# Regular scalar charged clouds around a Reissner-Nordstrom black hole and no-
hair theorems

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 reanalyze the possibility of finding bound states (scalar
clouds) of a test, charged and complex-valued scalar field with mass $\mu$ and
charge $q$ in the background of a Reissner-Nordstrom black hole (RNBH). In
order to determine the existence of such scalar clouds we impose suitable
regularity conditions for the scalar field at the event horizon. We find
numerical evidence for the absence of such clouds in the subextremal and
extremal RNBH when the field is massive but not self-interacting. More
importantly, we put forward a theorem that proves that such clouds cannot
exist. On the other hand, when a suitable self-interacting potential is
included, the theorem no longer applies, providing a heuristic justification
behind the existence of charged clouds (Q-clouds) that were reported recently.

###### pacs:

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

##  I Introduction

In a previous investigation Garcia , we presented numerical solutions that
represent bound states of a complex-valued massive scalar field $\Psi$ in the
test field limit around a subextremal Kerr black hole (BH) of mass $M$ and
angular momentum per unit mass $a$. This type of solutions, dubbed clouds,
were found originally by Herdeiro and Radu Herdeiro2014 ; Herdeiro2015  in the
test field limit and also when taking into account the backreaction of the
field in the spacetime. More recently in Garcia2 , we extended the analysis of
Garcia  by considering clouds in extremal Kerr black holes $(a=M)$. For the
latter, it was necessary to consider the extremality condition exactly and not
in the limit when $a\rightarrow M$, as in this limit the boundedness of the
radial derivative of the field was not secured. Thus, the extremal case
required a separate treatment and different regularity conditions. These
superregular conditions are different from those considered in the past by Hod
Hod2012  in that we demanded boundedness on the radial derivatives at the
horizon in addition of regularity of the field itself. The exact solutions
found by Hod in the extremal Kerr BH Hod2012  were further extended by the
author to the extremal Kerr-Newman black hole Hod2015  by considering a
charged and massive field $\Psi$. In this direction it is worth mentioning the
analysis in Benone2014  where the authors report numerical solutions of
charged scalar clouds around subextremal Kerr-Newman BH’s. Prior to those
solutions, Degollado $\&$ Herdeiro DegolladoHerdeiro2013  had shown that it is
possible to find scalar clouds around an extremal Reissner-Nordstrom BN (RNBH)
only when the mass $\mu$ of the scalar field $\Psi$ turns to be equal to its
electric charge (i.e. $\mu=|q|$), dubbed double extremal limit. From that
analysis it seems that the non-trivial solution is possible if the boundedness
condition for the radial derivative of the field is dropped. Otherwise the
solution reduces to the trivial one $\Psi=0$ in the domain of outer
communication of the RNBH.

In this work we reanalyze the possibility of finding non-trivial charged
scalar clouds in the subextremal and extremal RNBH when the boundedness of the
field and its radial derivatives are imposed at the horizon along the lines
proposed in our previous works Garcia ; Garcia2 . The numerical analysis shows
that such clouds do not exist when the field is massive but not self-
interacting. Moreover, we put forward a (no-hair) theorem based on standard
techniques which shows that such clouds cannot exist even when the boundedness
condition on the radial derivatives at the horizon is dropped, while keeping
the scalars formed from the derivatives of the field regular, casting doubts
on the significance of the purported regular configurations found in
DegolladoHerdeiro2013 . Finally, we consider a similar scenario but taking
into account a self-interacting potential for the field. In this case, the no-
hair theorem no longer applies, which allows us to understand heuristically
the existence of the so-called Q-clouds that were reported lately by several
authors Hong2020 ; Herdeiro2020 ; Brihaye .

##  II The boson clouds

We assume a RNBH described by the usual metric given in area coordinates 111We
use units where $G=c=1$.:

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}$
(1) $\displaystyle+\>r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2},$

where $M$ is the mass and $Q$ is the charge associated with the RNBH. Under
these coordinates,

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

provide the location of the external $(r_{+})$ and the internal horizon
$(r_{-})$ of the black hole, where the metric has coordinate singularities. We
will be interested solely in solving the differential equation for the boson
field $\Psi$ in the domain of outer communication of the RNBH while providing
regularity conditions for $\Psi$ at $r_{H}\equiv r_{+}$, in particular, in the
extremal case $r_{H}^{\rm ext}=r_{+}=r_{-}=M=|Q|$.

We consider a complex-valued, massive, charged scalar field $\Psi$ which has
the following energy-momentum tensor (EMT)

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

where $D_{a}\equiv\nabla_{a}-iqA_{a}$ stands for the covariant derivative
associated with the gauge field $A_{a}$, which in the present case is given in
terms of the electric potential

$A_{a}=-\Phi(dt)_{a}=-Q/r(dt)_{a}\,.$ (4)

associated with the RNBH solution; $q$ is the gauge coupling (i.e. electric
charge) for the field $\Psi$. The potential for a free massive field is given
by Eq.(6) provided below, but later in Sec. VI we analyze a scenario with
self-interaction terms. The EMT (3) is invariant under the $U(1)$ local
symmetry and the field $\Psi$ obeys the Klein-Gordon equation

$D^{a}D_{a}\Psi=\left(\nabla^{a}-iqA^{a}\right)\left(\nabla_{a}-iqA_{a}\right)\Psi=2\frac{\partial
U\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}\;,$ (5)

where

$U(\Psi^{*}\Psi)=\frac{1}{2}\mu^{2}\Psi^{*}\Psi.$ (6)

We are interested in finding “bound states” solutions and consider a scalar
field $\Psi(t,r,\theta,\varphi)$ with temporal and angular dependence of the
form,

$\Psi(t,r,\theta,\varphi)=\phi(r,\theta)e^{i(-\omega t+m\varphi)}\;,$ (7)

where $\phi(r,\theta)$ is a real-valued function, and $m$ is a positive
integer. The bound states correspond to a real-valued frequency $\omega$ equal
to the critical frequency $\omega_{c}\equiv q\Phi_{H}$ DegolladoHerdeiro2013 :

$\omega=\omega_{c}=q\Phi_{H}\;,$ (8)

where $\Phi_{H}$ is the electric potential at the horizon $r_{H}$:

$\Phi_{H}=\frac{Q}{r_{H}}\;.$ (9)

##  III The subextremal RNBH and regularity conditions

In order to solve the Klein-Gordon Eq. (5) for a free field we assume a mode
expansion in the form

$\Psi_{nlm}\left(t,r,\theta,\varphi\right)=R_{nlm}\left(r\right)S_{lm}\left(\theta,\varphi\right)e^{-i\omega
t},$ (10)

where the angular functions $S_{lm}(\theta,\varphi)$ obey the angular equation

$\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{dS_{lm}}{d\theta}\right)+\left(K_{l}-\frac{m^{2}}{\sin^{2}\theta}\right)S_{lm}=0\;,$
(11)

where $K_{l}$ are separations constants that relates the radial and angular
parts of the Klein-Gordon equation (5). We observe that Eq. (11) corresponds
to the Legendre equation, whose solutions are the spherical harmonics
$Y_{l}^{m}(\theta,\varphi)$, and the separation constants $K_{l}$ are provided
by

$K_{l}=l(l+1),$ (12)

where $l$ is a positive integer. We stress that in this scenario the
separation constants do not depend on the magnetic number $m$, in contrast
with clouds solutions around a Kerr BH Herdeiro2014 ; Herdeiro2015 ; Garcia .

Since the separation constants do not depend on the integer $m$ we can change
the notation of the radial function $R_{nlm}$ that appears in the Eq. (10) by
the form $R_{nl}$, to describe the radial functions that obey the radial
Teukolsky equation Teukolsky :

$\Delta\frac{d}{dr}\left(\Delta\frac{dR_{nl}}{dr}\right)+\left[\mathcal{H}^{2}-\left(K_{l}+\mu^{2}r^{2}\right)\Delta\right]R_{nl}=0\;,$
(13)

where

$\Delta=r^{2}-2Mr+Q^{2}\;,$ (14)

and

$\mathcal{H}\equiv\omega
r^{2}-qQr=\frac{qQr^{2}}{r_{H}}-qQr=qQr\left(\frac{r}{r_{H}}-1\right)\;,$ (15)

where we used Eqs. (8) and (9). Notice that $\mathcal{H}$ vanishes at
$r=r_{H}$.

Like in quantum mechanics, the integer parameters $(n,l,m)$ used to label the
scalar-field configurations correspond respectively to the number of nodes,
$n\geq 0$, for the radial function $R_{nl}$, the angular momentum $l\geq 0$,
and finally the “magnetic” number $m$ satisfies $|m|\leq l$. Given that the
background spacetime is spherically symmetric, intuitively one would not
expect the existence of cloud configurations with an angular dependence, for
instance, a dependence on $l$. Nevertheless, we keep this dependence
explicitly without assuming the value $l=0$ in the radial equation for
$R_{nl}$.

In order to find configurations that represent bound states we assume that
asymptotically the scalar field vanishes sufficiently fast. From (13) one
obtains that for $r_{H}\ll r$ the radial function behaves

$R_{nl}\sim\frac{e^{-\mu_{\rm eff}r}}{r}\,,$ (16)

where we introduced the effective mass

$\mu_{\rm eff}=\sqrt{\mu^{2}-\omega^{2}}=\sqrt{\mu^{2}-q^{2}\Phi_{H}^{2}}\,.$
(17)

Therefore we assume $\mu^{2}\geq\omega^{2}=\frac{q^{2}Q^{2}}{r_{H}^{2}}$222If
one allows the existence of configurations with $\mu^{2}<\omega^{2}$, then
asymptotically $R_{nl}\sim\frac{\pm e^{i||\mu_{\rm eff}||r}}{r}$. Therefore,
the radial gradients and the scalar-field potential would behave
asymptotically as $\sim 1/r^{2}$ and thus energy-momentum tensor would behave
asymptotically in this way too. As a consequence if one takes into account the
backreaction of the field into the spacetime, this kind of configurations
would not lead to an asymptotically flat spacetime, as the Komar mass would
diverge asymptotically as $\sim r$.. In Secs. IV and V we examine solutions
within the background of an extremal RNBH for which the strict equality
$\mu^{2}=\omega^{2}=q^{2}$ is considered in our attempt to recover the
solutions reported in DegolladoHerdeiro2013 .

Furthermore, for the bound state solutions to be physically meaningful we
impose regularity conditions on the scalar field $\Psi(t,r,\theta,\varphi)$ at
the BH horizon $r_{H}$. In particular, we impose that the field and its
derivatives are bounded at the horizon. More specifically, $R_{nl}(r)$,
$R^{\prime}_{nl}(r)$ and $R^{\prime\prime}_{nl}(r)$ have finite values at
$r=r_{H}$, where primes indicate the derivative with respect to the radial
coordinate. Thus, assuming that $R^{\prime\prime}_{nl}(r_{H})$ is bounded in
Eq. (13), the regularity condition for $R^{\prime}_{nl}(r_{H})$ in the
subextremal case ($|Q|<M$) turns out to be

$R^{\prime}_{nl}(r_{H})=\left[\frac{l(l+1)+\mu^{2}r_{H}^{2}}{2\left(r_{H}-M\right)}\right]R_{nl}\left(r_{H}\right)\;.$
(18)

The value $R_{nl}\left(r_{H}\right)$ is a priori arbitrary, and we can choose
for instance, $R_{nl}\left(r_{H}\right)\equiv 1$. To find
$R^{\prime\prime}_{nl}(r_{H})$ we need to differentiate Eq. (13) one more time
and demand that $R^{\prime\prime\prime}_{nl}(r_{H})$ is bounded. We find

$\displaystyle R^{\prime\prime}_{nl}(r_{H})=$
$\displaystyle\left[\frac{4\mu^{2}r_{H}\left(r_{H}-M\right)-q^{2}Q^{2}}{8\left(r_{H}-M\right)^{2}}\right]R_{nl}\left(r_{H}\right)$
(19) $\displaystyle+$
$\displaystyle\left[\frac{l(l+1)+\mu^{2}r_{H}^{2}-2}{4\left(r_{H}-M\right)}\right]R^{\prime}_{nl}\left(r_{H}\right)\;.$

We see that the radial derivatives in Eqs. (18) and (19) are finite on the
horizon $r_{H}=M+\sqrt{M^{2}-Q^{2}}$. However, we appreciate that in the
extremal RNBH one requires a separate analysis as in this case these
derivatives blow up when $r_{H}^{\rm ext}=M=|Q|$ (see Sec. IV).

Similar regularity conditions are obtained when analyzing clouds in the
background of a subextremal Kerr-Newman black hole Garciaetal , and when
considering the non-rotating limit $a=0$, we checked that they reduce to the
conditions (18) and (19).

We performed a numerical analysis to solve the radial Eq.(13) under the
regularity conditions (18) and (19), and find that the only solution that
vanishes asymptotically is the trivial one $R_{nl}=0$. Given that the
background spacetime is spherically symmetric one would expect cloud solutions
respecting symmetry. Nonetheless, spherically symmetric $(l=0)$ non-trivial
cloud solutions were not found either.

##  IV The extremal RNBH and regularity conditions

Let us now focus on the extremal RNBH associated with $r_{H}^{\rm ext}=|Q|=M$,
with metric

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

From Eq. (8) the critical frequency for the extremal case is

$\omega_{c}=\frac{qQ}{M}=\frac{qQ}{|Q|}=q\,{\rm sign(Q)}\;.$ (21)

For instance, $\omega_{c}=q$ when choosing $Q>0$, and thus $\Phi_{H}=1$.

The radial function $R^{\rm ext}_{nl}$ obey the equation

$\frac{d}{dr}\left(\Delta_{\rm ext}\frac{dR^{\rm
ext}_{nl}}{dr}\right)+\left[\frac{\mathcal{H}_{\rm ext}^{2}}{\Delta_{\rm
ext}}-\left(K_{l}^{\rm ext}+\mu^{2}r^{2}\right)\right]R^{\rm ext}_{nl}=0\;,$
(22)

where

$\displaystyle\Delta_{\rm ext}$ $\displaystyle=$
$\displaystyle\left(r-M\right)^{2}\;,$ (23) $\displaystyle\mathcal{H}_{\rm
ext}$ $\displaystyle\equiv$ $\displaystyle\omega
r^{2}-qQr=qr\left(r-Q\right)=qr\left(r-M\right)\;.$ (24)

Like in the subextremal scenario, $\mathcal{H}_{\rm ext}$ also vanishes at the
horizon $r=r_{H}=M$.

Assuming again boundedness of the field and the radial derivatives at the
horizon we find the following regularity conditions,

$R^{\rm
ext\prime}_{nl}(M)=\left[\frac{2M\left(q^{2}-\mu^{2}\right)}{2+M^{2}\left(q^{2}-\mu^{2}\right)-K^{\rm
ext}_{l}}\right]R^{\rm ext}_{nl}(M)\;.$ (25) $\displaystyle R^{\rm
ext\prime\prime}_{nl}(M)$ $\displaystyle=$
$\displaystyle-\left[\frac{2\left(q^{2}-\mu^{2}\right)}{6+M^{2}\left(q^{2}-\mu^{2}\right)-K^{\rm
ext}_{l}}\right]R^{\rm ext}_{nl}\left(M\right)$ (26) $\displaystyle-$
$\displaystyle\left[\frac{4M\left(q^{2}-\mu^{2}\right)}{6+M^{2}\left(q^{2}-\mu^{2}\right)-K^{\rm
ext}_{l}}\right]R^{\rm ext\prime}_{nl}\left(M\right)\;.$

Notice that Eqs. (25) and (26) are finite at the horizon $r_{H}^{\rm ext}=M$
assuming $|R^{\rm ext}_{nl}(M)|<\infty$.

The fact that $\Delta_{\rm ext}=(r-M)^{2}$ and its derivative
$\Delta^{\prime}_{\rm ext}=2(r-M)$ vanish at the horizon lead to the following
form for the separation constants $K_{l}^{\rm ext}$ in the extremal case while
assuming $R_{nl}^{\rm ext}(M)\neq 0$, otherwise the solution becomes the
trivial one by virtue of Eqs.(25) and (26):

$K_{l}^{\rm ext}=\left(q^{2}-\mu^{2}\right)r_{H}^{2}.$ (27)

These separation constants are different from those given by Eq.(12), which
are associated with the values required by the spherical harmonics (i.e. the
angular part of the field) to be well behaved. We thus face a similar
consistency problem that we found when analyzing clouds in the extremal Kerr
background Garcia2 . In particular, the separation constants given by (27) are
not even integers and are non-positive since $q^{2}\leq\mu^{2}$. Thus, both
types of the separation constants match

$l(l+1)=(q^{2}-\mu^{2})M^{2}\;,$ (28)

only if $\mu^{2}=q^{2}$, and therefore, only if $l=0$. The condition
$\mu=|q|=|\omega|$ (extremal test field) is precisely the one imposed by
Degollado & Herdeiro DegolladoHerdeiro2013  to report non-trivial cloud
solutions. Nevertheless, from the above considerations not only the angular
dependency is absent, but also non-trivial spherically symmetric solutions are
absent as well since the regularity conditions (25) and (26) reduce to

$R^{\rm ext\prime}_{nl}(M)=R^{\rm ext\prime\prime}_{nl}(M)=0,$ (29)

and the only possible radial regular solution is

$R^{\rm ext}_{nl}(r)\equiv{\rm const}.$ (30)

In particular, choosing ${\rm const}=0$, for the solution to vanish
asymptotically, we are led to the trivial solution

$\Psi(t,r,\theta,\varphi)\equiv 0,$ (31)

which indicates that it is not possible to find non-trivial scalar clouds or
bound states in the extremal RNBH under the scenario proposed in
DegolladoHerdeiro2013 .

This conclusion has, however, some caveats. Here we assumed regularity in the
radial derivatives for the field. This is a sufficient condition leading to a
well behaved scalars formed from the “kinetic” term
$g^{ab}(D_{a}\Psi)^{*}D_{b}\Psi$, but it is not necessary a priori. For
instance, given that in the subextremal scenario $R_{H}$ is a free parameter,
one could choose $R_{H}=(r_{H}-M)^{\beta}B$ where $B$ is a constant, that we
can take $B=1$. If $0<\beta<1$ then from Eq.(18) we see that in the extremal
limit $R_{H}\rightarrow 0$ and $R^{\prime}_{H}\rightarrow\infty$ and in this
way the trivial solution is avoided. Moreover, in such kinetic term appears
$g^{rr}(\Psi^{\prime})^{2}$, and since $g^{rr}=(r-M)^{2}/r^{2}$ in the
extremal RNBH, in principle, one could afford a divergence at $r=M$ in the
radial derivative of the type $\Psi^{\prime}\sim(r-M)^{-\alpha}$ with
$0<\alpha<1$, while still allowing for the kinetic scalars to be bounded at
the extremal horizon. This happens in the extremal Kerr cloud solutions found
by Hod Hod2012 , where the radial functions vanish at the horizon but the
derivatives blow-up there. But even with this caveat in mind, in the next
section we proof a no-hair theorem that excludes this possibility as well. In
fact, we have verified that if we propose the ansatz $R^{\rm
ext}_{nl}(r)=(r-M)^{\alpha}L(r)$ for solving Eq.(22) such that $R^{\rm
ext}_{nl}(r_{H})=0$ and $L(r_{H})\neq 0$, then we find an algebraic equation
for $\alpha$ that depends implicitly on $M$ and a differential equation for
$L(r)$ together with its regularity conditions333Originally, we implemented
this technique for the extremal Kerr BH in collaboration with P. Grandclément
and E. Gourgoulhon that we plan to report in a forthcoming report. In that
scenario, the resulting regular solutions for the equivalent of $L(r)$ have
$0<\alpha<1$ and hence, the radial solutions vanish at the horizon, and the
kinetic term turns out to be also bounded there, as we have discussed in the
main text for the extremal RNBH, despite the divergent behavior of $dR^{\rm
ext}_{nl}/dr$ at $r=M$.. However, we find that the only well behaved solutions
for $L(r)$ are those with $\alpha<0$, leading to bad-behaved solutions for
$R^{\rm ext}_{nl}(r)$ at the horizon. Thus, the only possibility for a regular
solution in the domain of outer communication of the extremal RNBH with a
vanishing field asymptotically is the trivial solution $R^{\rm
ext}_{nl}(r)\equiv 0$. This conclusion is further supported by a no-hair
theorem presented in the next section.

##  V No-hair theorem

As a complementary analysis we now present a more heuristic study to justify
the existence (or absence) of non-trivial boson clouds in the background of a
RNBH. This analysis is based upon an integral technique developed by
Bekenstein Bekenstein1972 , with variants provided by other authors to prove
no-hair theorems in different scalar-field theories Ayon2002 ; Heusler1996 ,
and which we implemented recently Garcia ; Garcia2  to analyze the existence
of non-charged clouds in the background of a Kerr BH.

Let us consider the Klein-Gordon equation for the charged boson field in the
form444In Gourgoulhon  they use a Klein-Gordon equation with the equivalent
form
${D}^{a}{D}_{a}\Psi=\frac{\partial\tilde{U}\left(|\Psi|^{2}\right)}{\partial|\Psi|^{2}}\Psi$.

${D}^{a}{D}_{a}\Psi=\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}},$
(32)

where $\tilde{U}\left(\Psi^{*}\Psi\right)$ 555Equation (32) is equivalent to
Eq. (5) when we consider the following relation between both
$U\left(\Psi^{*}\Psi\right)=\frac{\tilde{U}\left(\Psi^{*}\Psi\right)}{2}$. In
Gourgoulhon  they consider a energy-momentum tensor associated with the scalar
field
$T_{ab}=\nabla_{(a}\Psi\nabla_{b)}\Psi^{*}-\frac{1}{2}g_{ab}\left[\nabla_{c}\Psi\nabla^{c}\Psi^{*}+\tilde{U}\left(|\Psi|^{2}\right)\right]$.
is the potential associated with the complex scalar field
$\Psi=\phi(r,\theta)e^{-i(\omega t-m\varphi)}$. Multiplying both sides by
$\Psi^{*}$ in the last equation and integrating over a spacetime volume
$\mathcal{V}$ within the domain of outer communication of the BH we obtain

$\int_{\mathcal{V}}\Psi^{*}{D}^{a}{D}_{a}\Psi\sqrt{-g}d^{4}x=\int_{\mathcal{V}}\Psi^{*}\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}\sqrt{-g}d^{4}x\,.$

Integrating by parts the l.h.s and using the Gauss theorem, a straightforward
calculation leads to

$\displaystyle\int_{\partial\mathcal{V}}\Psi^{*}s^{a}D_{a}\Psi dS$
$\displaystyle=\int_{\mathcal{V}}\Big{[}(D^{a}\Psi)^{*}(D_{a}\Psi)+\Psi^{*}\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}\Big{]}\sqrt{-g}d^{4}x\,.$
(33)

The surface integral associated with the boundary $\partial\mathcal{V}$ has
four contributions: one at a portion of the BH horizon, one at spatial
infinity and two contributions corresponding to integrals over two spatial
hypersurfaces $\Sigma_{t_{1}}$ and $\Sigma_{t_{2}}$. The latter two cancel
each other because the spacetime is static, and the scalar-field contributions
stationary, and thus, these two integrals differ only by the normals to both
hypersurfaces, which are opposite. The surface integral associated with the
asymptotic region at spatial infinity vanishes when demanding that the field
$\Psi$ falls off sufficiently rapid, namely, exponentially due to the presence
of a mass term, which would produce an asymptotically flat spacetime if the
backreaction of the field were taken into account. Finally, it remains the
surface integral at the horizon, which is a null hypersurface, with normal
$s^{a}$ given by the timelike Killing field
$\xi^{a}=\left(\frac{\partial}{\partial t}\right)^{a}$ at the horizon.
Therefore, $g_{ab}\xi^{a}\xi^{b}$ vanishes at the horizon:
$g_{ab}\xi^{a}\xi^{b}|_{r_{H}}=g_{tt}|_{r_{H}}=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)|_{r_{H}}=0$.
Thus,
$\Psi^{*}s^{a}D_{a}\Psi|_{r_{H}}=-i\Psi^{*}\Psi\left(\omega+qA_{t}\right)|_{r_{H}}$.
Assuming that $\Psi^{*}\Psi$ is bounded but finite at the horizon, the surface
integral at $r_{H}$ vanishes due to the condition (8),
$\omega=-qA_{t}|_{r_{H}}=q\Phi_{H}$. We conclude

$\int_{\mathcal{V}}\Big{[}(D^{a}\Psi)^{*}(D_{a}\Psi)+\Psi^{*}\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}\Big{]}\sqrt{-g}d^{4}x=0\,.$
(34)

The first term in the integrand corresponds to the kinetic contribution:

$K=(D^{a}\Psi)^{*}(D_{a}\Psi)=g^{tt}(D_{t}\Psi)^{*}(D_{t}\Psi)+g^{ij}(D_{i}\Psi)^{*}(D_{j}\Psi)\,,$
(35)

where

$\displaystyle
g^{ij}(D_{i}\Psi)^{*}(D_{j}\Psi)=g^{ij}(\nabla_{i}\Psi)^{*}(\nabla_{j}\Psi)=g^{rr}(\nabla_{r}\phi)(\nabla_{r}\phi)$
(36) $\displaystyle+$ $\displaystyle
g^{\theta\theta}(\nabla_{\theta}\phi)(\nabla_{\theta}\phi)+g^{\varphi\varphi}(\nabla_{\varphi}\Psi^{*})(\nabla_{\varphi}\Psi)$
$\displaystyle=$ $\displaystyle
g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)+g^{\varphi\varphi}m^{2}\phi^{2}\,,$

which is non-negative in the domain of outer communication. Here lower-case
latin indices $i,j$ run $r,\theta,\phi$, and we used $D_{i}=\nabla_{j}$,
because $A_{a}$ has a component only in the time direction, and also used the
harmonic dependency of the field with respect to the angle $\varphi$ following
(7). Moreover, the indices $I,J$ run $r,\theta$. The term with time
derivatives in the kinetic term $K$ reads explicitly as follows,

$g^{tt}(D_{t}\Psi)^{*}(D_{t}\Psi)=g^{tt}\left(\omega+qA_{t}\right)^{2}\phi^{2}\,,$
(37)

where we used the harmonic time dependency of the field following (7).

Collecting these results, the integrand in (34), reads

$\displaystyle
I=g^{tt}\left(\omega+qA_{t}\right)^{2}\phi^{2}+\Psi^{*}\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}$
(38) $\displaystyle+$ $\displaystyle
g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)+m^{2}g^{\varphi\varphi}\phi^{2}\,.$

Below we present two scenarios: one analyzed by Degollado & Herdeiro
DegolladoHerdeiro2013  like in Sec. IV, where we show that the integrand (38)
is not negative, and thus, a no-hair theorem can be established, and another
one presented more recently by several authors Hong2020 ; Herdeiro2020 ;
Brihaye  where the integrand has no definite sign and thus, it is not possible
to establish a no-hair theorem.

The idea is to justify in an heuristic way the existence or absence of scalar
clouds around a charged, static and spherically symmetric black hole in these
two scenarios.

### Absence of charged clouds within a RNBH

We assume a RNBH where

$g^{tt}=-\frac{r^{2}}{\left(r-r_{H}\right)\left(r-r_{-}\right)}\,.$ (39)

In this case, the term (37) reads

$\displaystyle
g^{tt}\left(\omega+qA_{t}\right)^{2}\phi^{2}=-\frac{r^{2}\left(\omega+qA_{t}\right)^{2}\phi^{2}}{(r-r_{H})(r-r_{-})}$
(40) $\displaystyle=$
$\displaystyle-\frac{q^{2}Q^{2}\left(r-r_{H}\right)\phi^{2}}{r_{H}^{2}(r-r_{-})}$

where we used the condition (8) for $\omega$, and the electric potential (4)
like in DegolladoHerdeiro2013 .

Furthermore, we take the potential $\tilde{U}$ for a massive but free field as
follows,

$\tilde{U}(\Psi^{*}\Psi)=\mu^{2}|\Psi|^{2}\,.$ (41)

In this way, the integrand in (38), reads

$\displaystyle
I=\left[\mu^{2}-\frac{q^{2}Q^{2}\left(r-r_{H}\right)}{r_{H}^{2}(r-r_{-})}\right]\phi^{2}$
$\displaystyle+$ $\displaystyle
g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)+m^{2}g^{\varphi\varphi}\phi^{2}$
$\displaystyle=$
$\displaystyle\left[\mu^{2}-\frac{q^{2}Q^{2}}{r_{H}^{2}}+\frac{q^{2}Q^{2}\left(r_{H}-r_{-}\right)}{r_{H}^{2}(r-r_{-})}\right]\phi^{2}$
$\displaystyle+$ $\displaystyle
g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)+m^{2}g^{\varphi\varphi}\phi^{2}$

The term within the brackets is positive semi-definite (i.e. a non-negative
quantity) because $\mu^{2}\geq\omega^{2}=\frac{q^{2}Q^{2}}{r_{H}^{2}}$ in
order for the scalar-field to falls of asymptotically as in (16), and also
because the third term in the brackets is not negative since $r>r_{-}$ and
$r_{H}\geq r_{-}$ in the domain of outer communication of the RNBH. The
equalities $\omega^{2}=q^{2}$ and $r_{H}=r_{-}=|Q|=M$ occurs in the extremal
RNBH. The remaining terms of the integrand $I$ in (V) are non-negative in the
domain of outer communication. Thus, for an extremal RNBH the integrand
reduces to

$\displaystyle I$ $\displaystyle=$
$\displaystyle\left(\mu^{2}-q^{2}\right)\phi^{2}+g^{ij}(\nabla_{i}\Psi)^{*}(\nabla_{j}\Psi)$
(43) $\displaystyle=$
$\displaystyle\left(\mu^{2}-q^{2}+m^{2}g^{\varphi\varphi}\right)\phi^{2}+g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)\,.$

Therefore, in general the integral (34) becomes

$\displaystyle\int_{\mathcal{V}}\Big{[}\left(\mu^{2}-\frac{q^{2}Q^{2}}{r_{H}^{2}}+m^{2}g^{\varphi\varphi}+\frac{q^{2}Q^{2}\left(r_{H}-r_{-}\right)}{r_{H}^{2}(r-r_{-})}\right)\phi^{2}$
$\displaystyle+g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)\Big{]}\sqrt{-g}d^{4}x=0\,.$
(44)

So, in either scenario, the subextremal and extremal ones, the integrand in
the above integral is not negative and thus, in general the equality in (V)
holds only if the scalar-field vanishes identically, i.e.,
$\phi(r,\theta)\equiv 0$ and therefore $\Psi(t,r,\theta,\varphi)\equiv 0$. We
have thus proved that non-trivial regular charged clouds in the background of
a RNBH with a non self-interacting potential are not possible. In particular,
this conclusion holds also for the extremal scenario considered by Degollado &
Herdeiro DegolladoHerdeiro2013  where $\mu=|q|=|\omega|$, dubbed double
extremal limit. For that case the integral (V) reduces to

$\int_{\mathcal{V}}\Big{[}m^{2}g^{\varphi\varphi}\phi^{2}+g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)\Big{]}\sqrt{-g}d^{4}x=0\,,$
(45)

leading to $\phi(r)\equiv 0$ if $m\neq 0$ and $\phi(r)\equiv{\rm const}$ if
$m=0$. Nevertheless, since we demand that $\phi(r\rightarrow\infty)\rightarrow
0$ for the integral surface at spatial infinity to vanish, then $\phi(r)\equiv
0$ also for $m=0$. Thus, contrary to what it is claimed in
DegolladoHerdeiro2013 , non-trivial and regular charged clouds with
$\mu=|q|=|\omega|$ cannot exist in the background of an extremal RNBH, even if
$m=0$.

This conclusion is consistent with the one presented in Sec. IV, albeit more
general. For instance, in this analysis it was not necessary to impose the
boundedness of the radial derivative, $R^{\prime}_{nl}$, at the horizon. What
matters in this analysis is that each term in (V) is bounded in the domain of
outer communication, notably, at the horizon, in particular
$g^{IJ}(\nabla_{I}\phi)(\nabla_{J}\phi)$, namely,
$g^{rr}(\partial_{r}\phi)^{2}$. Thus, $\partial_{r}\phi$, or equivalently
$R^{\prime}_{nl}$ might diverge near the extremal horizon as
$R^{\prime}_{nl}\sim(r-M)^{-\alpha}$, with $0<\alpha<1$, so that
$g^{rr}(\partial_{r}\phi)^{2}=(r-M)^{2}(\partial_{r}\phi)^{2}/r^{2}\sim(r-M)^{2-2\alpha}/r^{2}$
is bounded at $r=M$.

Finally, this analysis also shows that the exact solution for charged clouds
obtained by Hod Hod2015  in the extremal Kerr-Newman do not admit the static
limit $a=0$, where $a$ is the Kerr parameter associated with the spin of the
black hole.

##  VI Charged Q-clouds

Several authors Hong2020 ; Herdeiro2020 ; Brihaye  analyzed the existence of
spherically symmetric scalar clouds with a self-interacting potential, within
the background of a RNBH, and also by taking into account the backreaction of
the boson field into the spacetime. For the latter case, a charged, static and
spherically symmetric black hole is assumed with a spacetime metric in the
following form

$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle\sigma^{2}(r)N(r)dt^{2}+\frac{1}{N(r)}dr^{2}$ (46)
$\displaystyle+$ $\displaystyle r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\,.$

For this kind of clouds a complex-valued and charged scalar field with no
angular dependency was considered,

$\Psi(t,r)=\psi(r)e^{-i\tilde{\omega}t},$ (47)

submitted to a potential $\tilde{U}$

$\displaystyle\tilde{U}(\Psi^{*}\Psi)$ $\displaystyle=$
$\displaystyle\mu^{2}\Psi^{*}\Psi-\lambda\left(\Psi^{*}\Psi\right)^{2}+\nu\left(\Psi^{*}\Psi\right)^{3}$
(48) $\displaystyle=$
$\displaystyle\mu^{2}\psi^{2}-\lambda\psi^{4}+\nu\psi^{6}\,,$

where $\lambda$ and $\nu$ are positive real numbers, with
$\nu>\lambda^{2}/4\mu^{2}$ for $\tilde{U}$ to be a true vacuum at $\psi=0$
Hong2020 . Figure 1 depicts this potential.

Figure 1: The scalar-field potential $\tilde{U}$ (48) as a function of $\psi$,
assuming the values $\mu=1=\lambda$ and $\nu=9/32$ as in Ref.Brihaye .

For the RNBH, and the test-field approximation, which is the only problem that
we analyze here, $\sigma(r)\equiv 1$, and

$N(r)=-g_{tt}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\,,$ (49)

given by (1).

At this point it is important to remark that the theory considered so far is
invariant with respect to a local phase transformation in the field $\Psi$.
This local transformation is compensated by the gauge transformation in the
electromagnetic potential. In particular, the theory is invariant with respect
to a transformation

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle{\tilde{\omega}}+\zeta\,,$ (50) $\displaystyle A_{t}$
$\displaystyle=$ $\displaystyle{\tilde{A}}_{t}-\zeta/q\,,$ (51)

where $\zeta$ is a constant. This can be appreciated by a direct substitution
in the full-fledged set of equations Herdeiro2020 . Nonetheless, this
invariance is apparent from the field equations of the full theory since
$A_{t}$ and $\omega$ appears always in a combination $\omega+qA_{t}$, which
remains invariant under the above transformation, and also because the radial
derivatives for $A_{t}$ are unaffected by this shift. As a consequence, one
can use a gauge different from the one of previous sections where

$\displaystyle\tilde{\omega}$ $\displaystyle=$ $\displaystyle 0\,,$ (52)
$\displaystyle{\tilde{A}}_{t}$ $\displaystyle=$ $\displaystyle
V(r)=\frac{Q}{r_{H}}-\frac{Q}{r}\,.$ (53)

This gauge was employed in Brihaye , and previously in Herdeiro2020 . Notice
that under this gauge, the electric potential $V(r)$ vanishes at the horizon,
but asymptotically it takes a non-zero value. In our case, whether one uses
this or the original gauge where $\omega\neq 0$, it is irrelevant since our
treatment is gauge invariant. Therefore the integrand (38) remains the same.
In particular the integrand (V) takes the same form, except that we have to
replace the mass term for the corresponding term obtained from the potential
(48), and also taking $m\equiv 0$, and $\nabla_{\theta}\phi=0$, since in this
scenario we are assuming only a time and radial dependency in the field. Thus,
the integral (V) becomes

$\displaystyle\int_{\mathcal{V}}\Big{[}\left(\mu^{2}-2\lambda\psi^{2}+3\nu\psi^{4}-\frac{q^{2}Q^{2}}{r_{H}^{2}}+\frac{q^{2}Q^{2}\left(r_{H}-r_{-}\right)}{r_{H}^{2}(r-r_{-})}\right)\psi^{2}$
$\displaystyle+g^{rr}(\psi^{\prime})^{2}\Big{]}\sqrt{-g}d^{4}x=0\,,$ (54)

where we used

$\displaystyle\Psi^{*}\frac{\partial\tilde{U}\left(\Psi^{*}\Psi\right)}{\partial\Psi^{*}}$
$\displaystyle=$
$\displaystyle\Psi^{*}\left[\mu^{2}\Psi-2\lambda\left(\Psi^{*}\Psi\right)\Psi+3\nu\left(\Psi^{*}\Psi\right)^{2}\Psi\right]$
(55) $\displaystyle=$
$\displaystyle\mu^{2}\psi^{2}-2\lambda\psi^{4}+3\nu\psi^{6}\,.$

Unlike the scenario with no self-interaction, the integrand in (VI) has not a
definite sign due to the presence of the self-interaction terms, notably,

$\Lambda(\psi)\equiv\psi^{2}\left(\mu^{2}_{\rm
eff,\infty}-2\lambda\psi^{2}+3\nu\psi^{4}\right)\,,$ (56)

where $\mu^{2}_{\rm eff,\infty}=\mu^{2}-\frac{q^{2}Q^{2}}{r_{H}^{2}}$
corresponds to the mass introduced in (17) for the non-self-interacting model,
and like in that model, $\mu^{2}_{\rm eff,\infty}\geq 0$ so that the boson
field also behaves asymptotically like in (16).

In this way the integral (VI) reads

$\int_{\mathcal{V}}\Big{[}\Lambda(\psi)+\frac{q^{2}Q^{2}\left(r_{H}-r_{-}\right)}{r_{H}^{2}(r-r_{-})}\psi^{2}+g^{rr}(\psi^{\prime})^{2}\Big{]}\sqrt{-g}d^{4}x=0\,,$
(57)

or even

$\int_{\mathcal{V}}\Big{[}\mu^{2}_{\rm
eff}(r)\psi^{2}+\Sigma(\psi)\psi^{2}+g^{rr}(\psi^{\prime})^{2}\Big{]}\sqrt{-g}d^{4}x=0\,,$
(58)

where

$\Sigma(\psi)\equiv\psi^{2}\left(-2\lambda+3\nu\psi^{2}\right)\,,$ (59)
$\mu^{2}_{\rm eff}(r)\equiv\mu^{2}_{\rm
eff,\infty}+\frac{q^{2}Q^{2}\left(r_{H}-r_{-}\right)}{r_{H}^{2}(r-r_{-})}\,,$
(60)

is a position-dependent effective squared mass.

The integrands in (57) or (58) are not positive semidefinite (i.e. they can be
negative) due to the presence of $\Lambda(\psi)$ and $\Sigma(\psi)$,
respectively. The terms containing the charges and the kinetic term (the one
with the radial derivative) are positive semidefinite for $r\geq r_{H}$. In
particular, $\Lambda$ is negative at the two minima corresponding to
$\psi_{\rm\pm}^{\rm\Lambda}=\pm\sqrt{\frac{2\lambda}{9\nu}\left[1+\sqrt{1-\frac{9\nu\mu^{2}_{\rm
eff,\infty}}{4\lambda^{2}}}\right]}$ and $\Sigma$ is also negative at the two
minima $\psi_{\pm}^{\Sigma}=\pm\sqrt{\frac{\lambda}{3\nu}}$. Thus, the
integrand in (VI) has no definite sign. Figures 2 and 3 depict $\Lambda$, and
$\Sigma$, respectively, for values of the parameters used in Brihaye , showing
that both quantities can be negative, namely, at the two minima666The specific
values of the parameters provided in this section amount to $M,r$ and $Q$
given in units of $1/\mu$, $q$ in units of $\mu$, and $\lambda$ and $\nu$ in
units of $\mu^{1/2}$, while $\psi$ is dimensionless (cf. Ref. Hong2020 ).. As
a consequence, a no-hair theorem cannot be established in this case. Not only
that theorem cannot be established for this kind of self-interacting scalar
field potential, but, as we stressed before, several authors have showed that
the presence of this kind of potential allows for the existence of non-trivial
charged clouds, termed Q-clouds Hong2020 ; Herdeiro2020 ; Brihaye  777At first
sight, it is puzzling that in Ref. Mayo1996  a no-hair theorem for a theory
similar to the one presented in this section was established. That is, a
static, spherically symmetric, asymptotically flat and charged subextremal
black hole within Einstein’s general relativity cannot support a non-trivial,
regular and charged complex-valued scalar field endowed with a positive
semidefinite scalar-field potential. However, in the proof of this theorem,
oddly enough, the mass term associated with the scalar field is not taken into
account, and as remarked in Hong2020b ; Herdeiro2020  it is precisely the mass
term that allows one to avoid such theorem.. As argued in Herdeiro2020 ;
Brihaye  these clouds can exist even in a Schwarzschild background, but in the
presence of a test electric field.

Figure 2: The function $\Lambda(\psi)$ (56) for different values of $q$,
taking $\mu=1=\lambda$ and $\nu=9/32$ as in Fig.1. Here we assume $Q=0.09$ and
$r_{H}=0.15$. The minima are located at $\psi_{\rm\pm}^{\rm\Lambda}$ where
$\Lambda(\psi)$ is negative. Figure 3: The function $\Sigma(\psi)$ (59) taking
$\mu=1=\lambda$ and $\nu=9/32$ as in Fig.1. The minima are located at
$\psi^{\Sigma}_{\pm}\approx\pm 1.0887$ where $\Sigma(\psi)$ is negative.

For the extremal RNBH $(r_{H}^{\rm ext}=M=|Q|)$ the integral (58) keeps the
same form, except that $\mu^{2}_{\rm eff}(r)=\mu^{2}_{\rm
eff,\infty,ext}=\mu^{2}-q^{2}$ is a non-negative constant, and

$N(r)=\frac{(r-M)^{2}}{r^{2}}=g^{rr},\qquad\sigma^{2}(r)=1.$ (61)

More specifically the integrand in (58) reduces to

$I=\left(\mu^{2}-q^{2}-2\lambda\psi^{2}+3\nu\psi^{4}\right)\psi^{2}+\frac{\left(r-M\right)^{2}}{r^{2}}\left(\partial_{r}\psi\right)^{2}\;,$
(62)

which does not have a definite sign. Therefore, one cannot establish a no-hair
theorem either in the extremal scenario. Figure 4 depicts the term

$\Lambda_{\rm
ext}(\psi)\equiv\left(\mu^{2}-q^{2}-2\lambda\psi^{2}+3\nu\psi^{4}\right)\psi^{2}\,,$
(63)

that appears in the integrand (62). We can rewrite this function as

$\Lambda_{\rm
ext}(\psi)=\left[3\nu\left(\psi^{2}-\frac{\lambda}{3\nu}\right)^{2}+\left(\mu^{2}_{\rm
eff,\infty,\rm ext}-\frac{\lambda^{2}}{3\nu}\right)\right]\psi^{2}.$ (64)

which indicates that when $\mu^{2}_{\rm eff,\infty,\rm
ext}\geqslant\lambda^{2}/3\nu$ the integrand of the integral (58) is positive
semidefinite, in which case, the only possible Q-cloud solution are the
trivial ones $\psi(r)\equiv 0$ and $\psi(r)=\pm\sqrt{\frac{\lambda}{3\nu}}$
when $\mu^{2}_{\rm eff,\infty,\rm ext}=\lambda^{2}/3\nu$ or $\psi(r)\equiv 0$
when $\mu^{2}_{\rm eff,\infty,\rm ext}>\lambda^{2}/3\nu$. These are trivial
solutions of Eq.(VI.1) (see below) when $M=Q$ which correspond to the three
minima ($\psi=\pm\sqrt{\frac{\lambda}{3\nu}}$, $\psi=0$) of the potential
which is introduced below in Eq.(68), and when assuming the extremal case
(denoted $U_{\rm eff}^{\rm ext}$ in the main text). This minima correspond
also to the zeros of $\Lambda_{\rm ext}(\psi)$. Notice that the two minima
$\psi_{\pm}^{\Lambda}$ associated with $\Lambda(\psi)$ degenerate in the
extremal case when $\mu^{2}_{\rm eff,\infty,\rm ext}=\lambda^{2}/3\nu$ and
become two zeros of $\Lambda_{\rm ext}(\psi)$ (cf. the discussion at the end
of Sec. VI.1). Notwithstanding, as we show below, there exist non-trivial
solutions in the near extremal scenario when $\mu^{2}_{\rm eff,\infty,\rm
ext}<\lambda^{2}/3\nu$.

Figure 4: The function $\Lambda_{\rm ext}(\psi)$, taking $\mu=1=\lambda$ and
$\nu=9/32$ for different values of $q$ in the extremal Reissner-Nordstrom
scenario.

###  VI.1 Subextremal Q-clouds solutions

In order to find Q-cloud solutions we solve numerically the radial equation
associated with the scalar field $\psi(r)$:

$\displaystyle
N^{2}(r)\psi^{\prime\prime}+\left[\frac{2}{r}N(r)+N^{\prime}(r)\right]N(r)\psi^{\prime}+\left(\tilde{\omega}+q\tilde{A}_{t}\right)^{2}\psi$
$\displaystyle=\frac{1}{2}\frac{\partial\tilde{U}}{\partial\psi}N(r)=\left(\mu^{2}\psi-2\lambda\psi^{3}+3\nu\psi^{5}\right)N(r)\,,$
(65)

in the background of a subextremal RNBH, where $N(r)$ is given by (49),
$\tilde{\omega}$ and $\tilde{A}_{t}$ by (52) and (53), respectively. Equation
(VI.1) is solved by implementing the following regularity conditions for first
and second derivatives at the horizon $r_{H}$:

$\displaystyle\psi^{\prime}_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2N^{\prime}(r)}\frac{\partial{\tilde{U}}}{\partial\psi}\Big{|}_{r=r_{H}}$
(66) $\displaystyle=$
$\displaystyle\frac{r_{H}^{3}\left(\mu^{2}\psi_{H}-2\lambda\psi^{3}_{H}+3\nu\psi^{5}_{H}\right)}{\left(r_{H}^{2}-Q^{2}\right)}\,,$
$\displaystyle\psi^{\prime\prime}_{H}$ $\displaystyle=$
$\displaystyle\frac{\psi^{\prime}_{H}}{4N^{\prime}(r_{H})}\left[\frac{\partial^{2}\tilde{U}}{\partial\psi^{2}}\Big{|}_{r=r_{H}}-\frac{4}{r_{H}}N^{\prime}(r_{H})-3N^{\prime\prime}(r_{H})\right]$
(67) $\displaystyle-$
$\displaystyle\frac{q^{2}Q^{2}\psi_{H}}{2\left[r^{2}_{H}N^{\prime}(r_{H})\right]^{2}}+\frac{N^{\prime\prime}(r_{H})}{2\left[2N^{\prime}(r_{H})\right]^{2}}\frac{\partial{\tilde{U}}}{\partial\psi}\Big{|}_{r=r_{H}}$
$\displaystyle=$
$\displaystyle\frac{\psi_{H}^{\prime}}{4\left[r_{H}^{2}-Q^{2}\right]}\Big{[}2\left(\mu^{2}-6\lambda\psi^{2}_{H}+15\nu\psi^{4}_{H}\right)r_{H}^{3}$
$\displaystyle+$ $\displaystyle
2r_{H}-\frac{8Q^{2}}{r_{H}}\Big{]}-\frac{q^{2}Q^{2}r_{H}^{2}\psi_{H}}{2\left[r_{H}^{2}-Q^{2}\right]^{2}}$
$\displaystyle-$
$\displaystyle\frac{\left(\mu^{2}\psi_{H}-2\lambda\psi^{3}_{H}+3\nu\psi^{5}_{H}\right)\left(r_{H}^{2}-2Q^{2}\right)r_{H}^{2}}{2\left[r_{H}^{2}-Q^{2}\right]^{2}}\;,$

where $\psi_{H}\equiv\psi(r_{H})$,
$\psi^{\prime}_{H}\equiv\psi^{\prime}(r_{H})$ and
$\psi^{\prime\prime}_{H}\equiv\psi^{\prime\prime}(r_{H})$.

Equation (VI.1) is solved numerically given the parameters $\lambda$, $\nu$,
$\mu$ and $q$, and fixing the value of the horizon $r_{H}$ and the charge $Q$
of the RNBH. The values for $\psi^{\prime}_{H}$ and $\psi^{\prime\prime}_{H}$
are determined once the specific value for $\psi_{H}$ is provided. This value
is found by a shooting method such that the field $\psi(r)$ vanishes
asymptotically. At this point, it is important to stress that the field
$\psi(r)$ is indeed submitted to an effective potential that in the asymptotic
region takes the form [cf. Eq.(VI.1)]

$U_{\rm eff}=\mu_{\rm eff,\infty}^{2}\psi^{2}-\lambda\psi^{4}+\nu\psi^{6},$
(68)

where $\mu_{\rm eff,\infty}^{2}=\mu^{2}-q^{2}Q^{2}/r_{H}^{2}$.

Figure 5: The effective potential $U_{\rm eff}$ (68) taking $\mu=1=\lambda$,
$\nu=9/32$, $Q=0.09$, and $r_{H}=0.15$, for different values of $q$.

Figure 5 depicts the effective potential (68) associated with different values
for $q$ taking $\mu=\lambda=1$, $\nu=9/32$, $r_{H}=0.15$ and $Q=0.09$. The
shooting method aims at the local minimum of this effective potential located
at $\psi=0$, where $U_{\rm eff}$ vanishes, starting from trial values
$\psi_{H}$ that depend on the value for $q$. In general a numerical
exploration shows that this trial values (assuming only positive ones for
concreteness) are such that $0<\psi_{H}<\psi_{\rm min}^{+}$ where $\psi_{\rm
min}^{+}$ is associated with one of the two global minima of $U_{\rm eff}$
given by $\psi_{\rm min}^{+}$, with $\psi_{\rm
min}^{\pm}=\pm\sqrt{\frac{\lambda}{3\nu}\left[1+\sqrt{1-\frac{3\mu_{\rm
eff,\infty}^{2}\nu}{\lambda^{2}}}\right]}$. As we stressed above, the local
minimum of $U_{\rm eff}$ is at $\psi=0$ and corresponds to the asymptotic
value of the field $\psi(r)$. Therefore for some $q$ the field $\psi(r)$ must
climb one of the local maximum of $U_{\rm eff}$ before reaching the local
minimum. A bad shooting can make the field to oscillate around any of the two
$\psi_{\rm max}$ associated with the local maxima of $U_{\rm eff}$ [cf. Fig. 8
below] or can make the field to go to $\pm\infty$. As remarked in Brihaye ,
giving $Q$, $\mu$ and $r_{H}$, the charge $q$ is limited from above by the
condition $\mu_{\rm eff,\infty}^{2}\geq 0$, which corresponds to $|q|\leq\mu
r_{H}/|Q|$. On the other hand, the zeros of $U_{\rm eff}$ are given by
$\psi=0$ and
$\psi=\pm\sqrt{\frac{\lambda}{2\nu}\left[1\pm\sqrt{1-\frac{4\mu_{\rm
eff,\infty}^{2}\nu}{\lambda^{2}}}\right]}$. So when $\mu_{\rm
eff,\infty}^{2}=\frac{\lambda^{2}}{4\nu}$, we see that the charge $q$ is
limited from below $|q|_{\rm min}\lesssim q$ where $|q|_{\rm
min}\approx\frac{r_{H}\mu}{|Q|}\sqrt{1-\frac{\lambda^{2}}{4\nu\mu^{2}}}$,
assuming $\nu>0$. All this analysis is qualitative, but gives a fair
description of the actual numerical study. In particular, the lower bound
$|q|_{\rm min}$ is approximate, since in this analysis we are neglecting the
contribution of the metric function $N(r)$ in $U_{\rm eff}$ and taking it as
if $N(r)=1$. Moreover, for this particular value of $\mu_{\rm eff,\infty}^{2}$
the potential $U_{\rm eff}$ “degenerate” in that the two global minima
$\psi_{\rm min}^{\pm}$ become also two of its three zeros at
$\psi=\pm\sqrt{\frac{\lambda}{2\nu}}$. In this degenerate situation
$\psi(r)=\psi_{\rm min}^{+}$ is an approximate solution. So when
$|q|\rightarrow|q|_{\rm min}$ the actual positive value
$\psi_{H}\rightarrow\psi_{\rm min}^{+}$, and since in this limit situation
$\psi_{\rm min}^{+}$ is an approximate solution for the field, then the field
$\psi(r)$ remains very close to the constant value $\psi_{\rm min}^{+}$ for a
relatively large values $r$ and then interpolates to the asymptotic value
$\psi=0$ associated with $\psi(r\rightarrow\infty)$, and the actual numerical
solution resembles a step function, as we can appreciate from Fig. 6. This
figure depicts some examples of Q-clouds solutions $\psi(r)$ for different
values of $q$ for a RNBH with charge $Q=0.09$, horizon $r_{H}=0.15$, and a
scalar field with mass $\mu=1$. As $q$ approaches its minimum value, we see
that the Q-cloud solution start looking like a step function. Our results are
in agreement with those obtained in Brihaye .

Figure 7 shows the quantity $\Lambda(\psi)$ that appears in Eq. (57), when
using the solutions $\psi(r)$ that are plotted in Fig. 6. From Fig. 7 we
appreciate that $\Lambda(\psi)$ has indeed negative contributions to the
integrand of the integral (57). The fact that the integrand has negative and
positive contributions allow us to understand why this integral vanishes when
$\psi(r)$ is not necessarily the trivial solution $\psi(r)\equiv 0$, in
contrast with the scenario of Sec. V where the self-interaction terms are
absent leading to an integrand which is never negative and therefore implying
that $\psi(r)\equiv 0$ is the only possible well behaved solution.

Figure 6: Q-cloud solutions $\psi(r)$ for a RNBH with $Q=0.09$, $r_{H}=0.15$,
and $\mu=1=\lambda$, $\nu=9/32$ taking different values for $q$. Figure 7: The
quantity $\Lambda(\psi)$ associated with the solutions of Fig. 6. Notice that
this quantity can be negative and contributes non-trivially to the integral
(57).

Figure 8 shows three numerical solutions for $\psi(r)$ with $q=1$ associated
with three different (albeit very similar) values $\psi_{H}$. The two
oscillating solutions correspond to the two values $\psi_{H}$ that undershoot
and overshoot the desired asymptotic value $\psi=0$, and which asymptotically
oscillate around $\psi_{\rm max}$ associated with the local maxima of $U_{\rm
eff}$, given by $\psi_{\rm max}\approx\pm 0.6175$ 888If the background were
not fixed, the equivalent of those two solutions would led to an spacetime
that is not asymptotically flat but perhaps asymptotically de Sitter (e.g. if
the oscillations falls-off sufficiently fast) with an effective cosmological
constant given by $U_{\rm eff}^{\rm max}=U_{\rm eff}(\psi_{\rm max})$.. These
two values are represented by the horizontal dotted lines. The non-oscillating
solution corresponds to the optimal shooting value $\psi_{H}$ leading to an
asymptotically vanishing solution.

Figure 8: Solutions $\psi(r)$ (solid lines) with $q=1$ for three different
values of $\psi_{H}$ within a RNBH with $Q=0.09$, $r_{H}=0.15$,
$\mu=1=\lambda$ and $\nu=9/32$. The three solutions correspond to $\psi_{H}$:
1.3820 (purple line), 1.381958 (green line) and 1.381959590 (red line). The
horizontal dotted lines represent the two values $\psi_{\rm max}\approx\pm
0.6175$ associated with the local maxima of the effective potential $U_{\rm
eff}$ depicted in Fig.5 for $q=1$. The solution corresponding to an
asymptotically vanishing field, which is the relevant for the current
analysis, is marked in red color.

We have also obtained Q-cloud solutions in the near extremal RNBH scenario
$Q\approx M$ which are consistent with those reported in Hong2020 . In this
scenario the charge $q$ is bounded as follows $q_{\rm min}\lesssim
q\lesssim\mu$, where $q_{\rm min}\approx 1/3$, and this value is obtained from
$|q_{\rm min}|$ in the near extremal limit and when $\nu$ saturates the bound
$\lambda^{2}/4\mu$ required for the scalar-field potential (48) to have a true
vacuum at $\Psi=0$. Figure 9 shows four solutions of this kind as $Q$
approaches $M$ and Fig. 10 shows the effective potential associated with these
solutions. From the regularity conditions (66) and (67) we appreciate that as
$Q\rightarrow M$ the derivatives diverge at the horizon, a feature that can be
appreciated also in Fig. 9. Due to this divergent behavior at the horizon, the
exact extremal case $(Q=M)$ requires a separate analysis that demands a
different numerical technique Garciaetal . Moreover, this analysis is also
necessary to prove that if a non-trivial physically meaningful solution exists
for the field $\psi(r)$ in the exact extremal scenario, then the kinetic term
in the integral (57), namely $g^{rr}(\psi^{\prime})^{2}$, remains well
behaved, notably at the horizon, despite a divergent $\psi^{\prime}_{H}$. At
this respect it is also interesting to remark that non-trivial Q-cloud
solutions in the extremal scenario with bounded derivatives at the horizon are
absent Garciaetal , and the only ones allowed that have bounded derivatives
are the trivial ones corresponding to the zeros of $\Lambda_{\rm ext}(\psi)$
(cf. Fig. 4) or equivalently, to the extrema (minima and maxima) of $U_{\rm
eff}^{\rm ext}$ depicted by Fig. 11. Thus, in the near extremal case when $q$
approaches $1/3$ something similar happens to the subextremal solutions when
$q\approx q_{\rm min}$. Namely, the solutions have a “step function” shape,
where $\psi(r)$ remains near the global minima of $U_{\rm eff}$ for larger
values of $r$ as $q\rightarrow 1/3$, and then interpolates to the local
minimum of $U_{\rm eff}$ associated with the asymptotic value $\psi\rightarrow
0$ passing through a local maximum. This behavior is depicted by Fig. 12.

Figure 9: Q-cloud solutions $\psi(r)$ for near extremal RNBH $Q\approx M$,
taking $M=2$, $\mu=1=\lambda$, $\nu=9/32$ and $q=0.8$. The charge of the black
hole is taken to be $Q=(1-\varepsilon)M$ with $\varepsilon=10^{-2}$,
$10^{-3}$, $10^{-4}$ and $10^{-5}$. Figure 10: The effective potential $U_{\rm
eff}$ (68) associated with the radial solutions for the near extremal RNBH
(Fig. 9) taking $\lambda=1$, $\nu=9/32$, $\mu=1$ and $q=0.8$. The charge of
the black hole is taken to be $Q=(1-\varepsilon)M$ with $\varepsilon=10^{-2}$,
$10^{-3}$, $10^{-4}$ and $10^{-5}$. For reference, the exact extremal case
$r_{H}=M=Q$ is depicted by the black dashed line. Figure 11: The effective
potential $U^{\rm ext}_{\rm eff}$ (68) taking $\mu=1=\lambda$, $\nu=9/32$, and
$Q=r_{H}=0.15$, for different values of $q$. Figure 12: Q-cloud solutions
$\psi(r)$ for near extremal RNBH $Q\approx M$, taking $M=2$, $\mu=1=\lambda$,
$\nu=9/32$. The charge of the black hole is taken to be $Q=(1-\varepsilon)M$
with $\varepsilon=10^{-4}$ and for values $q$ approaching $1/3$. Figure 13:
The effective potential $U_{\rm eff}$ (68) associated with the radial
solutions for the near extremal RNBH (Fig. 12) taking $\lambda=1$, $\nu=9/32$
and $\mu^{2}=1$ and for different values of $q$ that approach the positive
value $q_{\rm min}^{\rm ext}\approx 1/3$ with $Q=(1-\varepsilon)M$ and
$\varepsilon=10^{-4}$.

As mentioned before, in the exact extremal scenario $\mu^{2}_{\rm eff}(r)$
given by (60) reduces to $\mu^{2}_{\rm eff}(r)=\mu^{2}_{\rm
eff,\infty,ext}=\mu^{2}-q^{2}$ which is not position dependent anymore, and
then the extrema (minima and maxima) of $U_{\rm eff}^{\rm ext}$ become exact
but trivial Q-cloud solutions. Moreover, when $\mu^{2}_{\rm
eff,ext,\infty}=\frac{\lambda^{2}}{3\nu}$, the effective potential (68) with
$\mu^{2}_{\rm eff}=\mu^{2}_{\rm eff,ext,\infty}$, has only one global minima
at $\psi=0$ (see Fig. 14), which is the only possible solution since in this
case the function $\Lambda_{\rm ext}(\psi)$ is positive semidefinite and
therefore the integral (57) only holds if $\psi(r)\equiv 0$. In this
particular extremal scenario there is not even a real valued $q$ that
satisfies the condition $\mu^{2}_{\rm
eff,ext,\infty}=\mu^{2}-q^{2}=\frac{\lambda^{2}}{3\nu}$ unless
$\nu\geq\lambda^{2}/3\mu^{2}$.

Figure 14: The effective potential $U^{\rm ext}_{\rm eff}$ (68) for an
extremal RNBH ($r_{H}=|Q|=M$) taking $\mu=1=\lambda$, $\nu=9/32$, and for
$\mu_{\rm eff,ext,\infty}^{2}=\lambda^{2}/3\nu=32/27$.

##  VII Conclusion

A massive, charged and complex-valued scalar field coupled to a Reissner-
Nordstrom black hole has been studied for the subextremal and extremal
scenarios, by imposing regularity conditions on the field, notably, in the
radial part. When the scalar-field potential has no self-interactions terms it
is not possible to find non-trivial superregular numerical solutions or scalar
clouds (i.e. solutions where the field and its radial derivatives are bounded
in the domain of outer communication of the BH and at the BH horizon as well).
The extremal scenario with an extremal scalar field $(|q|=\mu)$, dubbed double
extremal, was previously studied by Degollado $\&$ Herdeiro
DegolladoHerdeiro2013  reporting non-trivial solutions, but it is unclear to
what extent those solutions are regular at the horizon, given that using an
integral method we have proven a theorem establishing that such non-trivial
configurations cannot exist even if one allows a certain singular behavior on
the radial derivative of the field at the extremal horizon while keeping the
kinetic scalars associated with the boson field bounded there. Therefore, this
conclusion casts doubts about the physical significance of the solutions found
in DegolladoHerdeiro2013 . On the other hand, by implementing the same
integral method to the case where the scalar-field potential has self-
interaction terms, it is not possible to prove a similar theorem, which in
turn provides a heuristic justification and understanding for the existence of
regular and spherically symmetric cloud solutions within this variant of the
theory (termed Q-clouds) which have been reported recently by several authors
Hong2020 ; Herdeiro2020 ; Brihaye  and that we have reproduced here in the
background of a subextremal (including a near extremal) RNBH. Finally, given
that the radial derivative of the field may diverge at the BH horizon in the
extremal scenarios (cf. Hod2012 ; Hod2015  for non-charged and charged clouds
in the backgrounds of an extremal Kerr and an extremal Kerr-Newman black
holes, respectively) while keeping the solutions for the clouds physically
meaningful Garciaetal , a more detailed study is in order for the numerical
analysis of Q-clouds in the presence of an exact extremal RN black hole and
not only in the near extremal limit Garciaetal .

## Acknowledgments

This work was supported partially by DGAPA–UNAM grant IN111719 and CONACYT
(FORDECYT-PRONACES) grant 140630. G.G. acknowledges CONACYT scholarship
291036. We are indebted to P. Grandclément, and E. Gourgoulhon for fruitful
discussions and valuable suggestions.

##  Appendix A Non-existence of bound states for a free scalar field around
Reissner-Nordstrom BH

It is instructive to recover the no-hair theorem presented in Sec. V in a much
more simplified fashion. We begin by considering Eq.(13) for the radial
function $R=R(r)$:

$\left(\Delta
R^{\prime}\right)^{\prime}=\left[\left(K_{l}+\mu^{2}r^{2}\right)-\frac{\mathcal{H}^{2}}{\Delta}\right]R,$
(69)

multiplying both sides by $R$ and integrating both sides from $r_{H}$ to
infinity, we obtain

$\int_{r_{H}}^{\infty}R\left(\Delta
R^{\prime}\right)^{\prime}dr=\int_{r_{H}}^{\infty}\left[K_{l}+\mu^{2}r^{2}-\frac{\mathcal{H}^{2}}{\Delta}\right]R^{2}dr\,.$
(70)

Integrating by parts the left-hand side of previous equation reads

$\int_{r_{H}}^{\infty}R\left(\Delta R^{\prime}\right)^{\prime}dr=\left.R\Delta
R^{\prime}\right|_{r_{H}}^{\infty}-\int_{r_{H}}^{\infty}\left(\Delta
R^{\prime}\right)R^{\prime}dr\,.$ (71)

Assuming the following conditions at the horizon and asymptotically [cf. Eq.
(16)]

$\displaystyle R(r_{H}),R^{\prime}(r_{H})<\infty\quad\text{(finite values at
$r_{H}$)},$ (72) $\displaystyle
R(r\rightarrow\infty),R^{\prime}(r\rightarrow\infty)\rightarrow 0\,,$ (73)

and given that $\Delta$ vanishes at the horizon,

$\Delta_{H}\equiv r_{H}^{2}-2Mr_{H}+Q^{2}=0,$

we conclude

$\left.R\Delta R^{\prime}\right|_{r_{H}}^{\infty}=0.$ (74)

Therefore Eq. (70) reduces to

$\int_{r_{H}}^{\infty}\Big{[}\Delta R^{\prime 2}+\alpha(r)R^{2}\Big{]}dr=0,$
(75)

where we have defined

$\alpha(r)\equiv K_{l}+\mu^{2}r^{2}-\frac{\mathcal{H}^{2}}{\Delta}.$ (76)

We observe that the first term $\Delta R^{\prime 2}$ is positive semi-
definite, however, the second term $\alpha(r)R^{2}$ does not have an apparent
definite sign. Nevertheless, below we prove that $\alpha(r)$ is not negative.
Since $\mathcal{H}$ is given by Eq.(15) and

$\Delta=\left(r-r_{H}\right)\left(r-r_{-}\right)\,,$ (77)

then

$\alpha(r)=K_{l}+\mu^{2}r^{2}-\frac{q^{2}Q^{2}}{r_{H}^{2}}\frac{\left(r-r_{H}\right)r^{2}}{\left(r-r_{-}\right)}.$
(78)

Using

$\frac{r-r_{H}}{r-r_{-}}=1+\frac{r_{-}-r_{H}}{r-r_{-}},$ (79)

it is possible to rewrite equation (76) as

$\alpha(r)=K_{l}+\left(\mu^{2}-\omega^{2}\right)r^{2}+\omega^{2}r^{2}\frac{\left(r_{H}-r_{-}\right)}{\left(r-r_{-}\right)}\,.$
(80)

We now appreciate that the function $\alpha(r)$ is positive semidefinite for
$r\geqslant r_{H}$ and $\mu^{2}\geq\omega^{2}$, and $r_{H}\geq r_{-}$, which
includes the extremal scenario. In the subextremal case $r_{H}>r_{-}$, the
function $\alpha(r)$ is strictly positive. Therefore the integrand that
appears in the integral (75) is non-negative, and for this integral to vanish
it is necessary that each term in the integrand vanishes identically for all
$r\geq r_{H}$. In particular, if $\mu^{2}>\omega^{2}$,

$R^{\prime}(r)\equiv 0\quad\text{and}\quad R(r)\equiv 0,$ (81)

which leads to the trivial solution $\Psi\equiv 0$. We conclude that there are
no bound states when the scalar field is coupled to a subextremal Reissner-
Nordstrom BH with $\mu^{2}>\omega^{2}$. In the extremal case $(r_{H}=M=Q)$ the
function $\alpha(r)$ reduces to

$\alpha(r)=K_{l}+\left(\mu^{2}-\omega^{2}\right)r^{2},$ (82)

where $\omega^{2}=q^{2}$ in this case. Again, if $\mu^{2}>\omega^{2}$, then
$\alpha(r)$ is strictly positive for all $r\geq r_{H}$, and the same
conclusion follows about the absence of non-trivial scalar clouds.

Finally, when focused on the scenario studied by Degollado and Herdeiro
DegolladoHerdeiro2013  about charged scalar clouds in the extremal RNBH with
an extremal test field with $\mu=|q|$ then

$\alpha(r)=K_{l}=l(l+1)\geqslant 0\,.$ (83)

Thus $\alpha(r)$ is nonzero except for $l=0$ (i.e. spherically symmetric
clouds), and then the integral (75) vanishes for all $l\geq 0$ only if the
radial function satisfies for all $r\geq r_{H}=M$:

$R^{\prime}(r)\equiv 0\quad\text{and}\quad R(r)\equiv 0,\quad\text{for}\quad
l\neq 0,$ (84)

and

$R^{\prime}(r)\equiv 0\quad\text{and}\quad R(r)\equiv{\rm
const.},\quad\text{for}\quad l=0\,.$ (85)

Since we demand that $R(r\rightarrow\infty)\rightarrow 0$, then $R(r)\equiv 0$
also for $l=0$. Thus, even in the particular extremal scenario of Ref.
DegolladoHerdeiro2013 , our analysis shows that non-trivial clouds are not
possible.

The final conclusion is that non-trivial regular bound states for a free
scalar field (massive and charged) in the background of a Reissner-Nordstrom
black hole, extremal or subextremal, are absent.

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