}).
\end{equation}

Note that for $n=0$ we have the normal wave equation with source function $f$. We also see that higher order corrections in pressure are waves that are generated by the higher orders of the curvature of the source function. In other words, if $f$ takes the form of a bump function we will end up adding sharper and more oscillatory corrections for each order.

We will now focus on solving the global Cauchy problem for Eq. (<ref>). To do that we will use the method of Green's functions. Before we do so, it is useful to simplify Eq. (<ref>) using a Fourier transform defined as

\begin{align}
\label{eq:fourier}
\hat{f}(\bm{k}) &= \int_{\mathbb{R}^4}d^4x\, f(\bm{x}) e^{-i \bm{k} \cdot \bm{x}}\\
\label{eq:fourier-inv}
f(\bm{x}) &= \int_{\mathbb{R}^4}\frac{d^4 k}{(2\pi)^4} f(\bm{k}) e^{i \bm{k} \cdot \bm{x}},
\end{align}

where $\bm{k} \cdot \bm{x}$ is the four-vector inner product defined by $\bm{k} \cdot \bm{x} = k^\alpha x_\alpha = \eta_{\alpha \beta} k_\alpha x_\beta$, where $\eta_{\alpha \beta} = \text{diag}(-1,1,1,1)$ is the Minkowski metric. Using this transformation Eq. (<ref>) becomes

\begin{equation}
\label{eq:wave-perturb-ft}
\hat{p}_n(\bm{k}) = \frac{1}{-\bm{k}^2} \left(\frac{i k_0\,\vec{k}^2}{-\bm{k}^2}\right)^n \hat{f}(\bm{k}),
\end{equation}

where $\vec{k}^2 = k_1^2+  k_2^2 + k_3^2$ and $\bm{k}^2 = -k_0^2 + \vec{k}^2$. This is a much-simplified formula that completely defines the frequency profile of each term given a source function $f$. Now we are ready to solve for the Green's functions.

§.§ Green's Functions

To provide closed form solutions to Eq. (<ref>) we study a general expression for the retarded propagator of Eq. (<ref>). We shall see that the retarded propagator is key to maintaining causality.

§.§.§ Using Potentials and Residues

The form of the propagator greatly simplifies if, instead of solving Eq. (<ref>) for the $n^{\text{th}}$ pressure component $p_n$ in the perturbative expansion, we solve it for a potential $\psi_{n}$ such that $\partial_t^n \psi_{n} = p_{n}$. With this substitution Eq. (<ref>) can be rewritten as

\begin{equation}
\label{eq:wave-perturb-potential}
\Box^{n+1} \psi_{n}(\bm{x}) = -(-\Delta)^n f(\bm{x}).
\end{equation}

To calculate the Green's function $G_{n}(\bm{x})$ for the $n$th order, we use the Fourier transform of (<ref>). By plugging in a $\delta$ source, $f(\bm{x}) = \delta (\bm{x})$, in Eq. (<ref>) and taking the transformation, we obtain an expression for the Fourier transformed Green's function $\hat{G}_n$

\begin{equation}
\label{eq:greens-ft}
\hat{G}_n(\bm{x}) = \frac{1}{-\bm{k}^2} \left(\frac{\vec{k}^2}{-\bm{k}^2}\right)^n.
\end{equation}

Integration Contour $\Gamma_\varepsilon$ of $k_0$ in the inverse Fourier transform of Eq. <ref> to obtain the retarded propagator.

Notice that the singularities of the order $n+1$ are at $\bm{k}^2 = 0$, i.e. for $k_0 = \pm |\vec{k}|$. To extract the retarded propagator in spatial coordinates from $\hat{G}$ we need to take the inverse Fourier transform defined in Eq. (<ref>) with the contour $\Gamma_\varepsilon$, shown in Fig. <ref>, for $\varepsilon \to 0$ to preserve causality. Specifically, such inverse Fourier transform is given by

\begin{equation}
\label{eq:greens-ift}
G_n(\bm{x}) = \lim_{\varepsilon \to 0} \int_{\Gamma_\varepsilon} \frac{dk_0}{2\pi}e^{-i k_0 t} \int_{\mathbb{R}^3} \frac{d^3 k}{(2\pi)^3} \hat{G}_n(\bm{k})\, e^{i \vec{k}\cdot \vec{x}}.
\end{equation}

We can rewrite this integral by “nudging" the singularities by $\varepsilon$. To do so, we do a coordinate transformation by changing $k_0 \to k_0 + i \varepsilon$ so that the integral becomes

\begin{equation}
\label{eq:greens-ift-2}
G_n(\bm{x}) = \lim_{\varepsilon \to 0}\int_{\mathbb{R}^4} \frac{d^4 k}{(2\pi)^4} \frac{(\vec{k}^2)^n\, e^{i \bm{k}\cdot \bm{x}}}{\left[(k_0 + i \varepsilon)^2 - \vec{k}^2\right]^{n+1}},
\end{equation}

where the singularities are at $k_0 = \pm |\vec{k}| - i\varepsilon$. We can now do the $k_0$ integral using the residue theorem. Since the analytic form of the residues is quite convoluted, we shall first look at some of their properties as follows.

§.§.§ Properties of Residues

As can be seen from Fig. <ref>, both singularities are on the lower half of the complex plane. For $t<0$, where the integral converges at the upper half plane, the contour encloses no singularities so that the integral is 0. For $t>0$ the contour includes the singularities and the integral must be calculated.
Let us denote the two residues by $\hat{g}_\pm(\vec{k}^2)$ corresponding to the singularities at $\pm |\vec{k}| + i \varepsilon$, respectively. Clearly, they are functions of $\vec{k}^2$ as well as $n$ and $\varepsilon$. By using the residue theorem around a circle  small enough to contain only one of the singularities, we can prove that the residues are related as follows

\begin{equation}
\label{eq:residues-relation}
\hat{g}_+(\vec{k}^2) = - \hat{g}_-^*(\vec{k}^2),
\end{equation}

where the $^*$ denotes the complex conjugate. This is really convenient as we can rewrite the integral in Eq. (<ref>) using the residue theorem as

\begin{equation}
\label{eq:greens-ift-3}
G_n(\bm{x}) = \Theta(t)\, \lim_{\varepsilon \to 0}\int_{\mathbb{R}^3} \frac{ d^3 k}{(2\pi)^3} \hat{g}_n(\vec{k}^2) e^{i \vec{k}\cdot \vec{x}},
\end{equation}

where $\Theta(t)$ is the Heaviside function, and $\hat{g}_n(\vec{k}^2)$ is the sum of the residues which is given by

\begin{equation}
\label{eq:lame-f}
\hat{g}_n(\vec{k}^2) = i \hat{g}_+ (\vec{k}^2) - i\hat{g}_+^*(\vec{k}^2) = - 2 \text{Im}{\hat{g}_+(\vec{k}^2)},
\end{equation}

where the last step is carried out using $\eqref{eq:residues-relation}$. In this way, not only have we managed to simplify the transform, but we have also shown that the Green's function is real for all orders $n$. That is because $\hat{g}_n$ is radially symmetric in $k$ and real-valued (since it is the imaginary part of a complex-valued function), so the inverse Fourier transform must be real.

§.§.§ Calculation of Residues and Green's Functions

We are now ready to calculate the analytic expression of the residues at the two singularities where $k_0 = k_{\pm} = \pm |\vec{k}| + i \varepsilon$. From what we have shown above, it is enough to calculate the residue at $k_0 = k_+$. We do so using the Laurent expansion of the integrand in Eq. (<ref>). By calculating a series representation centered at $k_0 = k_+$ for each term of the equation, and then multiplying them together, we obtain that the coefficient of the $(k_0 - k_+)$ term is given by

\begin{equation}
\label{eq:residue}
\begin{aligned}
\hat{g}_+ &= \lim_{\varepsilon \to 0} -e^{-\varepsilon t}\Theta(t)\,\text{Im}\, \frac{e^{i (\vec{k}\cdot \vec{x} - |\vec{k}| t)}}{2^n |\vec{k}|} P(|\vec{k}|)\\
&= - \Theta(t)\,\frac{\sin(\vec{k}\cdot{x} - |\vec{k}| t)}{2^n |\vec{k}|} P(|\vec{k}|),
\end{aligned}
\end{equation}

where $P(k)$ denotes a polynomial of $|\vec{k}|$ given by

\begin{equation}
\label{eq:polynomial}
P(|\vec{k}|) = (-1)^n \sum_{m=0}^{n} \frac{(2i|\vec{k}|t)^m}{m!}\binom{2n - m}{n}.
\end{equation}

We can now plug the expression of the residue into the integral using spherical coordinates to obtain the final expression of the Green's function $G_n$ for the potential $\psi_n$:

\begin{equation}
\label{eq:retarded-propagator-full}
\begin{aligned}
G_n(\bm{x}) = \frac{\Theta(t)}{4^{n+1}\pi r} \sum_{m=0}^{n} \frac{(-2t)^m}{m!} \binom{2n-m}{n}\\ \left[\delta^{(m)}(t-r) - \delta^{(m)}(t+r)\right],
\end{aligned}
\end{equation}

where $r = |\vec{x}|$. Using $\Theta(t)$ we can simplify the expression above and, by grouping the constants, we obtain

\begin{equation}
\label{eq:retarded-propagator}
G_n(\bm{x}) = \sum_{m=0}^n \frac{(-2)^m}{4^{n+1} \pi m!}\binom{2n-m}{n} G_{nm}(\bm{x}),
\end{equation}

where $G_{nm}(\bm{x})$ is given by

\begin{equation}
\label{eq:reatarded-propagator-terms}
G_{nm}(\bm{x}) = \frac{t^m}{r} \delta^{(m)}{(t-r)}.
\end{equation}

We managed to express $G_{nm}$ by using the fact that $\Theta(t) \delta(r+t) = 0$ $\forall r>0$. Using the above Green's functions leads to an expression for the pressure

\begin{equation}
\label{eq:greens-solutions}
p(\bm{x}) = \sum_{n=0}^{\infty} \partial_{t}^{n}\,(G_n * f)(\bm{x}),
\end{equation}

which is valid for any arbitrary source function $f$.

§.§ Causality

By factoring out a Heaviside theta in Eq. (<ref>) it becomes apparent that the propagator is indeed causal. The covolution of Eq. (<ref>) with the source function $f$ leads to the following integral

\begin{equation}
\label{eq:convolution-integral}
p_n(\bm{x}) = \int_{-\infty}^{t}dt' \int_{\mathbb{R}^3}d^3x'\, G_n(\bm{x} - \bm{x}') f(\bm{x}'),
\end{equation}

from which we can see that each perturbation order in pressure $p_n$ depends on the shape of the source before the time $t$.

§.§ Closed Form solution for Delta Source

Thanks to the analytic expression for the $n$-th order calculated in Eq. (<ref>), we can find a solution for a delta source function for $f$ in closed form. In particular, we want to describe the energy density of the source over space as a moving delta function of the form $\delta(x)\delta(y)\delta(z-v t)$ where $v$ is the speed of the particle in the units where the speed of sound $c$ is equal to 1. Assuming that the energy density that the particle deposits as a function of time is given by a function $q(t)$ we can proceed by defining $f$ as

\begin{equation}
\label{eq:delta-source}
f(\bm{x}) = q(t)\delta(x) \delta(y) \delta(z-vt).
\end{equation}

To find $p_n(\bm{x})$ using Eq. (<ref>), we need to calculate the convolution $(G_n * f)(\bm{x})$. Since the convolution is a linear operation, we can calculate it by obtaining the convolution for each component $G_{nm}$ in the expansion shown in Eq. (<ref>). Each term of the convolution can be computed by

\begin{equation}
\label{eq:source-convolution-1}
G_{nm}*f(\bm{x}) = \int_{\mathbb{R}^3} d^{3}x' \int_{\mathbb{R}}dt'\, G_{nm}(\bm{x}') f(\bm{x}-\bm{x}'),
\end{equation}

where the primed variables in $\bm{x}'$ are the convolution variables. Plugging the functions in, we obtain the following expression

\begin{multline}
\label{eq:source-convolution-2}
G_{nm}*f(\bm{x}) = \sum_{l=0}^{m}\sum_{z_k' \in S} (-1)^{l} l! \binom{m}{l}^2 v^{m-l}\\
\partial_{z'}^{m-l}\left[\frac{q(t-r)r^{m-l}}{r-vz'}\right](z_k'),
\end{multline}

where $v$ is the speed of the particle, $r = \sqrt{x^2 + y^2 + z'^2}$, and $S$ is the set of solutions $z_k'$ of the equation

\begin{equation}
\label{eq:particle-solutions}
z' - (z-v t) = v r(z'),
\end{equation}

where $r(z')$ is the radius as a function of $z'$ given by $r(z')~=~\sqrt{x^2 + y^2 + z'^2}$. We can use this expression to more compactly write the convolution $\psi_{n} = G_{n} * f$ so that $\psi_n (\bm{x})$ reads

\begin{equation}
\label{eq:psi_n-delta-source}
\psi_n (\bm{x}) = \sum_{m,l}^{n,m}\sum_{z_k' \in S} C_{nml}\, \partial_{z'}^{m-l}\left[\frac{q(t-r)(vr)^{m-l}}{r-vz'}\right](z_k'),
\end{equation}

where $C_{nml}$ is a constant factor given by

\begin{equation}
\label{eq:multiplier-delta-source}
C_{nml} := \frac{(-1)^{l+m}\,2^m l!}{4^{n+1} \pi m!} \binom{2n-m}{n}\binom{m}{l}^2.
\end{equation}

Equation (<ref>) shows that the potential is given by derivatives of the same function, evaluated at the roots of Eq. (<ref>). The physical meaning of the roots as well as their analytic form is described in the next section.

§.§ Sound sources for different particle speeds

In Eq. (<ref>) we have shown that the derivative is evaluated at certain “special” points that are the roots of Eq. (<ref>). Solving this equation, we obtain the following two solutions

\begin{equation}
\label{eq:z_k-primes}
z_\pm' = \gamma^2 (z-vt) \pm |v|\sqrt{\gamma^4 (z-vt)^2 + \gamma^2 \rho^2},
\end{equation}

where $\gamma^2 = 1/(1-v^2)$, and $\rho = \sqrt{x^2 + y^2}$ is the distance perpendicular to the path of the particle. However, as clearly shown by Eq. (<ref>) and Eq. (<ref>), these solutions do not always apply. To understand which solution is physically meaningful, we go back to the integral representation of the convolution with the source function in Eq. (<ref>).

The wave generated by a moving particle can be thought of as the sum of the sound waves generated by each point along its motion. The form of those sound waves is given by the retarded propagator we have calculated in Eq. (<ref>). Therefore, when an observer at spacetime point $\bm{x}$ observes a sound wave, we can trace it back to a point in the particle's track where it was generated.

An equivalent way to visualize this is sending particles along the path of a prototype wave. This is the approach we have implemented in the integral in Eq. (<ref>). The equivalence is made rigorous by the commutativity of the convolution. Therefore, Eq. (<ref>) predicts the points of intersection between a virtual particle passing through an observer located at $\bm{x}$ and a prototype wave propagating from the origin (and described by the propagator). This is clearly shown in Fig. <ref>.

Minkowski diagrams for sources moving slower (a) or faster (b) than the speed of sound $c=1$ in the medium. We can see that for slow particles (a) an observer $O$ observes the sound emitted from $P_+$. However, when the source moves faster than the speed of sound (b) the observer at $O$ observes the sound emitted from both $P_-$ and $P_+$. In Eq. (<ref>) we solve for the z-coordinate of these points.

The prototype generated wave on the $z-t$ plane as a family of curves for progressively larger perpendicular radii $\rho$ starting from 0. We can see that if $v>1$ there are observers between the solid and dashed lines, but for larger values of $\rho$ there are no intersections with the particle's path.

§.§.§ Slow Particles

To find out which of the two solutions in Eq. (<ref>) are applicable in the case of the particle speed being smaller than the speed of sound ($v<1$), we notice the following. Assuming $v<1$ implies that $\gamma^2 > 0$, hence $z_\pm'$ is defined for all values of $z,t,$ and $r$. However, Eq. (<ref>) imposes another condition which can be written as

\begin{equation}
\label{eq:z_k-condition}
z'-(z-vt) > 0.
\end{equation}

It follows that the only solution that satisfies Eq. (<ref>) is $z_+'$ ($S=\{z_+'\}$ for $v<1$), which is also schematically shown in Fig. <ref>.

§.§.§ Fast Particles

In the case of particles moving faster than the speed of sound ($v>1$) the considerations become more niche. Fig. <ref> illustrates that there are some solutions at positive times that the signal will never reach. Mathematically, this happens when the square root of $z_\pm'$ in Eq. (<ref>) becomes imaginary. This is possible since $v>1 \implies \gamma^2 <0$. In such case, the delta function in the convolution will have no zeroes and the signal would be zero. This happens when

\begin{equation}
\label{eq:v>1-condition}
vt < z + \rho \sqrt{v^2 -1}.
\end{equation}

When the condition in Eq. (<ref>) is satisfied, it can be shown that Eq. (<ref>) implies that, for $(z-vt)>0$, no solution exists and the pressure generated there is 0. However, when $(z-vt) < 0$, both solutions $z_\pm'$ hold and $S~=~\{z_+',z_-'\}$.

§.§ Leading Order Analytic Solution

To make our model a little less abstract we will explicitly provide a closed form solution for $n=0$, i.e. the undamped wave equation. The propagator is given by choosing $n=0$ in Eq. (<ref>):

\begin{equation}
\label{eq:solution-n=0}
p(\bm{x}) = \sum_{z_k' \in S}\partial_t\, \frac{q(t-r(z_k'))}{4\pi \left(r(z_k') - v z_k'\right)},
\end{equation}

where the set of solutions $S$ has been defined above depending on the value of $v$ and $\bm{x}$. Assuming that the particle starts depositing energy at time $t=0$ we can set $q(t) = \Theta(t)$ so that the solutions from Eq. (<ref>) simplify further to

\begin{equation}
\label{eq:solution-n=0-simple}
p(\bm{x}) = - \frac{ v \bar{z}\, \Theta( - \bar{z} - \rho \sqrt{v^2 -1} )}{\left[\bar{z}^2 + \rho^2 (1-v^2)\right]^{\frac{3}{2}}},
\end{equation}

where $\bar{z} = z - vt$, $\Theta$ is the Heaviside function, and $r = \sqrt{x^2 + y^2 + z^2}$, that is the radius of the observer from the point where the energy deposition starts. Supersonic particles ($v>1$) produce different signals than subsonic ($v<1$). This is a well-known result, and we can use the classical wave equation solution (term for $n=0$) to predict the point and time when the maximum signal occurs as a function of the distance  from the particle track $\rho$ (applying viscous corrections at a second stage).
For the subsonic case, this is at the point when the derivative of the multi-variable function vanishes in all of its components. This condition is given by

\begin{equation}
\label{eq:subsonic-max-pressure}
(z-vt)^2 = \frac{\rho^2}{2\gamma^2}.
\end{equation}

For supersonic particles, even at points away from $\rho = 0$, we have a singularity in the pressure at the zeroth order. This singularity is located within a shell of radius $\mathbb{R}^3$ defined by $\rho^2 + z^2 = t^2$. Hence, given a value of $\rho$, it follows that the maximum will be located at

\begin{equation}
\label{eq:supersonic-max-pressure}
\begin{split}
z^2 &= -\rho^2 \gamma^2\\
t &= r.
\end{split}
\end{equation}

It is worth noticing that, in the supersonic case, $v>1\implies\gamma^2<0$. We now exploit these results to examine the pressure around these special points and estimate the signal created by the particles.