# Regular scalar clouds around a Kerr-Newman black hole: subextremal and
extremal scenarios

Gustavo García<EMAIL_ADDRESS>Marcelo Salgado
<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad
Nacional Autónoma de México,
A.P. 70-543, CDMX 04510, México

###### Abstract

In this work we analyze the existence of electrically charged scalar clouds
which are bound states of a complex-valued massive scalar field in the
background of subextremal and extremal Kerr-Newman black holes (BH). In
particular, we reanalyze neutral (uncharged) clouds in extremal Kerr BH’s. For
the extremal scenarios we have implemented a novel technique which allows us
to obtain regular clouds at the BH horizon ${\cal H}^{+}$ which turn out to be
connected “continuously” with the cloud solutions in the subextremal case even
if some derivatives of the scalar field are unbounded at the horizon. In
particular, for subextremal BH’s we have established regularity conditions at
${\cal H}^{+}$, by demanding that the field and its radial derivatives are
bounded there, but in the extremal scenarios we relax this last condition
while demanding that some scalar invariants are well behaved at ${\cal
H}^{+}$. Furthermore we have implemented an integral technique to understand
and justify in a simple and heuristic way the existence of such cloud
configurations in those BH backgrounds.

###### pacs:

04.70.Bw, 03.50.-z, 97.60.Lf

##  I Introduction

Black holes (BH) are one of the most enigmatic and interesting predictions of
Einstein’s theory of general relativity (GR). Nowadays these objects are
considered to be beyond the realm of theoretical speculations as currently we
count with strong observational evidence about its presence in the universe,
namely, from the detection of gravitational waves by the LIGO-VIRGO-KAGRA
collaboration LIGOVirgo  which are emitted, among other sources, by the
inspirilling and collision of two BH’s, and also from the observation of the
shadows produced by massive black holes at the center of the galaxies M87
EHTM87  and the Milky Way EHTSagA , respectively.

From a theoretical perspective, BH’s seem to be very simple objects, since
according to the uniqueness theorems Uniqueness ; HeuslerBook  together with
the no-hair conjecture Carter1968 ; Ruffini1971 , they can be described by
only three parameters: mass ($M$), angular momentum ($J$) and electric charge
($Q$) Carter1968 ; Ruffini1971 . More specifically, those theorems establish
that all regular, stationary, axisymmetric and asymptotically flat BH
solutions (AFBH) of electrovacuum Einstein’s field equations belong to the
Kerr-Newman family of solutions which are characterized by those three
parameters.

Furthermore, these mathematical results and conjectures are supported by
several no-hair theorems showing that several non-trivial fields cannot be
present outside a BH. In particular, when one considers the Einstein-Klein-
Gordon (EKG) system, these theorems show that if some energy conditions are
satisfied Bekenstein1972 ; Bekenstein1995 ; Sudarsky1995 ; Pena1997 ;
Sudarsky1998 , the only possible scalar field $\Psi$ (real or complex)
solution present outside a static and spherically symmetric AFBH is the
trivial one $\Psi\equiv 0$, and so the only AFBH solution is the Schwarzschild
solution.

Remarkably, this state of affairs can change dramatically if some spacetime
symmetries are dropped and/or the energy conditions are abandoned in those
theorems. For instance, when one introduces rotation (while keeping the energy
conditions) it is possible to find non-trivial solutions for a complex-valued
scalar field outside a BH. For instance, Hod was perhaps the first to show the
existence of bound state (exact) solutions for massive but otherwise free
scalar fields of that kind in the background of extremal and near extremal
Kerr BH’s Hod2012 ; Hod2013 . Since the background was fixed and the scalar
field was a test field that does not backreact in the Kerr spacetime, such
solutions were termed scalar clouds and were not considered as genuine hairy
solutions, i.e., solutions of the full EKG system. Later, Hod himself extended
his results by finding exact nontrivial electrically charged scalar-cloud
solutions outside (extremal and near-extremal) Kerr-Newman BH’s (KNBH) Hod2014
; Hod2015 . Presumably motivated by Hod’s discoveries, Herdeiro & Radu
Herdeiro2014 ; Herdeiro2015  (hereafter referred to as HR) generalized those
results in several instances. First, they found numerically scalar clouds in
the background of subextremal Kerr BH’s, and then, more importantly, they
computed numerically genuine rotating, stationary, axisymmetric and AFBH hairy
solutions by taking into account the backreaction of the field in the
spacetime and thus, solving self-consistently the full EKG system under
suitable regularity conditions at the horizon ${\cal H}^{+}$. Moreover, they
showed that those hairy solutions are continuously connected with rotating
boson star solutions in the limit when the BH horizon shrink to ‘zero’.

Benone et al. Benone2014  extended Hod and HR cloud analysis by studying
electrically charged scalar-field clouds in the background of subextremal
KNBH. More recently, we also computed scalar clouds and rotating hairy AFBH
solutions by solving numerically the full EKG system using spectral methods
Grandclement2022 ; Garcia2023 , and confirmed the results reported by HR.

Delgado et al. Delgado2016  obtained hairy rotating charged AFBH solutions by
solving numerically the Einstein-Klein-Gordon-Maxwell system, which extends
the charged scalar-cloud analysis in a Kerr-Newman spacetime performed earlier
by Benone et al Benone2014 . All those BH solutions with a non-trivial scalar
hair represent some counterexamples to the no-hair conjecture.

Finally, let us mention that previously we analyzed the existence of clouds
Garcia2019  and lack thereof Garcia2021  in the background of Kerr and
Reissner-Nordstrom BH’s, respectively, by devising an integral method that is
usually employed in proving the no-hair theorems within, but not exclusively,
spherically symmetric scenarios. This technique provides a heuristic
understanding as to why the no-scalar-hair theorems in spherically symmetric
and static spacetimes cannot be extended to the axisymmetric and rotating
scenarios that leads precisely to the existence of cloud and hairy solutions
of the sort alluded above.

In this paper we extend the results of Refs.Garcia2019 ; Garcia2021  in
several respects. First, we analyze the existence of charged scalar clouds in
the background of subextremal KNBH, and then we study those clouds in the
extremal case by introducing a novel technique that allows us to impose
rigorous regularity conditions at the BH horizon ${\cal H}^{+}$ even when
working with Boyer-Lindquist (BL) type of coordinates that are inherently
singular at the horizon. In particular, we reanalyze the uncharged (neutral)
clouds in the extremal Kerr background. This novel treatment contrasts
drastically with a similar analysis performed previously by us in the extremal
Kerr spacetime Garcia2020  where superregularity conditions on the scalar-
field were considered by imposing boundedness in the radial derivatives of the
scalar field at ${\cal H}^{+}$. Those conditions, while reasonable, were not
necessary and led to inconsistencies that could only be remedy if the “quantum
numbers” $(n,l,m)$ that label the cloud solutions satisfied some Diophantine
equation of Pell type and for a large number of nodes in the radial part of
the field (i.e. $1\ll n$) Garcia2020 . In this paper we show instead that the
radial derivatives of the scalar field with respect to the BL coordinate $r$,
notably the first radial derivative, can be unbounded at ${\cal H}^{+}$ and
still leads to genuinely regular scalar-clouds without the need of imposing
any further conditions on the “quantum numbers” $(n,l,m)$ provided that the
divergence in the radial derivative is such that some invariant scalars formed
from first derivatives (e.g. the kinetic term for the field in the Lagrangian)
remain bounded, notably at ${\cal H}^{+}$.

As stressed above, this technique is applied to (electrically) charged cloud
solutions in the background of extremal KN spacetime as well. We check that
our solutions using that technique are in agreement with Hod’s exact solutions
in those two extremal backgrounds Hod2012 ; Hod2014 . Finally, we show that
those regular cloud solutions around exact extremal Kerr and KN BH’s, unlike
the superregular clouds in such extremal scenarios (cf. Ref.Garcia2020 ), can
be connected continuously with the cloud solutions around the corresponding
subextremal BH’s in the limit of extremality. This feature hints towards the
robustness of our method.

##  II Charged scalar clouds

In our study we consider a (test) massive, complex and charged scalar field
$\Psi$ around a KNBH, which in Boyer-Lindquist coordinates is described by the
following spacetime metric,

$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle\left(\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}$
(1) $\displaystyle-$
$\displaystyle\frac{2a\sin^{2}\theta\left(r^{2}+a^{2}-\Delta\right)}{\rho^{2}}dtd\varphi$
$\displaystyle+$ $\displaystyle\left(\frac{\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}\sin^{2}\theta}{\rho^{2}}\right)\sin^{2}\theta d\varphi^{2},$

where

$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta\;\;,\text{and}\;\;\Delta=r^{2}-2Mr+a^{2}+Q^{2}\;,$
(2)

where $M$ is the mass, $a$ the angular momentum per mass unit and $Q$ the
electric charge associated with the KNBH.

In this spacetime we can identify the presence of two horizons located at

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}-Q^{2}}\;,$ (3)

one at $r_{+}\equiv r_{H}$ that corresponds to the BH event horizon and
another one at $r_{-}$ which is an inner Cauchy horizon, such that
$\Delta(r_{\pm})=0$ . The existence of a KNBH requires $a^{2}+Q^{2}\leq
M^{2}$, where the equality is associated with an extremal KNBH, which we
consider in Sec. V. In particular, the Kerr spacetime is recovered when
$Q\equiv 0$. Due to the presence of the two horizons, it is convenient to
write

$\Delta=(r-r_{H})(r-r_{-})\;,$ (4)

where the values of $r_{H}$ and $r_{-}$ keep the following relationship

$r_{-}=\frac{a^{2}+Q^{2}}{r_{H}}\;.$ (5)

The angular velocity of the KNBH is given as follows:

$\Omega_{H}=\frac{a}{r_{H}^{2}+a^{2}}\;.$ (6)

From (2) and the fact that $\Delta(r_{H})=0$ one finds a relation between the
mass $M$ and the quantities $r_{H}$, $a$ and $Q$:

$M=\frac{r_{H}^{2}+a^{2}+Q^{2}}{2r_{H}}\;.$ (7)

Moreover, from Eqs.(5)–(7) we see that the quantities $r_{-}$, $\Omega_{H}$,
and $M$ can take a parametric form $r_{-}=r_{-}(r_{H},a,Q)$,
$\Omega_{H}=\Omega_{H}(r_{H},a,Q)$, and $M=M(r_{H},a,Q)$. These equations will
allow us to compute $r_{-}$, $\Omega_{H}$, and $M$ when finding the values for
$a$ that solve the eigenvalue problem for $\Psi$, once the values of $r_{H}$
and $Q$ are specified (cf. Sec. III).

The massive and charged scalar field $\Psi$ that we analyze for the existence
of scalar-cloud solutions has the following energy-momentum tensor (EMT):

$\displaystyle T_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\left(D_{a}\Psi\right)^{*}\left(D_{b}\Psi\right)+\left(D_{b}\Psi\right)^{*}\left(D_{a}\Psi\right)\Big{]}$
(8) $\displaystyle-$ $\displaystyle
g_{ab}\Big{[}\frac{1}{2}g^{cd}\left(D_{c}\Psi\right)^{*}\left(D_{d}\Psi\right)+U(\Psi^{*}\Psi)\Big{]}\;,$

where the operator

$D_{a}\equiv\nabla_{a}-iqA_{a}\;,$ (9)

represents the covariant derivative associated with the gauge field $A_{a}$,
which in the KNBH background is given by

$A_{a}=-\frac{Qr}{\rho^{2}}\left[(dt)_{a}-a\sin^{2}\theta(d\varphi)_{a}\right]\;,$
(10)

and the constant $q$ (i.e., the electrical charge) is the gauge coupling for
the scalar field $\Psi$. The operator $\nabla_{a}$ corresponds to the
covariant derivative compatible with the metric, in this case the KN metric.
For our study we focus only on the following potential.

$U(\Psi^{*}\Psi)=\frac{1}{2}\mu^{2}\Psi^{*}\Psi\;,$ (11)

which is associated with a massive but free field with mass $\mu$.

The dynamics of the charged and massive scalar field is given by the Klein-
Gordon (KG) equation coupled to the electromagnetic potential:

$\left(\nabla^{a}-iqA^{a}\right)\left(\nabla_{a}-iqA_{a}\right)\Psi=\mu^{2}\Psi\;.$
(12)

In order to find bound states for $\Psi$ in the domain of outer communication
(DOC) of the KNBH, including the horizon, we consider the following ansatz in
terms of the BL coordinates with temporal and angular dependence in the form,

$\Psi(t,r,\theta,\varphi)=\phi(r,\theta)e^{im\varphi}e^{-i\omega t}\;,$ (13)

where $\omega$ is the frequency of the scalar field and $m$ is an integer
number. This is the most general form that we can choose in such a way that
the energy-momentum tensor (8) respects the symmetries of the KN spacetime.

To ensure the existence of boson clouds, we impose the zero flux condition at
the BH horizon Herdeiro2014 ; Herdeiro2015 :

$\chi^{a}D_{a}\Psi\big{|}_{\mathcal{H}^{+}}=0;,$ (14)

where

$\chi^{a}\equiv\xi^{a}+\Omega_{H}\eta^{a}\;,$ (15)

is the helical Killing vector field given in terms of the timelike Killing
field $\xi^{a}=(\partial/\partial t)^{a}$ and the axial Killing field
$\eta^{a}=(\partial/\partial\varphi)^{a}$, which are associated with the time
and axial symmetries of the background spacetime. At the horizon $\chi^{a}$
becomes null and thus, it is tangent to the null geodesic generators of the
horizon. From (14) together with Eqs.(15), (10) and (13), we obtain the
following condition:

$\left(\omega-m\Omega_{H}-q\frac{Qr_{H}}{r_{H}^{2}+a^{2}}\right)\Psi_{H}=0$
(16)

Assuming that in general $\Psi_{H}\neq 0$, we conclude,

$\omega=m\Omega_{H}+q\Phi_{H}\;,$ (17)

where

$\Phi_{H}:=-A_{a}\chi^{a}|_{\mathcal{H}^{+}}=\frac{Qr_{H}}{r_{H}^{2}+a^{2}}=\frac{4\pi
Qr_{H}}{{\cal A}_{H}}\;,$ (18)

is the electric potential at the horizon as defined in terms of the helical
Killing field, and ${\cal A}_{H}=4\pi(r_{H}^{2}+a^{2})$ is the area of the BH
event horizon (cf. Ref. Hawking1976 ).

The bound states thus correspond to a field with a frequency given by (17),
which is called synchronicity condition Hod2014 ; Benone2014 , due to its
relationship with the BH’s angular velocity $\Omega_{H}$, in particular when
$Q\equiv 0$.

In fact, the condition (17) results also when imposing regularity of the field
$\Psi$ at the horizon, as we show below.

##  III Teukolsky radial equation and regularity conditions

When we substitute the scalar field ansatz (13) into the KG Eq.(12) we find
that the function $\phi(r,\theta)$ is separable, that is, it can be written as
the product of a function dependent on the variable $r$ and a function
dependent on the variable $\theta$ which allows us to expand the $\Psi$ field
in modes of the form

$\Psi_{nlm}=R_{nlm}(r)S_{lm}(\theta)e^{im\varphi}e^{-i\omega t}\;,$ (19)

where the angular functions $S_{lm}\left(\theta\right)$ (the spheroidal
harmonics) obey the following equation

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{dS_{lm}}{d\theta}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!+\left(K_{lm}+a^{2}(\mu^{2}-\omega^{2})\sin^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}\right)S_{lm}=0\,,$
(20)

and $K_{lm}$ are the separation constants ($|m|\leq l$) given by

$K_{lm}+a^{2}(\mu^{2}-\omega^{2})=l(l+1)+\sum_{k=1}^{\infty}c_{k}a^{2k}(\mu^{2}-\omega^{2})^{k}\;,$
(21)

which connects the angular and radial parts of the KG equation and ensures
that the angular functions $S_{lm}(\theta)$ are regular on the axis of
symmetry. As it occurs with the wave function of the hydrogen atom, the number
$l$ is a non-negative integer which is associated with the angular momentum of
the atom, while the integer $n$ ($n\geq 0$) labels the number of nodes in the
radial function $R_{nlm}(r)$. The expansion coefficients $c_{k}$ can be found
in Ref. Abramowitz .

The functions $R_{nlm}(r)$ obey a type of radial Teukolsky equation
Teukolsky1972 ,

$\displaystyle\Delta\frac{d}{dr}\left(\Delta\frac{dR_{nlm}}{dr}\right)+\left[\mathcal{H}^{2}+\left(2ma\omega-
K_{lm}\right.\right.$ (22) $\displaystyle-$
$\displaystyle\left.\left.\mu^{2}\left(r^{2}+a^{2}\right)\right)\Delta\right]R_{nlm}=0\;,$

where

$\mathcal{H}\equiv\left(r^{2}+a^{2}\right)\omega-am-qQr\;.$ (23)

Given the form of the frequency $\omega$ (17), we observe that the function
$\mathcal{H}$ vanishes at the horizon, which is precisely the regularity
condition at the horizon when demanding boundedness for the field and its
radial derivatives in the subextremal BH scenarios, as we discuss next.

In order to find cloud configurations, it is essential to establish regularity
conditions for the $\Psi$ field at the event horizon, namely, by demanding
that the field and some of its derivatives are bounded on the horizon. We thus
assume that $R_{nlm}(r)$ is $C^{3}$ on the horizon. To solve the differential
equation associated with the radial part (22) it is necessary to know the
values of the field and its radial derivatives at $r_{H}$:
$R^{\prime\prime}_{nlm}(r_{H})$, $R^{\prime}_{nlm}(r_{H})$ and
$R_{nlm}(r_{H})$. So, assuming that $R^{\prime\prime}_{nlm}(r_{H})$ is bounded
we find the regularity condition for $R^{\prime}_{nlm}(r_{H})$ from Eq.(22):

$\displaystyle R^{\prime}_{nlm}(r_{H})$ $\displaystyle=$
$\displaystyle-\frac{1}{2(r_{H}-M)}\left[2ma\omega-
K_{lm}-\mu^{2}\left(r_{H}^{2}+a^{2}\right)\right]$ (24) $\displaystyle\times
R_{nlm}(r_{H})\;.$

The value of $R_{nlm}(r_{H})$ is a free parameter, which we take
$R_{nlm}(r_{H})=1$, for simplicity and also to compare our numerical results
with those presented in Benone2014  where the same value is used. To find the
value for $R^{\prime\prime}_{nlm}(r_{H})$, we differentiate Eq. (22) once
more, and demand that $R^{\prime\prime\prime}_{nlm}(r_{H})$ is bounded, which
leads to111We can observe that, for $Q=0$ and $a\neq 0$ these regularity
conditions reduce to those obtained in the case of the Kerr metric (see Eqs.
(77)-(78) in Garcia2019 ). Alternatively, when $a=0$ and $Q\neq 0$, they
reduce to the regularity conditions for the field in the background of a
Reissner-Nordstrom (RN) black hole (see Eqs. (18)-(19) in Garcia2021 ).:

$\displaystyle R^{\prime\prime}_{nlm}(r_{H})$ $\displaystyle=$
$\displaystyle-\frac{1}{2(r_{H}-M)}\left[\frac{(M-r_{-})\left(2mar_{H}+qQ\left(r_{H}^{2}-a^{2}\right)\right)^{2}}{(r_{H}-r_{-})^{2}\left(r_{H}^{2}+a^{2}\right)^{2}}-\mu^{2}r_{H}\right]R_{nlm}(r_{H})$
(25) $\displaystyle-$
$\displaystyle\frac{1}{4(r_{H}-M)}\left[2(1+ma\omega)-K_{lm}-\mu^{2}\left(r_{H}^{2}+a^{2}\right)\right]R^{\prime}_{nlm}(r_{H})\;.$

Both regularity conditions (24) and (25) are valid only in the subextremal
case $r_{H}\neq M$. Thus, making use of these two regularity conditions we can
find cloud solutions close to the extremal background only in the limit
$r_{H}\rightarrow M$, but we cannot treat exactly the extremal scenario with
such conditions. Clouds in the exact extremal background with $r_{H}=M=r_{-}$
will be analyzed separately in Sec.V below.

In order to find charged cloud solutions around a subextremal KNBH Eq.(22) was
solved numerically with their respective regularity conditions (24) and (25),
using a 4th order Runge-Kutta algorithm and integrating in the DOC from
$r=r_{H}$ outwards. As a part of the numerical scheme, the values of $r_{H}$,
$Q$, $\mu$ and $q$ were fixed, and selecting some values for the integers $n$,
$l$ and $m$. Then the BH angular momentum per mass unit $a$ can be used as an
eigenvalue such that the radial function $R_{nlm}(r)$ vanishes asymptotically
as we did in our previous study for clouds around a subextremal Kerr BH
Garcia2019 .

Figure 1 shows the existence lines associated with the numerical solutions of
Eq.(22), with fixed charge $\mu Q=0.1$ and different values for $r_{H}$. We
have taken the quantum numbers $n=0$ and $m=l=1,2,3$, respectively, as well as
a scalar field $\Psi$ with $q/\mu=1$.

In Tables 4-6 of Appendix A are displayed the numerical eigenvalues $a_{nlm}$
(i.e., the spectrum) associated with the eigenvalue problem of Eq.(22). Our
results depicted in Figure 1 are consistent with the those reported in
Benone2014 .

Figure 1: Existence (black dotted) lines for charged scalar clouds ($n=0$ and
$l=m$ with $m=1,2,3$) coupled to a subextremal Kerr-Newman black hole with
charge $\mu Q=0.1$. The relation between the mass and the charge of the scalar
field is $q/\mu=1$. The solid blue line corresponds to an extremal KNBH
$a^{2}+Q^{2}=M^{2}$ (BH solutions do not exist above this line).

Figure 2 plots a family of radial solutions $R_{nlm}(r)$ with $n=0$ and
$m=l=1$, using different values for $r_{H}$. The corresponding values for the
spectrum $a_{nlm}$ are shown in Table 4 of Appendix A.

Figure 2: Radial solutions $R_{11}$ ($R_{011}$) with $n=0$ and $l=1=m$
associated with charged boson clouds around a subextremal Kerr-Newman black
hole and the corresponding location of the event horizon $\mu r_{H}$ taking
$\mu Q=0.1$ and $q/\mu=1.0$.

Cloud solutions around KNBH close to extremality $a^{2}+Q^{2}\approx M^{2}$
(i.e. $r_{H}\approx M$) are depicted in Fig. 3 for the fundamental mode
($n=0$) and $m=l=1$. The numerical values for the spectrum $a_{nlm}$ can be
found in the last four rows and second column of Table 4 (see Appendix A). For
clouds similar to these ones, Tables 5 and 6 display more values $a_{nlm}$
when $m=l=2$ and $m=l=3$, respectively. Note from Fig. 3 that as the solutions
approach the extremal scenario the slope $R^{\prime}$ at the horizon increases
(cf. Eq.(24) ). At this respect, cloud solutions for the exact extremal
scenario around Kerr and Kerr-Newman BH’s are analyzed below in Sec.V since,
as stressed before, the radial part $R(r)$ for the boson field $\Psi$ requires
different kind of regularity conditions at the horizon because their
derivatives (24) and (25) blows-up when $r_{H}=M$.

Figure 3: Radial solutions $R_{11}$ ($R_{011}$) with $n=0$ and $l=1=m$
associated with charged boson clouds around a Kerr-Newman black hole close to
extremality $r_{H}\approx M$. Here $\mu Q=0.1$ and $q/\mu=1.0$ are fixed.

Figure 4 shows some examples of radial solutions $R_{nlm}$ with different
nodes ($n=0,1,2$) associated with a fixed value of the horizon radius $\mu
r_{H}=0.5$. In particular, the figures depicts $R_{n11}$ and $R_{n22}$, i.e.,
taking $m=l=1$ and $m=l=2$, respectively.

Figure 4: Radial solutions $R_{nlm}$ with principal number $n=0,1,2$ (number
of nodes) around a KNBH with the horizon located at $\mu r_{H}=0.5$, and
taking integers $m=l=1$ (top panel) and $m=l=2$ (bottom panel), respectively.
Here $\mu Q=0.1$ and $q/\mu=1.0$.

Figure 5 shows the angular function $S_{lm}(\theta)$ and the 3D spheroidal
harmonic $|S_{lm}(\theta,\varphi)|$ for $m=l=2$, associated with the charged
scalar field $\Psi$ coupled to a Kerr-Newman black hole with $\mu r_{H}=0.5$
and $\mu Q=0.1$.

Figure 5: Angular part $S_{lm}(\theta)$ of the field $\Psi$ (top panel) and 3D
spheroidal harmonic $|S_{22}(\theta,\varphi)|$ (bottom panel) for $m=l=$2.

##  IV No-hair-theorem obstructions

In this Section we justify in a more heuristic way the existence of charged
scalar clouds in the presence of a KNBH. The treatment developed here is
similar to the one presented in Garcia2019 ; Garcia2021 , which analyze,
respectively, the existence of neutral scalar clouds around a Kerr BH, and
their absence thereof around a RNBH. To do so, we use a technique initially
implemented by Bekenstein Bekenstein1972 , and then employed, mutatis
mutandis, in several studies that prove no-hair theorems in several kind of
scalar-field theories Heusler1996 ; Ayon2002 .

First, let us consider the Klein-Gordon equation (12) in the form

$D^{a}D_{a}\Psi=\mu^{2}\Psi\;.$ (26)

Multiplying both sides of this equation by $\Psi^{*}$ and integrating over a
suitable volume $\mathcal{V}$ contained within the DOC of the KNBH (bounded by
two spatial hypersurfaces $t=const$, a section of the horizon and a timelike
surface $r=const$ when $r\rightarrow\infty$) we find,

$\int_{\mathcal{V}}\Psi^{*}D^{a}D_{a}\Psi\sqrt{-g}d^{4}x=\int_{\mathcal{V}}\mu^{2}\Psi^{*}\Psi\sqrt{-g}d^{4}x\;.$
(27)

Integrating by parts the l.h.s of the previous equation and using the Gauss
theorem we obtain,

$\int_{\partial\mathcal{V}}\Psi^{*}s^{a}D_{a}\Psi
dS=\int_{\mathcal{V}}\Big{[}(D^{a}\Psi)^{*}(D_{a}\Psi)+\mu^{2}\Psi^{*}\Psi\Big{]}\sqrt{-g}d^{4}x$
(28)

In this case we have that the surface integral that correspond to the boundary
$\partial\mathcal{V}$ is composed by four regions: the two spacelike
hypersurfaces $\Sigma_{1}$ and $\Sigma_{2}$, a portion of the black hole
horizon and, finally, the asymptotic region corresponding to spatial infinity
$i^{0}$. The contributions associated with $\Sigma_{1}$ and $\Sigma_{2}$
cancel each other because the integrals are identical except for the fact that
their normal vectors have opposite signs (i.e.
$s^{a}_{\Sigma_{1}}=-s^{a}_{\Sigma_{2}}$). The surface integral associated
with spatial infinity vanishes when assuming that the scalar field $\Psi$
decays exponentially, and thus it vanishes asymptotically. Finally, at the
horizon (a null hypersurface) the normal $s^{a}$ is equal to the Killing
vector $\chi^{a}$. Furthermore, assuming that $\Psi^{*}$ is bounded at the
horizon and using the zero-flux condition (14) the integral associated with
the portion of horizon vanishes as well. We conclude that

$\int_{\partial\mathcal{V}}\Psi^{*}s^{a}D_{a}\Psi dS=0\;,$ (29)

and therefore

$\int_{\mathcal{V}}\Big{[}(D^{a}\Psi)^{*}(D_{a}\Psi)+\mu^{2}\Psi^{*}\Psi\Big{]}\sqrt{-g}d^{4}x=0\,.$
(30)

Given the harmonic dependence of the field $\Psi$ (13) one obtains the
following expression for the kinetic term

$\displaystyle K$ $\displaystyle\equiv$
$\displaystyle\left(D^{a}\Psi\right)^{*}\left(D_{a}\Psi\right)$ (31)
$\displaystyle=$ $\displaystyle
g^{tt}\left(D_{t}\Psi\right)^{*}\left(D_{t}\Psi\right)+g^{t\varphi}\left(D_{t}\Psi\right)^{*}\left(D_{\varphi}\Psi\right)$
$\displaystyle+$ $\displaystyle g^{\varphi
t}\left(D_{\varphi}\Psi\right)^{*}\left(D_{t}\Psi\right)+g^{\varphi\varphi}\left(D_{\varphi}\Psi\right)^{*}\left(D_{\varphi}\Psi\right)$
$\displaystyle+$ $\displaystyle
g^{rr}\left(D_{r}\Psi\right)^{*}\left(D_{r}\Psi\right)+g^{\theta\theta}\left(D_{\theta}\Psi\right)^{*}\left(D_{\theta}\Psi\right)$
$\displaystyle=$
$\displaystyle\mathcal{R}\phi^{2}+g^{rr}\left(\partial_{r}\phi\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}\phi\right)^{2}\;,$

where we have defined

$\displaystyle\mathcal{R}$ $\displaystyle\equiv$ $\displaystyle
g^{tt}\hat{\omega}^{2}-2g^{t\varphi}\hat{m}\hat{\omega}+g^{\varphi\varphi}\hat{m}^{2}\;,$
(32) $\displaystyle\hat{\omega}$ $\displaystyle\equiv$
$\displaystyle\omega+qA_{t}\;,$ (33) $\displaystyle\hat{m}$
$\displaystyle\equiv$ $\displaystyle m-qA_{\varphi}\;.$ (34)

Considering the frequency of the field $\Psi$ (17) and the electromagnetic
potential $A_{a}$ (10), it is possible to express Eqs. (33) and (34) as
follows:

$\displaystyle\hat{\omega}$ $\displaystyle=$ $\displaystyle
m\Omega_{H}+q\Phi_{H}-\frac{qQr}{\rho^{2}}\;,$ (35) $\displaystyle\hat{m}$
$\displaystyle=$ $\displaystyle m-\frac{aqQr\sin^{2}\theta}{\rho^{2}}\;.$ (36)

From Eqs. (35) and (36) together with (17) it is easy to find the following
relationship between $\hat{\omega}$ and $\hat{m}$:

$\hat{m}=m+(\hat{\omega}-\omega)a\sin^{2}\theta\;.$ (37)

If one substitutes the explicit form of the metric components $g^{tt}$,
$g^{t\varphi}$, and $g^{\varphi\varphi}$ into Eq.(32), one obtains the
following expression for $\mathcal{R}$:

$\displaystyle\mathcal{R}=$ $\displaystyle-$
$\displaystyle\frac{1}{\Delta\rho^{2}}\left[\left(r^{2}+a^{2}\right)\hat{\omega}-a\hat{m}\right]^{2}$
(38) $\displaystyle+$
$\displaystyle\frac{1}{\rho^{2}\sin^{2}\theta}\left[a\hat{\omega}\sin^{2}\theta-\hat{m}\right]^{2}\;,$

or equivalently

$\displaystyle\mathcal{R}=$ $\displaystyle-$
$\displaystyle\frac{(r-r_{H})}{\rho^{2}(r-r_{-})}\left[\omega
r+\left(mr_{H}-aqQ\right)\Omega_{H}\right]^{2}$ (39) $\displaystyle+$
$\displaystyle\frac{1}{\rho^{2}\sin^{2}\theta}\left[a\omega\sin^{2}\theta-m\right]^{2}\;.$

From this last equation, we observe that $\mathcal{R}$ contains a contribution
that is never positive (the one with the factor $r-r_{H}$ in the first line)
and another contribution that is positive defined in the DOC (including the
horizon, in the second line).

Therefore, the kinetic term $K$ is represented by the following expression:

$\displaystyle K=$ $\displaystyle-$
$\displaystyle\frac{(r-r_{H})\phi^{2}}{\rho^{2}(r-r_{-})}\left[\omega
r+\left(mr_{H}-aqQ\right)\Omega_{H}\right]^{2}$ (40) $\displaystyle+$
$\displaystyle\frac{\phi^{2}}{\rho^{2}}\left[\frac{a\omega\sin^{2}\theta-m}{\sin\theta}\right]^{2}$
$\displaystyle+$ $\displaystyle
g^{rr}\left(\partial_{r}\phi\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}\phi\right)^{2}\;.$

Therefore, the integrand of the integral (30) has the following form

$\displaystyle I$ $\displaystyle\equiv$ $\displaystyle K+\mu^{2}\Psi^{*}\Psi$
(41) $\displaystyle=$
$\displaystyle-\,\frac{(r-r_{H})\phi^{2}}{\rho^{2}(r-r_{-})}\left[\omega
r+\left(mr_{H}-aqQ\right)\Omega_{H}\right]^{2}$
$\displaystyle+\,\frac{\phi^{2}}{\rho^{2}}\left[\frac{a\omega\sin^{2}\theta-m}{\sin\theta}\right]^{2}$
$\displaystyle+\;\;\frac{\Delta}{\rho^{2}}\left(\partial_{r}\phi\right)^{2}+\frac{1}{\rho^{2}}\left(\partial_{\theta}\phi\right)^{2}+\mu^{2}\phi^{2}\;.$

If nontrivial regular charged boson clouds exist (i.e. $\phi(r,\theta)\neq 0$
in general), the following inequality

$\mathcal{R}=g^{tt}\hat{\omega}^{2}-2g^{t\varphi}\hat{m}\hat{\omega}+g^{\varphi\varphi}\hat{m}^{2}\leq
0\;,$ (42)

must hold in a region of the KN spacetime so that the (non positive) term in
the first line of (41) compensates the non-negative contributions associated
with the quadratic terms
$g^{rr}\left(\partial_{r}\phi\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}\phi\right)^{2}$
and $\mu^{2}\phi^{2}$, and in this way the volume integral (30) is satisfied
for non-trivial clouds. We show that this is the case in some particular
instances, and then provide numerical evidence in more general cases.

First, considering Eq.(39) we find the following behaviors at the horizon
$r_{H}$ and asymptotically (when $r\rightarrow\infty$), respectively:

$\mathcal{R}_{H}=\frac{\left[aqQr_{H}\sin^{2}\theta-\left(r_{H}^{2}+a^{2}\cos^{2}\theta\right)m\right]^{2}}{(r_{H}^{2}+a^{2}\cos^{2}\theta)(r_{H}^{2}+a^{2})^{2}\sin^{2}\theta}\;,$
(43) $\mathcal{R}_{\infty}\sim-\omega^{2}\;.$ (44)

In particular, at the equatorial plane $\theta=\pi/2$, Eq.(43) reduces to

$\mathcal{R}_{H}=\left[\frac{aqQ-mr_{H}}{r_{H}^{2}+a^{2}}\right]^{2}\;.$ (45)

Clearly, this shows that $\mathcal{R}$ must interpolate between a positive
value at the horizon (45) and a negative value at spatial infinity (44).

Furthermore, since in the kinetic term appears the combination
$\phi^{2}\mathcal{R}$, we introduce the quantity $\Lambda$,

$\Lambda\equiv\phi^{2}\mathcal{R}\;,$ (46)

which presents the following behaviors at the horizon and the equatorial plane
and asymptotically ($r_{H}\ll r$), respectively

$\displaystyle\Lambda\big{|}_{H}$ $\displaystyle=$
$\displaystyle\phi^{2}_{H}\mathcal{R}_{H}=m^{2}\phi^{2}_{H}\left[\frac{aqQ-
mr_{H}}{r_{H}^{2}+a^{2}}\right]^{2}\;,$ (47) $\displaystyle\Lambda_{\infty}$
$\displaystyle=$
$\displaystyle\phi^{2}_{\infty}\mathcal{R}_{\infty}\sim-\phi^{2}_{\infty}\left(m\Omega_{H}+q\Phi_{H}\right)^{2}\;.$
(48)

In particular, $\Lambda_{\infty}$ vanishes as $\phi\rightarrow 0$
asymptotically.

On the other hand, when considering an extremal KNBH, $a^{2}+Q^{2}=M^{2}$, the
quantity $\mathcal{R}$ given in Eq.(39) reduces to

$\displaystyle\mathcal{R}^{\rm ext}=$ $\displaystyle-$
$\displaystyle\frac{1}{\rho^{2}}\left[\omega r+\left(mM-
aqQ\right)\Omega_{H}\right]^{2}$ (49) $\displaystyle+$
$\displaystyle\frac{1}{\rho^{2}}\left[\frac{a\omega\sin^{2}\theta-m}{\sin\theta}\right]^{2}\;.$

Then, the frequency of the scalar field is given by,

$\omega=\frac{ma}{M^{2}+a^{2}}+\frac{qQM}{M^{2}+a^{2}}=\frac{ma+qQM}{M^{2}+a^{2}}\;.$
(50)

In this case, the value of $\mathcal{R}^{\rm ext}$ at the horizon $r=r_{H}=M$
takes the form,

$\displaystyle\mathcal{R}^{\rm
ext}_{H}=-\frac{1}{M^{2}+a^{2}\cos^{2}\theta}\left\\{\left[\frac{qQ^{3}+2maM}{M^{2}+a^{2}}\right]^{2}\right.$
(51) $\displaystyle+$
$\displaystyle\left.\left[\frac{ma^{2}\cos^{2}\theta+mM^{2}-aqQM\sin^{2}\theta}{(M^{2}+a^{2})\sin\theta}\right]^{2}\right\\}\;,$

which, at the equatorial plane ($\theta=\pi/2$) reduces to

$\mathcal{R}^{\rm
ext}_{H}=-\frac{1}{M^{2}}\left[\frac{qQ^{3}+2maM}{M^{2}+a^{2}}\right]^{2}+\left[\frac{aqQ-
mM}{M^{2}+a^{2}}\right]^{2}\;.$ (52)

We note that when $Q=0$, the above equation is simply

$\mathcal{R}^{\rm ext}_{H}=-\frac{3m^{2}}{4M^{2}}\;,$ (53)

this value is similar the one that appears in the equation (61) of Garcia2019
, and which corresponds to the extremal Kerr scenario222In Garcia2019  we use
a slightly different notation and the equivalent expression for (53) does not
include the factor $m^{2}$..

The asymptotic behavior for $\mathcal{R}^{\rm ext}$ when $M\ll r$, coincides
with the asymptotic behavior for $\mathcal{R}$ given by Eq. (44).

In view of this, even if $\mathcal{R}^{\rm ext}_{H}$ were positive for some
values of the parameters, $\mathcal{R}^{\rm ext}$ must interpolate from a
positive to a negative value, proving that this quantity is also negative in a
region of the DOC.

We conclude that the contribution associated with Eq.(47) to the kinetic term
(40) is positive at the horizon (in the subextremal case) and then it becomes
negative. In the extremal case something similar happens, except that at the
horizon $\mathcal{R}^{\rm ext}$ can be even less positive (cf. blue dashed-
line of top panel of Fig.6 which corresponds to a near extremal solution).

Since the field vanishes asymptotically, it follows that the $\Lambda$ term
(in fact the entire kinetic term) will also vanish asymptotically. As in the
Kerr scenario Garcia2019 , we observe that the existence of nontrivial
localized solutions for $\phi(r,\theta)$ in a Kerr-Newman background implies
that the inequality (42) must hold in some region of the DOC.

Figure 6 shows the rotational part of the kinetic term (31) that appears in
the integral (30) for some of the numerical solutions depicted in Figs. 2 and
3 with parameters displayed in Tables 4-6, and evaluated at $\theta=\pi/2$,
for simplicity.

Figure 6: Rotational contribution [cf. Eq. (46)] $\Lambda/\Psi^{*}\Psi={\cal
R}$ to the kinetic term $K=(\nabla_{c}\Psi^{*})(\nabla^{c}\Psi)$ given by Eq.
(40) computed from the regular cloud solutions in the background of a Kerr-
Newman BH and evaluated at the equatorial plane ($\theta=\pi/2$) for different
locations of the horizon $r_{H}$, and with “quantum numbers” $n=0$ and $m=l=1$
(top panel), $m=l=2$ (middle panel), and $m=l=3$ (bottom panel).

We observe that on the horizon this quantity is positive but then becomes
negative. This rotational part has been normalized with the square of the
field amplitude $\Psi^{*}\Psi=\phi^{2}$, which is positive. Furthermore, due
to this normalization, the quantity $\mathcal{R}=\Lambda/\phi^{2}$ does not
vanish asymptotically, but it is close to the negative constant value given by
Eq.(44)
$\mathcal{R}=\Lambda/\phi^{2}\rightarrow-\left(m\Omega_{H}+q\Phi_{H}\right)^{2}=-\omega^{2}$.

Table 1 shows some numerical values of the quantity $\Lambda/\Psi^{*}\Psi$ at
the horizon and asymptotically $(r\rightarrow\infty)$ corroborating that the
numerical results are consistent with the analytical expectations.

Kinetic term (rotational contribution)
---
$m$ | $\mu r_{H}$ | $\dfrac{\Lambda}{\Psi^{*}\Psi}\left(r=r_{H}\right)$ | $\dfrac{\Lambda}{\Psi^{*}\Psi}\left(r\rightarrow\infty\right)$
1 | 0.15 | 43.77526399 | -0.99998181
0.40 | 4.63855257 | -0.99516743
0.57 | 0.75238174 | -0.93941249
2 | 0.50 | 14.36173016 | -0.99668771
0.95 | 2.400727382 | -0.96461025
1.17 | 0.67682839 | -0.80623325
3 | 0.50 | 34.38453473 | -0.99829089
1.00 | 7.11593895 | -0.98607769
1.86 | 0.61079886 | -0.69454971
Table 1: Values of the rotational contribution to the kinetic term at
$r=r_{H}$ and $r\rightarrow\infty$ associated with Figure 6.

The fact that the rotational contribution is negative in most of the DOC for
the KNBH indicates that the integral (30) vanishes due to the presence of such
a negative quantity, without the need for the field $\Psi(t,r,\theta,\varphi)$
to vanishes identically, something that does occur in the spherically
symmetric scenario Bekenstein1972 ; Bekenstein1995 ; Sudarsky1995 ; Pena1997 ;
Sudarsky1998 , and which leads to the no-hair theorems in such scenario. This
situation is quite similar to what is found in the background of a Kerr black
hole Garcia2019 . These results allow us to understand in a simple and
heuristic way the existence of nontrivial charged scalar clouds and the
explain why the no-hair theorems cannot be extended for rotating BH’s
precisely due to the presence of the rotational contribution $\Lambda$ in the
integrand of Eq.(30) which is not positive (semi) definite when non-trivial
cloud solutions exist.

##  V Extremal scenarios: a new perspective

In a previous investigation Garcia2020  we analyzed the existence of
(uncharged) clouds in the background of an exact extremal Kerr BH. In order to
do so, we imposed some superregularity conditions at the horizon $r_{H}^{\rm
ext}=M$ on the radial part $R(r)$ of the field $\Psi$. Such conditions
consisted in assuming that $R(r)$ and its derivatives were bounded at the
horizon, and contrary to what happens in the subextremal scenario, those
conditions constrained the separation constants $K_{lm}$ in such a way that a
priori they did not coincide with the values (21) required for the angular
part $S_{lm}(\theta)$ of the spheroidal harmonics to be regular on the axis of
symmetry $\theta=\\{0,\pi\\}$. These inconsistencies forced us to conclude
that regular clouds could not exist around exact extremal Kerr BH’s unless the
pairs of numbers $(l,m)$ satisfied a Diophantine equation of Pell type and
only if the numbers of nodes $n$ were arbitrarily large. Notwithstanding, even
in that case, those clouds seemed not to coincide with the clouds obtained
from the subextremal scenario in the limit of extremality $M\rightarrow a$:
both solutions were not connected in a “continuous” way, and as mentioned, the
exact extremal ones required the existence of a large number of nodes, and an
additional restriction on the numbers $(l,m)$ (a Pell-Diophantine equation),
something that does not occur for cloud solutions around subextremal Kerr
BH’s.

If we follow the same strategy for exact extremal KNBH we encounter similar
kind of problems, which allow us to conclude that assuming superregularity
conditions for the radial part $R(r)$ is a too restrictive a condition, which
as we show next, it is not necessary in order to have well behaved clouds.

Now, if the derivatives of $R(r)$ are not bounded at the horizon, the question
is, how to solve the problem, and clearly, how can the corresponding clouds be
regular at the horizon. The answer to these questions are analyzed in this
section by implementing a new method that leads to regular clouds even if the
radial function $R(r)$ has unbounded derivatives at the horizon. Moreover,
these cloud solutions approach the cloud solutions from the subextremal
scenario in the limit of extremality, all without the need of constraining the
separation constants and without requiring a large number of nodes, as we will
see.

The key aspect to take into account is that first derivatives $R^{\prime}(r)$
that appear whether in the trace of the energy-momentum tensor for the scalar
field or in the kinetic part $K$ of the action functional of the theory in the
form $g^{ab}(D_{a}\Psi)^{*}D_{b}\Psi=g^{ab}\nabla_{a}\Psi^{*}\nabla_{b}\Psi$
(when $Q=0$), which has an invariant (coordinate independent) meaning, show up
only in the combination $g^{rr}(R^{\prime}(r))^{2}$ [times some angular
functions; cf. Eq.(31)]. In particular, for extremal BH’s $g^{rr}$ has a
factor $(r-M)^{2}$, thus $(r-M)^{2}R^{\prime 2}$ can be bounded at the horizon
even if $R^{\prime}$ blows up there, provided that near the horizon
$R^{\prime}\sim(r-M)^{\sigma}$ with $\sigma\geq-1$. That is, provided that
near the horizon $R(r)\sim(r-M)^{\alpha}$ with $\alpha>0$, where
$\sigma=\alpha-1$. In this way, not only the scalar field is bounded at the
horizon $r=M$, but the scalar invariants formed from first derivatives of the
field are bounded there as well.

Since in this section we focus only on extremal BH’s, we consider the metric
(1) but taking $M^{2}=a^{2}+Q^{2}$, i.e., $r_{H}^{\rm ext}=M$ which leads to

$\Delta=(r-M)^{2}\;.$ (54)

Thus, the complex-valued scalar field $\Psi$ obeys the Klein-Gordon Eq.(12) in
the extremal Kerr background which is solved again using separation of
variables with the mode expansion like in Eq.(19), similar to the subextremal
scenario.

In order to obtain bound states in the extremal scenarios we now assume the
following ansatz factorization for the radial function $R_{nlm}$:

$R_{nlm}(r)=(r-M)^{\alpha}L_{nlm}(r)\;,$ (55)

where the exponent $\alpha$ will be determined from the radial equation for
$R_{nlm}(r)$, but it will be required to be positive (i.e. $\alpha>0$) such
that $R(r_{H})=0$ and $L(r_{H})$ is bounded. Otherwise, if $\alpha<0$,
$R(r_{H})$ is unbounded and regular clouds are not possible. Moreover even if
$0<\alpha$, the radial derivatives may still diverge at the horizon if
$0<\alpha<1$. However, this divergence is not physically meaningful as the
physically meaningful clouds are those with a bounded kinetic term,
$K=g^{ab}(D_{a}\Psi)^{*}D_{b}\Psi$ in the DOC, notably at the horizon, in
particular, the term $g^{rr}(R^{\prime})^{2}$ (where for convenience, we omit
the mode labels $(n,m,l)$). More specifically, given (55), and for simplicity
$Q=0$, the first radial derivative reads,

$R^{\prime}(r)=\alpha(r-M)^{\alpha-1}L(r)+(r-M)^{\alpha}L^{\prime}(r)\;,$ (56)

which as we emphasized before, it diverges at $r=M$ if $0<\alpha<1$.
Nevertheless, when considering the term $g^{rr}R^{\prime 2}$ that appears in
the kinetic term $K$, and using the form of the derivative $R^{\prime}(r)$ as
above, we find

$\displaystyle g^{rr}R^{\prime 2}S^{2}(\theta)$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{\alpha^{2}}{\rho^{2}}\left[(r-M)^{\alpha}L(r)\right]^{2}$
(57) $\displaystyle+$
$\displaystyle\frac{2\alpha}{\rho^{2}}(r-M)^{2\alpha+1}L(r)L^{\prime}(r)$
$\displaystyle+$
$\displaystyle\frac{1}{\rho^{2}}(r-M)^{2\alpha+2}[L^{\prime}(r)]^{2}\Big{\\}}S^{2}(\theta)\;,$

which is bounded in the DOC for any $\alpha>0$ as far as $L(r)$ and its first
derivative are bounded there. When this is the case, in particular, at the
horizon $r=M$, the term (57) vanishes there, but it diverges if $\alpha<0$.

As we will show below, notably, for the extremal Kerr and the extremal KN
BH’s, we will be looking for radial solutions of the form (55) with $0<\alpha$
and with $L(r)$ and its derivatives bounded such that the term (57) remains
bounded in the DOC, despite the fact that $R^{\prime}$ itself may blow up at
the horizon $r=M$. In this way, the physically meaningful solutions will have
$\psi$ and the kinetic term bounded in the DOC. To achieve this, the angular
part for $\psi$, which corresponds to the spheroidal harmonics, also satisfies
suitable regularity conditions, namely, at the axis of symmetry.

###  V.1 Extremal Kerr

First we analyze solutions for a non-charged field $\Psi$ in the background of
a extremal Kerr BH ($|a|=M$; for concreteness we take $a=M$) with the metric
given in Boyer-Lindquist coordinates by

$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle\left(\frac{\Delta-M^{2}\sin^{2}\theta}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}$
(58) $\displaystyle-$
$\displaystyle\frac{2M\sin^{2}\theta\left(r^{2}+M^{2}-\Delta\right)}{\rho^{2}}dtd\varphi$
$\displaystyle+$ $\displaystyle\left(\frac{\left(r^{2}+M^{2}\right)^{2}-\Delta
M^{2}\sin^{2}\theta}{\rho^{2}}\right)\sin^{2}\theta d\varphi^{2},$

where

$\rho^{2}=r^{2}+M^{2}\cos^{2}\theta,$ (59)

and $M$ is the mass associated with the Kerr BH. In this scenario $q\equiv 0$
and the operator Eq.(9) reduces to the covariant derivative compatible with
the spacetime metric:

$D_{a}=\nabla_{a}\;.$ (60)

The radial function $R_{nlm}$ obeys then a simplified version of Eq. (22)
given by

$\displaystyle\Delta\frac{d}{dr}\left(\Delta\frac{dR_{nlm}}{dr}\right)$ (61)
$\displaystyle+$ $\displaystyle\left[\mathcal{H}_{\rm
Kerr}^{2}+\left(2mM\omega-
K_{lm}-\mu^{2}\left(r^{2}+M^{2}\right)\right)\Delta\right]$
$\displaystyle\times$ $\displaystyle R_{nlm}=0\;,$

where

$\mathcal{H}_{\rm Kerr}\equiv\left(r^{2}+M^{2}\right)\omega-Mm\;.$ (62)

The frequency (17) associated with an extremal Kerr BH and its angular
velocity become,

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle m\Omega_{H}^{\rm
ext}\;,$ (63) $\displaystyle\Omega_{H}^{\rm ext}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}\;.$ (64)

When replacing the ansatz (55) in the radial Eq.(61) we obtain the following
differential equation for $L(r)$ (for brevity we omit the labels $(n,l,m)$ ):

$\left(r-M\right)^{2}L^{\prime\prime}(r)+2\left(\alpha+1\right)(r-M)L^{\prime}(r)+\left[\frac{m^{2}}{4M^{2}}\left(r+M\right)^{2}+\alpha(\alpha+1)+m^{2}-K_{lm}-\mu^{2}\left(r^{2}+M^{2}\right)\right]L(r)=0\;.$
(65)

Differentiating Eq. (65) once and twice and assuming boundedness of higher
derivatives for $L(r)$ at the horizon we obtain the values (regularity
conditions) for $L^{\prime}(r_{H})$ and $L^{\prime\prime}(r_{H})$, which are
are given by

$L^{\prime}(r_{H})=\frac{\left(2\mu^{2}M^{2}-m^{2}\right)}{2(\alpha+1)M}L(r_{H})\;,$
(66)

and

$L^{\prime\prime}(r_{H})=\frac{1}{(4\alpha+6)}\left[\left(4\mu^{2}M-\frac{2m^{2}}{M}\right)L^{\prime}(r_{H})+\left(2\mu^{2}-\frac{m^{2}}{2M^{2}}\right)L(r_{H})\right].$
(67)

Moreover, given those regularity conditions, the boundedness of $L(r)$ at the
horizon and assuming $L(r_{H})\neq 0$ we find, when evaluating (65) at $r=M$,
that the exponent $\alpha$ satisfies the following simple quadratic algebraic
equation

$\alpha^{2}+\alpha+2m^{2}-2\mu^{2}M^{2}-K_{lm}=0\;,$ (68)

which has the following solutions:

$\alpha_{\pm}=\frac{-1\pm\sqrt{1+4\left[K_{lm}+2\left(\mu^{2}M^{2}-m^{2}\right)\right]}}{2}\;.$
(69)

We choose the sign $(+)$ for avoiding a negative definite $\alpha$ and thus to
consider only potentially bounded radial solutions at the horizon. This will
be possible provided the positive square root is larger than $-1$. As we show
below, this depends on the values of the BH parameter $\mu M$ (i.e. $M$ in
units of $1/\mu$) and the numbers $(n,l,m)$, notably, the value $m$. Figure 7
shows solutions for the radial function $L(r)$ (taking $L(r_{H}^{\rm ext})=1$
for simplicity) associated with an extremal Kerr BH with “quantum numbers”
$m=l=1$, $m=l=2$ and $n=0,1,2$. Figure 8 depicts the corresponding radial part
$R(r)$ (55). Each solution has an exponent $\alpha_{+}$ that satisfies the
condition $0<\alpha_{+}$ which leads to physically meaningful regular clouds
with a bounded kinetic term, despite the fact that $R^{\prime}(M)$ diverges
given that $0<\alpha_{+}<1$ (see the figure’s caption).

Figure 7: (Extremal Kerr scenario) Functions $L(r)$ with $m=l=1$ (top panel)
and $m=l=2$ (bottom panel), for $n=0,1,2$. The functions are bounded and well
behaved in the DOC, in particular, at the horizon $r=M$ where $L(M)=1$.

Figure 8: Extremal Kerr scenario: (Top panel) Radial Functions ($m=l=1$) for
$n=0,1,2$. The values $\alpha_{n}$ for each $n$ are $\alpha_{0}=0.38396168$,
$\alpha_{1}=0.37307025$ and $\alpha_{2}=0.36957870$, respectively. (Bottom
panel) Radial Functions ($m=l=2$) for $n=0,1,2$ with values $\alpha_{n}$ given
by $\alpha_{0}=0.27115038$, $\alpha_{1}=0.15291680$ and
$\alpha_{2}=0.09129800$, respectively. Notice that $R(r)$ vanishes at the
horizon $r=M$ and $R^{\prime}$ blows up there.

A remarkable fact is that Hod Hod2012  had found exact solutions to the radial
Eq.(22) associated with this scenario (uncharged clouds in the background of
an extremal Kerr BH, $Q=0$, $a=M$). These solutions are,

$R(z)=Az^{-\frac{1}{2}+\beta}e^{-\frac{1}{2}z}L_{n}^{(2\beta)}(z)\;,$ (70)

where

$z\equiv 2\sqrt{\mu^{2}-\frac{m^{2}}{4M^{2}}}(r-M)\;,$ (71)

and

$\beta^{2}\equiv K_{lm}+\frac{1}{4}-2m^{2}+2\mu^{2}M^{2}\;,$ (72)

$L_{n}^{(2\beta)}(z)$ are the generalized Laguerre polynomials and $A$ is a
normalization constant. By comparing Eqs. (69) and (72) we find that the
exponent $\alpha_{+}$ and Hod’s $\beta$ maintain the following relation:

$\beta=\alpha_{+}+\frac{1}{2}\;.$ (73)

Figure 9 compares our numerical solution for the radial function $R_{nlm}$
using $r_{H}=M=a$ with that obtained by Hod analytically taking $l=m=1$ and
different values of $n$. Figure 10 shows the comparison between both solutions
for $l=m=2$. The numerical and the exact solutions show an excellent
agreement, proving that our approach is robust.

Figure 9: Radial solutions $R_{nlm}$ for the extremal Kerr scenario with
principal numbers $n=0,1$ and $m=l=1$. The red dots correspond to the
numerical solutions obtained by solving Eq. (22) and the blue continuous line
corresponds to the exact solution computed by Hod (Eq.(17) in Hod2012 ). The
agreement between both solutions is excellent.

Figure 10: Same as Fig.9 but taking $n=0,1$ and $m=l=2$.

Figure 11 shows the angular functions $S_{lm}(\theta)$ associated with the
values $\mu M=0.52550879$ ($m=l=1$) and $\mu M=1.14093944$ ($m=l=2$). As we
can appreciate, the functions are perfectly regular on the axis of symmetry at
$\theta=\\{0,\pi\\}$. In this way, we have succeeded in finding numerically
neutral scalar clouds around exact extremal Kerr BH’s which are perfectly
regular at the horizon and on the axis of symmetry. Namely, given the values
obtained for the exponent $\alpha$ in each solution, the invariant (coordinate
independent) scalar $g^{ab}\nabla_{a}\Psi^{*}\nabla_{b}\Psi$ turns to be
bounded in the DOC, notably, at the horizon even if the radial derivative
$R^{\prime}(r)$ is unbounded there.

Figure 11: Spheroidal harmonics $S_{lm}(\theta)$ associated with the angular
part of the field $\Psi$ for $l=m=1$ (top panel) and $l=m=2$ (bottom panel),
respectively. (Extremal Kerr scenario).

###  V.2 Extremal Reissner-Nordstrom

### Case $q^{2}<\mu^{2}$

Next we consider a charged field $\Psi$ coupled with an extremal RNBH with the
metric

$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle\frac{\left(r-M\right)^{2}}{r^{2}}dt^{2}+\frac{r^{2}}{\left(r-M\right)^{2}}dr^{2}$
(74) $\displaystyle+$ $\displaystyle r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2},$

where $M=|Q|$ is the mass associated with the extremal RNBH. For this case we
consider the derivative operator (9) with

$A_{a}=-\frac{Q}{r}dt_{a}\;,$ (75)

and the scalar-field potential (11) for a massive free field, like in the Kerr
BH of Sec. V.1. The radial Eq. (22) takes the form

$\Delta\frac{d}{dr}\left(\Delta\frac{dR_{nlm}}{dr}\right)+\left[\mathcal{H}_{\rm
RN}^{2}-\left(K_{lm}+\mu^{2}r^{2}\right)\Delta\right]R_{nlm}=0$ (76)

with

$\mathcal{H}_{\rm RN}\equiv\omega r^{2}-qMr\;,$ (77)

and

$K_{lm}=l(l+1)\;.$ (78)

In this case, the boson-field frequency (17) associated with an extremal RNBH
is

$\omega=q\;.$ (79)

The radial equation for the function $L(r)$ in this scenario is

$\displaystyle(r-M)^{2}L^{\prime\prime}+2(\alpha+1)(r-M)L^{\prime}$ (80)
$\displaystyle+$
$\displaystyle\left[\alpha(\alpha+1)+\left(q^{2}-\mu^{2}\right)r^{2}-l(l+1)\right]L=0\,,$

subject to the following regularity conditions:

$L^{\prime}(r_{H})=\frac{\left(\mu^{2}-q^{2}\right)M}{\alpha+1}L(r_{H})\;,$
(81)

and

$L^{\prime\prime}(r_{H})=\frac{\left(\mu^{2}-q^{2}\right)}{2\alpha+3}\left[2ML^{\prime}(r_{H})+L(r_{H})\right]\;,$
(82)

with $\alpha$ satisfying the following quadratic equation

$\alpha^{2}+\alpha+\left(q^{2}-\mu^{2}\right)M^{2}-l(l+1)=0\;,$ (83)

with solutions

$\alpha_{\pm}=\frac{-1\pm\sqrt{1+4\left[l(l+1)+\left(\mu^{2}-q^{2}\right)M^{2}\right]}}{2}\;.$
(84)

As before, we take only the sign $(+)$ in order for $\alpha$ to be positive
and the radial function $R(r)$ to be bounded at $r=M$.

Using the value for $\alpha_{+}$ in Eq. (84) the radial Eq. (80) for $L(r)$
reads as follows:

$(r-M)L^{\prime\prime}+2(\alpha_{+}+1)L^{\prime}+\left(q^{2}-\mu^{2}\right)(r+M)L=0\;.$
(85)

When we solve numerically Eq.(85) using the regularity conditions (81) and
(82) we find that the only acceptable solution is

$L(r)=\text{const}\;.$ (86)

Since we demand that the boson field vanishes asymptotically we conclude that
the function $L(r)$ is the trivial solution $L(r)\equiv 0$, so that the radial
functions $R_{nlm}(r)$ is well behaved asymptotically. Thus, under such
circumstances $R_{nlm}(r)$ also vanishes everywhere and the only solution to
the radial Eq.(80) in this scenario is

$R_{nlm}(r)\equiv 0\;.$ (87)

These numerical results corroborate that it is not possible to find charged
scalar clouds around an extremal Reissner-Nordstrom black hole, as it was
concluded previously in Ref.Garcia2021 . There exists, however, a way to avoid
this conclusion, but it requires a scalar-field potential that includes a
self-interaction term Garcia2021 ; Hong2020 ; Herdeiro2020 ; Brihaye2021 .
Under such conditions it is possible to find numerically charged clouds, known
in the literature as Q-clouds, which as we just concluded, are absent when the
field is only a massive but free.

### Double extremal case $q^{2}=\mu^{2}$

Considering the case $q^{2}=\mu^{2}$ analyzed in DegolladoHerdeiro2013 ;
Garcia2021  Eq. (85) reads

$(r-M)L^{\prime\prime}+2(\alpha_{+}+1)L^{\prime}=0\,.$ (88)

However, we see that in this double extremal scenario Eq. (84) leads to the
following possible values for $\alpha_{+}$

$\alpha_{+}=l\;.$ (89)

So Eq. (88) takes the form

$(r-M)L^{\prime\prime}+2(l+1)L^{\prime}=0\;,$ (90)

whose solution is of the form

$L(r)=\frac{const.}{(2l+1)\left(r-M\right)^{2l+1}}\;,$ (91)

so the radial function $R(r)=(r-M)^{\alpha}L(r)$ is

$R(r)=\frac{const.}{(2l+1)\left(r-M\right)^{l+1}}\;.$ (92)

We can see that these solutions are not regular at horizon $r=M$ for any
$l\geqslant 0$, and therefore, the only possibility is again $R(r)\equiv 0$, a
result that is in a agreement with the results reported in Garcia2021 . In
fact, from (81) and (82) we already notice that when $q^{2}=\mu^{2}$, then
$L^{\prime}(r_{H})\equiv 0\equiv L^{\prime\prime}(r_{H})$, and thus, the
solution (91) requires the constant to be identically zero, leading also to
$R(r)\equiv 0$.

###  V.3 Extremal Kerr-Newman

We now consider a charged field $\Psi$ coupled to an extremal KNBH with the
metric (1) but taking $\Delta=(r-M)^{2}$ and $M^{2}=a^{2}+Q^{2}$. We assume
again the scalar-field potential for a free but massive field (11).

The equation associated with the radial part of the scalar field is provided
by Eq. (22). The frequency for an extremal KNBH is the the same as in (17)
except that the BH’s angular velocity is

$\Omega_{H}\equiv\frac{a}{M^{2}+a^{2}}\;,$ (93)

and the electric potential at the horizon is

$\Phi_{H}\equiv\frac{QM}{M^{2}+a^{2}}\;.$ (94)

As before, the integer $m$ is the azimutal harmonic index and $q$ is the
charge coupling constant of the scalar field. The radial equation for the
function $L(r)$ now satisfies the following differential equation:

$(r-M)^{2}L^{\prime\prime}+2(\alpha+1)(r-M)L^{\prime}+\left\\{\alpha(\alpha+1)+\left[\frac{am(r+M)+qQ\left(Mr-a^{2}\right)}{M^{2}+a^{2}}\right]^{2}+2ma\omega-
K_{lm}-\mu^{2}\left(r^{2}+a^{2}\right)\right\\}L=0\;.$ (95)

The regularity conditions for the first and second derivatives of $L(r)$ at
the horizon are

$L^{\prime}(r_{H})=\frac{\mu^{2}M\left(M^{2}+a^{2}\right)^{2}-(am+qQM)(2amM+qQ^{3})}{(\alpha+1)\left(M^{2}+a^{2}\right)^{2}}L(r_{H})\;,$
(96) $\displaystyle L^{\prime\prime}(r_{H})=$
$\displaystyle\frac{1}{(2\alpha+3)\left(M^{2}+a^{2}\right)^{2}}\left\\{\left[\mu^{2}\left(M^{2}+a^{2}\right)^{2}-(am+qQM)^{2}\right]L(r_{H})\right.$
(97) $\displaystyle+$ $\displaystyle
2\left.\left[\mu^{2}M\left(M^{2}+a^{2}\right)^{2}-\left(2maM+qQ^{3}\right)\left(am+qQM\right)\right]L^{\prime}(r_{H})\right\\}\,,$

respectively, with $\alpha$ satisfying the following equation

$\alpha^{2}+\alpha+\left[\frac{2amM+qQ^{3}}{M^{2}+a^{2}}\right]^{2}+2ma\omega-
K_{lm}-\mu^{2}\left(M^{2}+a^{2}\right)=0\;,$ (98)

whose solutions are

$\alpha_{\pm}=\frac{-1\pm\sqrt{1+4\left\\{K_{lm}+\mu^{2}\left(M^{2}+a^{2}\right)-\left[2ma\omega\left(M^{2}+a^{2}\right)^{2}+\left(2amM+qQ^{3}\right)^{2}\right]/\left(M^{2}+a^{2}\right)^{2}\right\\}}}{2}\;.$
(99)

Again, we take the sign $(+)$ for $\alpha$ to be positive. We appreciate that
when taking $Q=0$ but $a\neq 0$ in the regularity conditions (96) and (97) and
also in Eq. (99) for $\alpha$, they reduce to Eqs. (66), (67) and (69),
respectively, which are associated with the extremal Kerr BH. On the other
hand, when we take $a=0$ and $Q\neq 0$ in those equations, we recover
Eqs.(81), (82) and (84), respectively, corresponding to the extremal RNBH.

Figure 12 depicts the radial function $L(r)$ associated with an extremal KNBH
for $m=l=1$ and $m=l=2$, with number of nodes $n=0,1,2$.

Figure 12: Extremal Kerr-Newman scenario $q/\mu=1$ and $\mu Q=0.1$: Functions
$L(r)$ with $m=l=1$ (top panel) and $m=l=2$ (bottom panel), for number of
nodes $n=0,1,2$. Like in the extremal Kerr scenario, these functions are
bounded and well behaved in the DOC, in particular, at the horizon $r=M$ where
$L(M)=1$.

Figure 13 shows the complete radial part $R(r)$ (55) of the scalar field
$\Psi$ around an extremal KNBH corresponding the functions $L(r)$ of Fig. 12
for the integers $m=l=1$ and $m=l=2$, and nodes $n=0,1,2$. These solutions
have $0<\alpha_{+}<1$ (see the caption) and thus leads to regular clouds with
a bounded kinetic term even if $R^{\prime}$ blows up at the extremal horizon
$r=M$.

Figure 13: Extremal Kerr-Newman scenario $q/\mu=1$ and $\mu Q=0.1$: (Top
panel) Radial Functions $R$ with $m=l=1$, for $n=0,1,2$. The values of the
exponent $\alpha_{n}$ for each $n$ are $\alpha_{0}=0.38914753$ $(\mu
a=0.56475594)$, $\alpha_{1}=0.37873576$ $(\mu a=0.55077117)$ and
$\alpha_{2}=0.37547687$ $(\mu a=0.54634993)$, respectively. (Bottom panel)
Radial Functions with $m=l=2$, for $n=0,1,2$. In this case
$\alpha_{0}=0.27624214$ $(\mu a=1.17996470)$, $\alpha_{1}=0.16035789$ $(\mu
a=1.11618248)$ and $\alpha_{2}=0.10040099$ $(\mu a=1.08597093)$, respectively.
Like in the extremal Kerr scenario, notice that $R(r)$ vanishes at the horizon
$r=M$ and $R^{\prime}$ blows up there.

Like in the extremal Kerr BH, Hod Hod2015  also found exact solutions for
clouds around extremal KNBH. The solutions to the radial Eq. (22) reported by
Hod are,

$R(\chi)=A\chi^{-\frac{1}{2}+\beta}e^{-\epsilon\chi}L_{n}^{(2\beta)}(2\epsilon\chi),$
(100)

where

$\chi\equiv\frac{r-M}{M},$ (101) $\epsilon\equiv
M\sqrt{\mu^{2}-\omega^{2}}\,,$ (102) $\beta^{2}\equiv
K_{lm}+\frac{1}{4}-2am\omega-(2M\omega-
qQ)^{2}+\mu^{2}\left(M^{2}+a^{2}\right)\,,$ (103)

$L_{n}^{(2\beta)}(2\epsilon\chi)$ are the generalized Laguerre polynomials and
$A$ is a normalization constant. The exponent $\alpha$ and Hod’s $\beta$
maintain the same relationship as in the extremal Kerr,

$\beta=\alpha_{+}+\frac{1}{2}\;,$ (104)

except that the $\alpha_{+}$ (99) includes the electric charges $Q$ and $q$,
and it also includes the separation constants $K_{lm}$ corresponding to the
charged extremal scenario.

Fig. 14 compares our numerical solution for the radial function $R_{nlm}$
(with $r_{H}^{\rm ext}=M$) with Hod’s (100) taking $l=m=1$ and different
values for $n$. Figure 15 depicts something similar but taking $l=m=2$.

Figure 14: Radial solutions $R_{nlm}$ for an extremal KNBH ($q/\mu=1$ and $\mu
Q=0.1$) with principal numbers $n=0,1$ and $m=l=1$. The red dots correspond to
the numerical solutions obtained by solving the Eq.(22) and the blue
continuous line shows the exact solution given by Hod (Eq.(49) in Hod2015 ).
Notice the excellent agreement between the analytic and the numerical
solutions.

Figure 15: Radial solutions $R_{nlm}$ for the extremal KNBH ($q/\mu=1$ and
$\mu Q=0.1$) with principal numbers $n=0,1$ and $m=l=2$. The red dots
correspond to the numerical solutions obtained from solving the equation (22)
and the blue continuous line corresponds to the exact solution given by Hod
(Eq.(49) in Hod2015 ). The agreement between the analytic and the numerical
solutions is excellent.

Figure 16 shows the angular functions $S_{lm}(\theta)$ associated with the
charged scalar field $\Psi$ for the values $\mu M=0.57354100$ ($m=l=1$) and
$\mu M=1.18419453$ ($m=l=2$). From the figure one can appreciate the
regularity of those functions on the axis of symmetry. In this way we have
obtained a complete regular solution (namely, regular at the horizon and on
the axis of symmetry) for the charged scalar field $\Psi$ around an extremal
KNBH, which in addition leads to a well behaved kinetic term
$g^{ab}(D_{a}\Psi)^{*}D_{b}\Psi$ given by Eq.(31) despite the fact that its
radial derivative is unbounded at the BH’s horizon333The unboundedness of the
radial derivatives of $R(r)$ at the horizon, together that $R(r)$ vanishes
there, are features that can be appreciated also from the solutions (70) and
(100) found by Hod Hod2012 ; Hod2015  for clouds around extremal Kerr and KN
BH’s, respectively, and more vividly from Figs. 9, 10, 14 and 15..

Figure 16: Spheroidal harmonics $S_{lm}(\theta)$ associated with the angular
part of the charged scalar field $\Psi$ corresponding to take $l=m=1$ (top
panel) and $l=m=2$ (bottom panel) respectively. (Extremal Kerr-Newman
scenario).

In summary, by using the decomposition for the radial function with the ansatz
(55) it is possible to obtain both neutral and electrically charged regular
scalar cloud solutions around (exact) extremal Kerr and extremal KNBH,
respectively.

To conclude this section, a final analysis is in order. Let us remind that in
the subextremal scenarios the value of the radial function at the horizon
$R(r_{H})$ is a free parameter that for simplicity and convenience we take it
as $R(r_{H})=1$. However, if we want that the radial solutions approach as
continuously as possible to the corresponding solutions of extremal BH’s we
need to modify $R(r_{H})$ such that as $r_{H}\rightarrow r_{H}^{\rm ext}=M$,
$R(r_{H})\rightarrow 0$, since in the extremal scenarios by construction the
radial functions vanish $R(r_{H}^{\rm ext})\equiv 0$. To achieve this, we
propose the following modification for the value $R(r_{H})$:

$R(r_{H})=C\times(r_{H}-M)^{\alpha}\;,$ (105)

where $0<\alpha<1$ is the same exponent as computed for the respective radial
solutions for extremal Kerr and KNBH (i.e. using the same numbers $(l,m,n)$)
that we want to compare with. Here $C$ is a constant that we take $C=1$.
Moreover, taking into account the regularity condition Eq.(24) we observe that
in the extremal limit ($r_{H}\rightarrow M$) and with the modification (105)
we will have

$R(r_{H})\rightarrow
0\;,\;\;\;\text{and}\;\;\;R^{\prime}(r_{H})\rightarrow\infty\;,$ (106)

which, as a bonus, allows us to avoid a trivial solution that we would obtain
if the exponent was fixed with a value $\alpha>1$ since then
$R^{\prime}(r_{H})\equiv 0$. In this way, the regularity conditions are
respected but are changed suitably for each cloud computed around subextremal
BH’s (Kerr and Kerr-Newman). In particular, the value $R(r_{H})$ is not fixed
anymore to $R(r_{H})=1$, but changes with $r_{H}$ and $M$ according to (105).

Using this improved method we obtain a “smooth” match between the radial
solutions associated with the subextremal and extremal scenarios in the limit
where the former approach the latter, something, that as emphasized in the
Introduction, did not happen with the treatment carried out by us in
Garcia2020 .

Figure 17 shows a sample of radial cloud solutions $R_{nlm}$ that are
recomputed for the subextremal Kerr BH (top panel) and the subextremal KNBH
(bottom panel) with the new value $R(r_{H})$ fixed according to the
prescription (105) where $\alpha$ is taken, respectively, as $\alpha_{+}$ of
Eq.(69) and $\alpha_{+}$ of Eq.(99). In each panel, the radial function is
approaching smoothly to the corresponding solution for the exact extremal case
(and the maximum amplitude decreases as a consequence of this) as
$r_{H}\rightarrow M$. The latter are depicted in both panels by the red line
solutions.

Figure 17: Radial functions $R_{nlm}$ associated with a scalar field $\Psi$
around a subextremal Kerr BH (top panel) and KNBH (bottom panel) with $n=0$,
$m=1$, $l=1$ for different values of the BH angular momentum $a$ that converge
to the extremal values $r_{H}^{\rm ext,Kerr}=M=a\approx 0.525/\mu$ (top panel)
and $r_{H}^{\rm ext,KN}=M=\sqrt{a^{2}+Q^{2}}\approx 0.564/\mu$ (bottom panel),
with $Q=0.1/\mu$. Here $R(r_{H})=(r_{H}-M)^{\alpha}$. Notice that the slope as
well as the value $R(r_{H})$ change as the extremal cases are reached. For
reference, the corresponding radial solutions around the exact extremal BH’s
(red curves) are included in both panels.

Figure 18 shows the radial functions $R_{nlm}$ that are represented in Fig. 17
but focusing on their behavior near the event horizon. We see how the value of
the radial function on the horizon, $R(r_{H})$, changes depending on the
location of $r_{H}$, unlike what was presented in Sec. III, where all cloud
configurations for the Kerr-Newman scenario have a fixed value $R(r_{H})=1$
(see Figure 3). For the Kerr scenario see Fig. 3 of Ref.Garcia2019 .

Figure 18: Behavior of the radial functions $R_{nlm}$ that appears in the Fig.
17, but depicted close to the event horizon.

###  V.4 Configurations for $m=l\geq 3$

The (uncharged and charged) scalar cloud configurations around extremal Kerr
and KN BH’s presented in Sections V.1 and V.3, respectively, correspond to
values of $m=l=1,2$ only. This is because numerically we observe that for
$m=l\geq 3$ the values of the exponent $\alpha_{+}$ appearing in the radial
function (55) become negative in both scenarios, which immediately implies
that the field $\Psi$ becomes divergent at the event horizon including their
radial derivatives.

Something to note is that $\alpha_{+}\equiv 0$ when

$K_{lm}=2m^{2}-2\mu^{2}M^{2}\;,$ (107)

for the extremal Kerr scenario [cf. Eq.(69)], and

$\displaystyle K_{lm}$ $\displaystyle=$
$\displaystyle\frac{2m^{2}a^{2}+2qmaQM}{M^{2}+a^{2}}+\frac{\left(2amM+qQ^{3}\right)^{2}}{\left(M^{2}+a^{2}\right)^{2}}$
(108) $\displaystyle-$ $\displaystyle\mu^{2}\left(M^{2}+a^{2}\right)\;,$

for a KNBH [cf. Eq. (99)]444When $Q=0$ in (108) the constants $K_{lm}$ reduce
to those of Eq. (107), where $a=M$ for an extremal Kerr BH.. Focusing on the
case of an extremal Kerr BH (58) and based on the study presented in
Garcia2020  we see that if we take the separation constants as in Eq.(107),
one obtains radial solutions and its derivatives that are well behaved on the
horizon but that diverge on the axis of symmetry. Something completely
analogous happens when one studies the Kerr-Newman scenario and chooses the
separation constants that appear in (108). It is worth mentioning that this
choice of the separation constants arises precisely when superregularity
conditions are imposed on the radial functions in extremal scenarios
Garcia2020 , but then the separation constants (which incidentally do not
depend explicitly on the number $l$) given by (107) and (108) do not coincide
in general with the actual separation constants (21) which makes the
spheroidal harmonics to be well behaved in the axis of symmetry555See
Garcia2020  for more details.. Thus, we must disregard this possibility and
look for $\alpha_{+}>0$.

To understand what happens when $m\geq 3$, we analyze first the following term
associated with the Kerr scenario

$\Theta\equiv 1+4\left[K_{lm}+2\left(\mu^{2}M^{2}-m^{2}\right)\right]\;,$
(109)

which appears as subradical in Eq.(69). Using Eq. (21) in the previous
equation we find

$\Theta=4\left(l+\frac{1}{2}\right)^{2}+4\mu^{2}M^{2}-7m^{2}+4\lambda\;,$
(110)

where

$\lambda\equiv\sum_{k=1}^{\infty}c_{k}(\mu^{2}M^{2}-m^{2}/4)^{k}\;.$ (111)

Table 2 shows some values of certain quantities associated with the parameter
$\Theta$ for the optimal values of $\mu M$ that we found for the extremal Kerr
BH leading to asymptotically vanishing clouds. The table includes the values
$m=l=3$ alluded above leading to bad behaved clouds at the horizon
($\alpha_{+}<0$), and also some values $m<l$ leading to well behaved clouds
everywhere in the DOC.

$m$ | $l$ | $\mu M$ | $\sqrt{\Theta}$ | $\lambda$ | $\alpha_{+}$
---|---|---|---|---|---
1 | 1 | 0.52550879 | 1.76792337 | 0.00522877 | 0.38396168
2 | 0.50806701 | 4.36422594 | 0.00348492 | 1.68211297
3 | 0.50420923 | 6.55932909 | 0.00197260 | 2.77966454
2 | 2 | 1.14093944 | 1.54230077 | 0.04293009 | 0.27115039
3 | 1.04424934 | 5.04800576 | 0.03013387 | 2.02400474
4 | 1.02432966 | 7.56811183 | 0.01982790 | 3.28405591
3 | 3 | 1.87316979 | 0.76699072 | 0.13830364 | -0.11650463
4 | 1.61385150 | 5.36694663 | 0.09651237 | 2.18347331
Table 2: Quantities $\mu M$, $\Theta$ and $\lambda$ that contributes to the
parameter $\alpha_{+}$ given in Eq.(69). These values correspond to $n=0$.
Clouds are not regular when $\alpha_{+}<0$, namely, when $m=l=3$ as one
notices in the second-last row.

In Hod2012  Hod finds that the existence of bound states implies that666See
Eq. (18) in Hod2012 .

$\frac{m}{2}<\mu M<\frac{m}{\sqrt{2}}\;.$ (112)

Using the inequalities (112) in $\Theta$ given by Eq.(110) it is possible to
observe the corresponding inequalities:

$\Theta_{\rm min}<\Theta<\Theta_{\rm max}\;,$ (113)

where777For $m=l$ we have that $\Theta_{\rm min}=2m(2-m)+1$.

$\Theta_{\rm min}=4\left(l+\frac{1}{2}\right)^{2}-6m^{2}\;,$ (114)

and

$\Theta_{\rm
max}=4\left(l+\frac{1}{2}\right)^{2}-5m^{2}+\sum_{k=1}^{\infty}c_{k}\left(\frac{m^{2}}{4}\right)^{k}\;.$
(115)

Now, in order for $\alpha_{+}>0$, we require $\Theta>1.$ If $\Theta$ saturates
the lower bound of (113) and in addition $\Theta_{\rm min}\geq 1$, then these
conditions are sufficient for $\alpha_{+}>0$. From Eq.(114) we find that
$1\leq\Theta_{\rm min}$ if

$|m|\leq\sqrt{\frac{2l(l+1)}{3}}\;.$ (116)

For example, taking $l=1$, $|m|<\sqrt{4/3}\approx 1.15$; for $l=2$, $|m|\leq
2$; for $l=3$, $|m|\leq\sqrt{8}\approx 2.83$; and for $l=4$,
$|m|\leq\sqrt{40/3}\approx 3.65$, and so on888For $m=l$, $\Theta_{\rm min}\geq
1$ implies $2m(2-m)\geq 0$ and then $0\leq m\leq 2$.. In the case $l=3$, and
given that $m$ is an integer, the condition $|m|\leq\sqrt{8}$ can be only
satisfied if for instance, $|m|=2$, but not for $|m|=3$. This trend is
confirmed by the numerical analysis as shown in Table 2. From the table we
appreciate that for $l=3=m$, $\sqrt{\Theta}<1$, and thus $\alpha_{+}<0$, which
leads to singular scalar clouds at the horizon. This problem persists for
$l\geq 4$ which forces $|m|<l$ when $l\geq 3$. Notwithstanding, when $|m|<l$,
the numerical evidence shows that $\alpha_{+}>1$, which leads not only to a
field $\Psi$ that vanishes at the horizon, but its radial derivative also
vanishes there (cf. Eq.(56)).

We thus conjecture that if $m=l\geq 3$ the scalar cloud configurations are not
regular at the horizon. On the other hand, when $|m|<l$ the scalar clouds
around an extremal Kerr BH are regular everywhere and the field and its radial
derivative $R^{\prime}$ vanish at the horizon $r=M$ since in this situation
$\alpha_{+}>1$. Even $R^{\prime\prime}$ vanish at the horizon when
$\alpha_{+}>2$. A similar behavior occurs in extremal KNBH (see Table 3) and
we extend our conjecture to that scenario as well.

$l$ | $m$ | $\mu a$ | $\mu M$ | $\alpha_{+}$
---|---|---|---|---
3 | 2 | 1.08761968 | 1.09220720 | 2.02617256
3 | 3 | 1.90941874 | 1.91203554 | -0.11159253
4 | 3 | 1.65721537 | 1.66022974 | 2.18544718
Table 3: Behavior of the exponent $\alpha_{+}$ given in Eq.(99) for some
values of the parameters ($n=0$, $\mu Q=0.1$ and $q/\mu=1$) leading to charged
clouds around a KNBH. Only the clouds with $\alpha_{+}>0$ are regular in the
DOC, including the horizon.

Figure 19 depicts some solutions for the radial function $R(r)$ with $m<l$,
around extremal Kerr BH’s (top panel) and extremal KNBH (bottom panel). These
radial functions have $\alpha_{+}>0$, and thus, provide clouds that are
regular in the DOC. The plots confirm that $R(M)=0=R^{\prime}(M)$.

In this regard, it is important to emphasize that our parametrization (55)
allows us to find numerically nontrivial regular clouds even in cases where
$R(M)=0=R^{\prime}(M)=R^{\prime\prime}(M)$ as otherwise treating directly the
differential equation for $R(r)$ with those conditions at the horizon would
not necessarily lead to the actual non-trivial solution but rather to the
trivial one $R(r)\equiv 0$.

Figure 19: Radial part $R_{nlm}$ of the scalar field $\Psi$ associated with
integer numbers $n=0$ and $l\neq m$, considering $l=\\{4,3\\}$ and
$m=\\{3,2\\}$, respectively, around an extremal Kerr BH (top panel) and an
extremal Kerr-Newman BH (bottom panel) with $\mu Q=0.1$.

##  VI Conclusion

We analyzed the existence of scalar clouds, electrically charged and non-
charged, in the backgrounds of Kerr and Kerr-Newman BH’s (extremal and
subextremal), respectively. The electrically charged cloud solutions presented
in this work are consistent with those reported in Ref. Benone2014 . Some of
the results presented in Garcia2019  about the existence of clouds around a
Kerr BH’s were extended here as to include the electric charge in both the
clouds and the BH by establishing an integral method that allows us to
understand and justify in a simple and heuristic way the existence of such
non-trivial cloud solutions in this type of scenario. Since a similar method
is precisely used to prove no-hair theorems in static spacetimes, we
illustrate that when rotating BH’s are present there exist obstructions to
extend those theorems due to the presence of non-positive definite terms
associated with the rotation which does not allow us to conclude that only
trivial fields are possible around rotating BH’s.

Moreover, a new technique based on factorizing the radial function in terms of
a regular part and a potentially irregular one at the horizon of the form
$(r-M)^{\alpha}$, allows us to analyze clouds around exact extremal Kerr and
KN BH’s. In this way we can analyze the exponent $\alpha$ and when $\alpha>0$
the cloud solutions turn to be well behaved in the DOC (including the
horizon), where the scalar field associated with the clouds vanishes.
Remarkably even if the radial derivatives are unbounded at the horizon when
$0<\alpha<1$, the cloud solutions are still regular since the covariant
(coordinate invariant) kinetic term that contains radial derivatives turns to
be bounded due to the presence of an (inverse) metric component that vanishes
at the horizon. This treatment closes a gap left in a previous analysis by us
Garcia2020  where boundedness of radial derivatives at the horizon
(superregularity conditions) were imposed leading to some inconsistencies.
Furthermore, we checked that the cloud solutions without the superregularity
restriction are in excellent agreement with the exact solutions found in the
past by Hod Hod2012 ; Hod2015 . Finally, the factorization technique made
possible to find cloud solutions when $m<l$, even for large $l$ or when $m=l$
for $l=1,2$ while providing us with some insight about the impossibility to
find regular solutions when $l=m\geq 3$ which are associated with $\alpha<0$.
It is possible that a similar factorization technique might be implemented
when dealing with hairy-rotating black holes (at the full non-linear level)
like those analyzed recently in Garcia2023 , which may help us to deal with
the extremal scenario in quasi-isotropic coordinates (as opposed to the Boyer-
Lindquist coordinates used in this work) which are inherently singular at the
horizon located at $r_{\rm qiso}=0$ Garcia2023 ; Garcia2023b .

## Acknowledgments

This work was supported partially by DGAPA–UNAM grant IN105223 and CONACYT
(FORDECYT-PRONACES) grant 140630. We are indebted to P. Grandclément and E.
Gourgoulhon for their helpful comments and discussions.

##  Appendix A Data Kerr-Newman-Bosonic field

The following Tables 4-6 show the eigenvalues $\mu a$ found for different
values of $\mu r_{H}$ that leads to cloud solutions associated with a massive
and charged scalar field around Kerr-Newman black holes with electric charge
$\mu Q=0.1$. The numerical data corresponds to the fundamental mode with $n=0$
(nodeless).

$\boldsymbol{\mu r_{H}}$ | $\boldsymbol{\mu a}$ | $\boldsymbol{\omega/\mu}$ | $\boldsymbol{\Omega_{H}/\mu}$ | $\boldsymbol{\mu M}$
---|---|---|---|---
0.12 | 0.0024057827214 | 0.9999996481207 | 0.1670011217984 | 0.1016907824604
0.13 | 0.0039153033832 | 0.9999984500873 | 0.2314648000216 | 0.1035204984637
0.14 | 0.0056316381056 | 0.9999960654455 | 0.2868642910689 | 0.1058275548134
0.15 | 0.0075569016729 | 0.9999909079654 | 0.3350120098438 | 0.1085236892096
0.16 | 0.0096935363204 | 0.9999833297078 | 0.3772689989226 | 0.1115436395199
0.18 | 0.0146121358341 | 0.9999577267443 | 0.4480392883781 | 0.1183708736489
0.20 | 0.0204132614973 | 0.9999148929166 | 0.5050699554542 | 0.1260417531124
0.22 | 0.0271286580352 | 0.9998513094893 | 0.5521140887208 | 0.1343999183790
0.24 | 0.0347969078252 | 0.9997633604195 | 0.5916752221561 | 0.1433558849879
0.26 | 0.0434646737536 | 0.9996473679561 | 0.6254883866728 | 0.1528638035856
0.28 | 0.0531883296210 | 0.9994993316724 | 0.6547948474511 | 0.1629089257283
0.30 | 0.0640361280605 | 0.9993145886544 | 0.6805069316617 | 0.1735010428283
0.32 | 0.0760911342624 | 0.9990874703943 | 0.7033111398859 | 0.1846716573645
0.34 | 0.0894552748854 | 0.9988109332642 | 0.7237350261189 | 0.1964738914776
0.36 | 0.1042551145940 | 0.9984764316070 | 0.7421923620962 | 0.2089849012764
0.38 | 0.1206497816675 | 0.9980712433904 | 0.7590119339466 | 0.2223110129163
0.40 | 0.1388439245643 | 0.9975807867885 | 0.7744631630351 | 0.2365970442355
0.42 | 0.1591062367041 | 0.9969830777394 | 0.7887683055310 | 0.2520414220930
0.44 | 0.1818009833365 | 0.9962470407675 | 0.8021164876315 | 0.2689222699342
0.45 | 0.1942129892664 | 0.9958134150449 | 0.8084841073244 | 0.2780207613331
0.46 | 0.2074420085504 | 0.9953258919251 | 0.8146731607918 | 0.2876436814255
0.48 | 0.2367940433120 | 0.9941447016175 | 0.8265887193270 | 0.3088243947375
0.50 | 0.2710838133913 | 0.9925727318373 | 0.8380067446339 | 0.3334864338828
0.52 | 0.3125126793142 | 0.9903508799232 | 0.8490711695634 | 0.3635232449347
0.54 | 0.3658458319769 | 0.9868435054170 | 0.8599170356659 | 0.4031881229396
0.55 | 0.4005968950143 | 0.9840576024849 | 0.8652614282156 | 0.4299798839046
0.56 | 0.4464066171732 | 0.9795814346444 | 0.8703938593518 | 0.4668561320143
0.57 | 0.5216314025034 | 0.9692329338467 | 0.8737554381729 | 0.5324555439278
0.571 | 0.5331035527382 | 0.9671656526322 | 0.8735959705977 | 0.5431176864642
0.572 | 0.5456459308501 | 0.9646870300545 | 0.8731543592753 | 0.5549943023193
0.573 | 0.5589214827882 | 0.9617607440179 | 0.8723304138988 | 0.5678204397227
Table 4: Eigenvalues associated with charged scalar clouds $(q/\mu=1)$ around a Kerr-Newman black hole $(\mu Q=0.1)$ with parameters $l=m=1$, which are indicated by the dotted line labeled $m=1$ in Figure 1. $\boldsymbol{\mu r_{H}}$ | $\boldsymbol{\mu a}$ | $\boldsymbol{\omega/\mu}$ | $\boldsymbol{\Omega_{H}/\mu}$ | $\boldsymbol{\mu M}$
---|---|---|---|---
0.11 | 0.0005501512807 | 0.9999999913311 | 0.0454659107242 | 0.1004559212110
0.12 | 0.0012007197534 | 0.9999998457823 | 0.0833749687127 | 0.1016726738663
0.13 | 0.0019518996880 | 0.9999993766910 | 0.1154709913424 | 0.1034761919707
0.14 | 0.0028039130146 | 0.9999981691570 | 0.1429994267048 | 0.1057423640292
0.15 | 0.0037570136356 | 0.9999960966621 | 0.1668736972497 | 0.1083803838381
0.20 | 0.0100497880790 | 0.9999645374804 | 0.2506119179064 | 0.1252524956010
0.25 | 0.0189254314487 | 0.9998837214336 | 0.3010814800992 | 0.1457163439110
0.30 | 0.0304518478210 | 0.9997402387422 | 0.3349031895315 | 0.1682121917261
0.35 | 0.0447203722441 | 0.9995247610664 | 0.3592000318380 | 0.1921427309909
0.40 | 0.0618494070021 | 0.9992276217155 | 0.3775325816450 | 0.2172816864331
0.45 | 0.0819895860755 | 0.9988379819796 | 0.3918778692444 | 0.2435803246942
0.50 | 0.1053309163894 | 0.9983424813634 | 0.4034205058387 | 0.2710946019474
0.53 | 0.1209694101014 | 0.9979874437927 | 0.4093254061340 | 0.2882392435663
0.56 | 0.1379073613568 | 0.9975836649289 | 0.4146112922233 | 0.3059093217110
0.59 | 0.1562164495946 | 0.9971257190680 | 0.4193689895868 | 0.3241555755287
0.62 | 0.1759797916451 | 0.9966071031775 | 0.4236710960409 | 0.3430394250544
0.65 | 0.1972942730077 | 0.9960199728535 | 0.4275760093436 | 0.3626346385859
0.68 | 0.2202735266691 | 0.9953547106563 | 0.4311307891055 | 0.3830297254053
0.71 | 0.2450518675579 | 0.9945995206569 | 0.4343732802970 | 0.4043312801363
0.74 | 0.2717894364571 | 0.9937396366685 | 0.4373335094200 | 0.4266685795741
0.77 | 0.3006791637884 | 0.9927563671831 | 0.4400346280063 | 0.4501999737249
0.80 | 0.3319562973226 | 0.9916256588933 | 0.4424933579909 | 0.4751218645825
0.83 | 0.3659117265932 | 0.9903159676282 | 0.4447199257343 | 0.5016815612400
0.86 | 0.4029110404802 | 0.9887850014551 | 0.4467173467138 | 0.5301961084539
0.89 | 0.4434224612577 | 0.9869745617414 | 0.4484797525389 | 0.5610806062628
0.92 | 0.4880588597134 | 0.9848020384726 | 0.4499891268523 | 0.5948920926873
0.95 | 0.5376425365021 | 0.9821457307799 | 0.4512091404838 | 0.6323997352928
0.98 | 0.5933068329668 | 0.9788181524010 | 0.4520732679808 | 0.6747005092067
1.01 | 0.6566536849725 | 0.9745146810617 | 0.4524608087165 | 0.7234129019742
1.04 | 0.7299691681541 | 0.9687097809656 | 0.4521458785027 | 0.7809879742575
1.07 | 0.8163174734726 | 0.9604481365331 | 0.4506868510505 | 0.8510627184563
1.10 | 0.9182234019219 | 0.9480368057917 | 0.4472300908723 | 0.9377882799259
1.13 | 1.0301793052790 | 0.9295128956888 | 0.4405922448678 | 1.0390130093032
1.16 | 1.1292776607708 | 0.9060186509682 | 0.4308792528429 | 1.1339948427224
1.17 | 1.1554073083968 | 0.8979049159343 | 0.4273167707862 | 1.1597718155114
1.171 | 1.1577936094314 | 0.8970988529260 | 0.4269580202881 | 1.1621379342614
1.172 | 1.1601361476599 | 0.8962943195009 | 0.4265990801174 | 1.1644624066156
1.173 | 1.1624338345399 | 0.8954914592236 | 0.4262400096121 | 1.1667439981599
1.174 | 1.1646852618309 | 0.8946904231618 | 0.4258808660970 | 1.1689811580605
1.175 | 1.1668885219775 | 0.8938913763719 | 0.4255217049233 | 1.1711718394565
1.176 | 1.1690408371188 | 0.8930945110393 | 0.4251625794030 | 1.1733131287633
1.177 | 1.1711375085258 | 0.8923000828668 | 0.4248035401426 | 1.1754001970586
1.178 | 1.1731655545607 | 0.8915086268446 | 0.4244446282652 | 1.1774199568793
Table 5: Similar to Table 4 but for the mode $l=m=2$ with $n=0$. These eigenvalues are associated with the dashed line labeled $m=2$ in Figure 1. $\boldsymbol{\mu r_{H}}$ | $\boldsymbol{\mu a}$ | $\boldsymbol{\omega/\mu}$ | $\boldsymbol{\Omega_{H}/\mu}$ | $\boldsymbol{\mu M}$
---|---|---|---|---
0.11 | 0.0003667114669 | 0.9999999936894 | 0.0303063959882 | 0.1004551567150
0.12 | 0.0008002130296 | 0.9999999130702 | 0.0555678782846 | 0.1016693347537
0.13 | 0.0013005618428 | 0.9999996490018 | 0.0769486205492 | 0.1034680440811
0.14 | 0.0018678229089 | 0.9999989779648 | 0.0952801275132 | 0.1057267455800
0.15 | 0.0025020704490 | 0.9999978224055 | 0.1111721987292 | 0.1083542011884
0.20 | 0.0066812847781 | 0.9999803910845 | 0.1668459212137 | 0.1251115989157
0.25 | 0.0125511715278 | 0.9999358844349 | 0.2003138512792 | 0.1453150638134
0.30 | 0.0201308031580 | 0.9998579754837 | 0.2226729457434 | 0.1673420820596
0.35 | 0.0294451011920 | 0.9997429587765 | 0.2386788837717 | 0.1905243056917
0.40 | 0.0405252189993 | 0.9995877540750 | 0.2507092553147 | 0.2145528667186
0.45 | 0.0534090512068 | 0.9993894736426 | 0.2600847043470 | 0.2392805852786
0.50 | 0.0681418718515 | 0.9991450820340 | 0.2675973328890 | 0.2646433146994
0.55 | 0.0847771353082 | 0.9988512383161 | 0.2737508864655 | 0.2906246933373
0.60 | 0.1033774554613 | 0.9985040423163 | 0.2788807911368 | 0.3172390819147
0.65 | 0.1240158270189 | 0.9980989210825 | 0.2832188000384 | 0.3445230195009
0.70 | 0.1467771341929 | 0.9976304566891 | 0.2869299021704 | 0.3725310908013
0.75 | 0.1717600232967 | 0.9970921767998 | 0.2901344366039 | 0.4013343370686
0.80 | 0.1990792424803 | 0.9964763056687 | 0.2929218796355 | 0.4310203404916
0.85 | 0.2288685825797 | 0.9957734399481 | 0.2953597176268 | 0.4616946047600
0.90 | 0.2612846044529 | 0.9949721236604 | 0.2974992999951 | 0.4934831358467
0.95 | 0.2965114078248 | 0.9940582809832 | 0.2993797608315 | 0.5265363236685
1.00 | 0.3347667970329 | 0.9930144454746 | 0.3010306507165 | 0.5610344041978
1.05 | 0.3763103452734 | 0.9918186941452 | 0.3024736589061 | 0.5971949885522
1.10 | 0.4214540724942 | 0.9904431444100 | 0.3037236410283 | 0.6352834251008
1.15 | 0.4705767659242 | 0.9888517901817 | 0.3047890502212 | 0.6756271707076
1.20 | 0.5241434279212 | 0.9869973133089 | 0.3056717668726 | 0.7186359720970
1.25 | 0.5827319765302 | 0.9848162647386 | 0.3063662051602 | 0.7648306225883
1.30 | 0.6470701365174 | 0.9822215790490 | 0.3068574048848 | 0.8148845236817
1.35 | 0.7180861622466 | 0.9790906071036 | 0.3071175319952 | 0.8696843468185
1.40 | 0.7969763040270 | 0.9752454418509 | 0.3070996992667 | 0.9304182961359
1.45 | 0.8852846376814 | 0.9704198870107 | 0.3067271070690 | 0.9986996171429
1.50 | 0.9849584254600 | 0.9642041300870 | 0.3058741164314 | 1.0767143666282
1.55 | 1.0982215461176 | 0.9559590030595 | 0.3043353150022 | 1.1672872788248
1.60 | 1.2267292840044 | 0.9447270945530 | 0.3017884637265 | 1.2733952300731
1.65 | 1.3687597323441 | 0.9293464357228 | 0.2978152258224 | 1.3957585469354
1.70 | 1.5144783472102 | 0.9092897655990 | 0.2921647692556 | 1.5275425482848
1.75 | 1.6474738004273 | 0.8858774119710 | 0.2851943805618 | 1.6533342637413
1.80 | 1.7569780179411 | 0.8615391793175 | 0.2776965167271 | 1.7602699320911
1.85 | 1.8416061126351 | 0.8379523602049 | 0.2702674756578 | 1.8443278578635
1.86 | 1.8554705618497 | 0.8333964779116 | 0.2688164052998 | 1.8581642488953
1.861 | 1.8567869929913 | 0.8329443673848 | 0.2686720665517 | 1.8594784893448
1.862 | 1.8580878534479 | 0.8324929349477 | 0.2685278700756 | 1.8607772478868
1.863 | 1.8593800880717 | 0.8320420880535 | 0.2683838187323 | 1.8620674481796
1.864 | 1.8606487211222 | 0.8315920316224 | 0.2682399090332 | 1.8633341371817
1.865 | 1.8618953389739 | 0.8311427465329 | 0.2680961426494 | 1.8645788882822
1.866 | 1.8631094879049 | 0.8306943781580 | 0.2679525183212 | 1.8657912550700
Table 6: Similar to Table 4 but for the mode $l=m=3$ with $n=0$. These
eigenvalues are associated with the dotted line labeled $m=3$ in Figure 1.

## References

  * (1) LIGO Scientific Collaboration & Virgo Collaboration, Phys. Rev. Lett. 116, 061102 (2019).
  * (2) The Event Horizon Telescope Collaboration et. al., Phys. Rev. D 875, L1-L5 (2019).
  * (3) The Event Horizon Telescope Collaboration et. al., Phys. Rev. D 930, L12 (2022).
  * (4) B. Carter, Phys. Rev. 174, 1559 (1968).
  * (5) R. Ruffini and J. A. Wheeler, Phys. Today 24(1), 30 (1971).
  * (6) W. Israel, Phys. Rev. 164, 1776 (1967); Commun. Math. Phys. 8, 245 (1971); B. Carter, Phys. Rev. Lett. 26, 331 (1971); R. Wald, ibid. 26, 1653 (1971); D. C. Robinson, ibid. 34, 905 (1977); P. Q. Mazur, J. Phys. A 15, 3173 (1982); Phys. Lett. A 100, 341 (1984).
  * (7) M. Heusler, Black Hole Uniqueness Theorems, Cambridge University Press, 1996.
  * (8) J. D. Bekenstein, Phys. Rev. Lett. 28, 452 (1972); idem Phys. Rev. D 5, 1239 (1972); idem Phys. Rev. D 5, 2403 (1972).
  * (9) J. D. Bekenstein, Phys. Rev. D 51, R6608 (1995).
  * (10) D. Sudarsky, Class. Quant. Grav. 12, 579 (1995).
  * (11) I. Peña, and D. Sudarsky, Class. Quant. Grav. 14, 3131 (1997).
  * (12) D. Sudarsky, and T. Zannias, Phys. Rev. D 58, 087502 (1998).
  * (13) S. Hod, Phys. Rev. D 86, 104026 (2012); Phys. Rev. D 86, 129902(E) (2012).
  * (14) S. Hod, Eur. Phys. J. C 73, 2378 (2013).
  * (15) S. Hod, Phys. Rev. D 90, 024051 (2014)
  * (16) S. Hod, Phys. Lett. B 751, 177-183 (2015).
  * (17) C. Herdeiro, and E. Radu, Phys. Rev. Lett. 112, 221101 (2014).
  * (18) C. Herdeiro, and E. Radu,Class. Quant. Grav. 32, 144001 (2015).
  * (19) C. Benone, L. C. B. Crispino, C. Herdeiro, and E. Radu, Phys. Rev. D 90, 104024 (2014).
  * (20) P. Grandclément, and Jordan Nicoules, Phys. Rev. D 105, 104011 (2022).
  * (21) G. García, E. Gourgoulhon, P. Grandclément and M. Salgado, Phys. Rev. D 107, 084047 (2023).
  * (22) J. F. M. Delgado, C. A. R. Herdeiro, E. Radu, and H. Rúnarsson Phys. Lett. B 761, 234 (2016).
  * (23) G. García and M. Salgado, Phys. Rev. D 99, 044036 (2019).
  * (24) G. García and M. Salgado, Phys. Rev. D 104, 064054 (2021).
  * (25) G. García and M. Salgado, Phys. Rev. D 101, 044040 (2020).
  * (26) S. W. Hawking, Phys. Rev. D 191, 044040 (1976).
  * (27) M. Heuler, Helv. Phys. Acta 69, 501 (1996)
  * (28) E. Ayón-Beato, Class. Quant. Grav. 19, 5465 (2002)
  * (29) M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graps, and Mathematical Tables, Dover, 1964.
  * (30) S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972).
  * (31) J.P Hong, M. Suzuki, and M. Yamada, Phys. Lett. B 803, 135324 (2020).
  * (32) C.A.R. Herdeiro, and E. Radu, Eur. Phys. J. C 80, 390 (2020).
  * (33) Y. Brihaye and B. Hartmann, Class. Quant. Grav. 38, 06LT01 (2021).
  * (34) J. C. Degollado and C. A. R. Herdeiro, Gen Relativ Gravit 45, 2483-2492 (2013).
  * (35) G. García, E. Gourgoulhon, P. Grandclément and M. Salgado, in preparation.