The interplay of WGC and WCCC via charged scalar field fluxes in the RPST
framework
Mohammad Reza Alipoura111Email<EMAIL_ADDRESS>Jafar Sadeghi
a222Email<EMAIL_ADDRESS>
aDepartment of Physics, Faculty of Basic Sciences, University of Mazandaran
P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

In this paper, we investigate the weak cosmic censorship conjecture (WCCC) for
the Reissner-Nordstrom (R-N) AdS black hole in a restricted phase space
thermodynamics (RPST). Also here, we consider energy flux and equivalence
mass-energy principle and examine the weak gravity conjecture (WGC) and the
weak cosmic censorship conjecture. The incoming and outgoing energy flux leads
to changes in the black hole. In that case, by applying the first law, we
examined whether the second law of thermodynamics is valid. And also one can
say that, in the case where absorption and superradiance are in the saturated
to an equilibrium. Also, by using the thermodynamics of black holes in the
restricted phase space, we show that if the black hole is in an extreme or
close to an extreme state with radiation and particle absorption, the weak
cosmic censorship conjecture is established. In addition, with the help of
equivalence mass and energy principle and second-order approximation, in the
near extremity, we find that when the black hole radiates and its central
charge is greater than the scaled electric charge, the superradiance particles
obey the weak gravity conjecture, and this causes the black hole to move
further away from its extreme state. But when the particles that obey the weak
gravity conjecture are attracted to the black hole when the black hole is very
small. Then, in this case we note that the black hole becomes closer to its
extreme state.

Keywords: R-N AdS Black Hole, Weak Gravity Conjecture, Weak Cosmic Censorship
Conjecture, Restricted Phase Space.

###### Contents

  1. 1 Introduction
  2. 2 RPS thermodynamics and charged Ads black hole
  3. 3 Charged massive scalar field in charged Ads black hole
  4. 4 Charged Scalar Field and RPS Thermodynamics
  5. 5  WCCC in Near-Extremal and Extremal Charged AdS black hole
  6. 6 WGC with equivalence of mass and energy
  7. 7 Conclusions
  8. 8 Acknowledgments

##  1 Introduction

Many attempts are made to establish a connection between quantum theory and
gravity. Among these efforts is the swampland program, used for the
phenomenology of quantum gravity (string theory). In the swampland program,
effective field theories that are compatible with quantum gravity are
considered landscape and those that are not compatible are considered
swampland [1, 2, 3]. Also, one of the important conjectures of this swampland
program, which defines the border between the landscape and the swampland, is
the weak gravity conjecture, which is considered the upper limit of mass,
hence gravity is the weakest force [4, 5]. A lot of work has been done in this
field [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24]. One of the most important phenomena in cosmology is the black hole. Since
the black hole is the solution to Einstein’s equation, we can use the weak
gravity conjecture derived from the string theory for the black hole to find a
connection between these two theories.

Since general relativity breaks down near the fundamental singularity, Penrose
proposed the weak cosmic censoring conjecture to solve this problem, where the
singularity is hidden from distant observers by the black hole’s event horizon
[25, 26, 27]. Therefore, people proposed experiments to validate the
conjecture of weak cosmic censorship by absorbing or throwing up particles
(the field). By doing this, they find out whether this event horizon remains a
black hole or whether it is destroyed, in this case, the validity of the weak
cosmic censorship conjecture is determined [28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39]. Among these works, it can be mentioned [36] that by using the
thermodynamics of black holes in the extended phase space, they showed that
for an extremal and near-extremal black hole despite the dispersion of charged
fields in many times small, stable black hole event horizon and weak cosmic
censorship conjecture will hold. Note that they did the calculations by
considering an isobaric process, i.e. $d\ell=0$. In this article, we examine
the weak cosmic censorship conjecture by altering the phase space through
restricted phase space thermodynamics, to determine if such changes impact the
conjecture’s validity.

The study of black hole thermodynamics in different phase spaces offers a rich
landscape for exploring the fundamental aspects of gravitational physics. In
extended phase space, thermodynamic variables are expanded to include the
cosmological constant, treated as a dynamical pressure, which allows for a
deeper understanding of the thermodynamic properties of black holes,
especially in Anti-de Sitter (AdS) spacetimes. This framework has led to
insights into the microstructure of black holes and the connection between
classical thermodynamics and quantum physics [40, 41, 42]. Conversely, the
restricted phase space approach focuses on a more constrained set of
variables, often excluding the cosmological constant as a variable, thereby
simplifying the thermodynamic study of black holes. This formalism is
particularly useful in the context of holography, where it can reveal
interesting behaviors of AdS black holes without the complexity introduced by
the extended phase space variables [44, 45]. The concept of restricted phase
space thermodynamics (RPST) for AdS black holes via holography is a relatively
new formalism proposed by [44, 45]. It’s based on Visser’s holographic
thermodynamics but significantly modified [43]: The AdS radius $(\ell)$ is
held fixed as a constant. This means that the RPST does not involve the
pressure $P$ and volume $V$ variables typically found in black hole
thermodynamics. Instead, RPST introduces the central charge $(C)$ and chemical
potential $\mu$ as a new pair of conjugate thermodynamic variables [44, 45,
46, 47, 48, 49, 50, 51]. Recently, efforts have been made to establish the
weak gravity conjecture and the weak cosmic censoring conjecture in various
black holes. These findings demonstrate that the weak cosmic censoring
conjecture remains valid for black holes even when $Q>M$ [52, 53, 54, 55, 56].
Now, in this article, we investigate the weak gravity conjecture for particles
entering and leaving the black hole using the principle of equivalence of mass
and energy to find their effect on the weak cosmic censorship conjecture.

So, the above information give us motivation to arrange paper as: In section 2
we show in extended phase space, the cosmological constant is equivalence to
thermodynamic pressure and black hole’s volume as its conjugate thermodynamic
variable. Also, in this space, the mass of the black hole is regarded as
enthalpy. Also in this section, we study the first law of thermodynamics and
Smarr for the charged AdS black hole in the extended phase space. By using
dual theory of holography, the central charge, also has a conjugate
thermodynamic quantity as a chemical potential. We study the first law of
thermodynamics and Smarr’s relation for a charged AdS black hole in restricted
phase space. In section 3 , we generally examine the charged scalar field near
the event horizon of corresponding black hole. In this study, we examine the
scattering behavior of the coupled massive scalar field possessing an electric
charge interacting with the charged AdS black hole within a 4-dimensional
spacetime framework. The charged massive scalar field in charged Ads black
hole is discussed in section 4. In section 5, we investigate WCCC in near-
extremal and extremal charged AdS black hole with the above mentioned
information. In section 6 we study the WGC for the corresponding black hole
with help of equivalence of mass and energy principle. Finally we have some
results and conclusion in section 7.

##  2 RPS thermodynamics and charged Ads black hole

In the extended phase space, the cosmological constant $(\Lambda)$ is
equivalence to thermodynamic pressure of black hole. As we know, the change of
AdS radius $(\ell)$ will change the black hole’s thermodynamic pressure, with
the black hole’s volume as its conjugate thermodynamic variable [57],

$\begin{split}P=-\frac{\Lambda}{8\pi}=\frac{(d-1)(d-2)}{16\pi\ell^{2}},\qquad
V=\bigg{(}\frac{\partial M}{\partial P}\bigg{)}_{S,Q},\end{split}$ (1)

where $d$ and $r_{h}$ are the dimension and radius of the event horizon of the
black hole, respectively. Also, in this space, the mass of the black hole is
regarded as enthalpy [36, 58], and we have,

$\begin{split}M=E+PV,\end{split}$ (2)

where $(E)$ is the internal energy of the black hole. In addition, the
important relations of the first law of thermodynamics and Smarr for the
charged AdS black hole in the extended phase space can be expressed by the
following expressions,

$\begin{split}dM=TdS+\Phi dQ+VdP\end{split}$ (3)

and

$\begin{split}M=\frac{d-2}{d-3}TS+\Phi Q-\frac{2}{d-3}PV,\end{split}$ (4)

here, $S$, $\Phi$ and $T$ are the entropy, electric potential and Hawking
temperature of the black hole respectively. In restricted phase space,
Newton’s constant $(G)$ varies and the AdS radius $(\ell)$ is considered
fixed, thus the black hole’s mass is seen as its internal energy $(E)$ [44,
45, 49, 54],

$\begin{split}E=M.\end{split}$ (5)

By using dual theory of holography, the central charge $(C)$ as an extensive
variable, also it has conjugate thermodynamic quantity as a chemical potential
$(\mu)$ [44, 45, 43],

$\begin{split}C=\frac{\ell^{d-2}}{G},\qquad\mu=\bigg{(}\frac{\partial
E}{\partial C}\bigg{)}_{S,\hat{Q}},\end{split}$ (6)

Also here, the electric charge and corresponding potential are rewritten by
following equation,

$\begin{split}\hat{Q}=Q\sqrt{C},\qquad\hat{\Phi}=\frac{\Phi(r_{h})}{\sqrt{C}}=\frac{\hat{Q}}{r_{h}C}\end{split}$
(7)

The first law of thermodynamics and Smarr’s relation for a charged AdS black
hole in restricted phase space are obtained by,

$\begin{split}dE=TdS+\hat{\Phi}d\hat{Q}+\mu dC\end{split}$ (8)

and

$\begin{split}E=TS+\hat{\Phi}\hat{Q}+\mu C\end{split}$ (9)

Next, we obtain the metric, entropy, mass, and temperature of the charged
black hole in the restricted phase space. So, in that case, the metric
background of charged AdS black hole in 4-dimension will be following form,

$\begin{split}ds^{2}=-h(r)dt^{2}+\frac{1}{h(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\end{split}$ (10)

where [44]

$\begin{split}h(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}+\frac{r^{2}}{\ell^{2}}\qquad
or\qquad
h(r)=1-\frac{2\ell^{2}M}{Cr}+\frac{\ell^{2}\hat{Q}^{2}}{C^{2}r^{2}}+\frac{r^{2}}{\ell^{2}}\end{split}$
(11)

here, we assume units as $k_{B}=c=\hbar=1$. When $\frac{Q}{M}<\sqrt{G}$, the
black hole has two internal and external event horizons, we denote the
external event horizon by $r_{h}$. Also, in the case of
$\frac{Q}{M}=\sqrt{G}$, the black hole is in an extreme state with only one
event horizon. Also, the entropy of charged black hole in the corresponding
space is obtained by following equation,

$\begin{split}S=\frac{\pi r_{h}^{2}}{G},\qquad r_{h}=\ell\bigg{(}\frac{S}{\pi
C}\bigg{)}^{\frac{1}{2}}\end{split}$ (12)

By using equations (7),(11) and (12), one can obtain the mass of black hole
as,

$M=\frac{\pi CS+\pi^{2}\hat{Q}^{2}+S^{2}}{2\pi^{3/2}\ell\sqrt{CS}}$ (13)

Also, Hawking temperature and chemical potential are obtained by,

$T=\bigg{(}\frac{\partial M}{\partial S}\bigg{)}_{\hat{Q},C}=\frac{\pi
CS-\pi^{2}\hat{Q}^{2}+3S^{2}}{4\pi^{3/2}\ell S\sqrt{CS}}$ (14)

and

$\mu=\bigg{(}\frac{\partial M}{\partial C}\bigg{)}_{S,\hat{Q}}=\frac{\pi
CS-\pi^{2}\hat{Q}^{2}-S^{2}}{4\pi^{3/2}C\ell\sqrt{CS}}$ (15)

In the next section, we are going to examine the charged scalar field near the
event horizon of corresponding black hole.

##  3 Charged massive scalar field in charged Ads black hole

Black holes are capable of acquiring conserved quantities, including energy,
momenta and electric charge, through the interaction with an external field.
The obtained conserved quantities from black hole corresponds to the fluxes of
the external field. Through these fluxes, the black hole undergoes alterations
in its state as it engages with the external field. By analyzing the fluxes,
we can approximate the transformations occurring within the black hole over
infinitesimal time intervals. In this study, we examine the scattering
behavior of the coupled massive scalar field possessing an electric charge
interacting with the charged AdS black hole within a 4-dimensional spacetime
framework. The equation of the massive charged scalar field near the charged
black hole is given by [36, 58, 59],

$\frac{1}{\sqrt{-g}}\partial_{\mu}(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\Psi)-2iqg^{\mu\nu}A_{\mu}\partial_{\nu}\Psi-q^{2}g^{\mu\nu}A_{\mu}A_{\nu}\Psi-\mu_{s}^{2}\Psi=0$
(16)

where $A^{\mu}=A^{\nu}=(\Phi(r),0,0,0)$, $g^{\mu\nu}$ , $\mu_{s}$, $q$ and
$\Psi$ are the electric potential, black hole tensor metric, mass, electric
charge and charged massive scalar field, respectively. To obtain the scalar
field, we use separation of variables,

$\Psi(t,r,\theta,\phi)=e^{-i\omega t}R(r)Y_{lm}(\theta,\phi)$ (17)

where $R(r)$, $Y_{lm}(\theta,\phi)$ and $\omega$ are radial functions,
spherical harmonic functions and energy of particle, respectively.
Furthermore, the integer parameters $l$ and $m$ denote the spherical and
azimuthal harmonic indices of the resonant eigenvalues that define the
characteristics of the charged massive scalar fields within the charged black-
hole spacetime. We get two radial and angular parts equations by putting
relations (10) and (17) into equation (16),

$h(r)\frac{d}{dr}\bigg{[}r^{2}h(r)\frac{dR(r)}{dr}\bigg{]}+\bigg{[}r^{2}\bigg{(}\omega-\frac{qQ}{r}\bigg{)}-h(r)\bigg{(}l(l+1)+\mu_{s}r^{2}\bigg{)}\bigg{]}R(r)=0$
(18)
$\bigg{[}\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\bigg{(}\sin\theta\frac{\partial}{\partial\theta}\bigg{)}\bigg{]}Y_{lm}(\theta,\phi)=-l(l+1)Y_{lm}(\theta,\phi)$
(19)

To solve the radial equation, it is common practice to introduce the tortoise
coordinate $\frac{dr}{dy}=h(r)$. In this case, the radial range of $r_{h}\leq
r<+\infty$ becomes that of $-\infty<y\leq 0$ and the equation of the radial
part becomes as,

$\frac{d^{2}R}{dy^{2}}+\frac{2h(r)}{r}\frac{dR}{dy}+\bigg{[}(\omega-\frac{qQ}{r})^{2}-\frac{h(r)}{r^{2}}\bigg{(}l(l+1)-\mu_{s}r^{2}\bigg{)}\bigg{]}R=0$
(20)

Since the flux of the scalar field into the black hole is important, we check
equation (20) near the event horizon of the black hole $(r\rightarrow r_{h})$,
in which case we have a Schrödinger-like equation,

$\frac{d^{2}R}{dy^{2}}+\bigg{(}\omega-\frac{qQ}{r_{h}}\bigg{)}^{2}R=0$ (21)

Therefore, the radial part of the scalar function is obtained,

$R(r)=e^{\pm i(\omega-\frac{qQ}{r_{h}})^{2}y}=e^{\pm
i[\omega-q\Phi(r_{h})]^{2}y}$ (22)

Here, $+/-$ sign indicates the outgoing and ingoing of the fields. Therefore,
the wave function of the charged scalar field near the event horizon of the
charged black hole takes the following form,

$\Psi(t,r,\theta,\phi)=e^{-i\omega t}e^{\pm
i[\omega-q\Phi(r_{h})]^{2}y}Y_{lm}(\theta,\phi)$ (23)

In this study, we examine alterations in a charged AdS black hole resulting
from incoming/outgoing fluxes of the charged scalar field. As the scalar field
enters/exits the black hole, changes in energy and electric charge will mirror
those of the black hole itself. Consequently, variations in the properties of
the black hole, contingent upon the energy and charge, are anticipated to
adhere to a particular relationship. The energy and electric charge conveyed
by the scalar field are represented by their fluxes at the outer horizon,
which are derived from the energy-momentum tensor,

$T^{\mu}_{\nu}=\frac{1}{2}\mathcal{D}^{\mu}\Psi\partial_{\nu}\Psi^{*}+\frac{1}{2}\mathcal{D}^{*\mu}\Psi^{*}\partial_{\nu}\Psi-\delta^{\mu}_{\nu}\bigg{(}\frac{1}{2}\mathcal{D}_{\rho}\Psi\mathcal{D}^{*\rho}\Psi^{*}-\frac{\mu_{s}^{2}}{2}\Psi\Psi^{*}\bigg{)}$
(24)

with $D_{\mu}=\partial_{\mu}-iqA_{\mu}$. Also, the electric current of the
charged scalar field is,

$j^{\alpha}=\frac{iq}{2}\Psi\mathcal{D}^{*\alpha}\Psi^{*}-\frac{iq}{2}\Psi^{*}\mathcal{D}^{\alpha}\Psi$
(25)

Therefore, from equations (24) and (25), the energy flux and electric charge
near the event horizon are obtained by [36],

$\frac{dE}{dt}=\int\sqrt{-g}T^{r}_{t}d\phi
d\theta=\omega(\omega-q\Phi_{h})r_{h}^{2}$ (26)
$\frac{dQ}{dt}=-\int\sqrt{-g}j^{r}d\phi
d\theta=\frac{q}{\omega}\frac{dE}{dt}=q(\omega-q\Phi_{h})r_{h}^{2}.$ (27)

From the above equations, when $\omega>q\Phi_{h}$, the energy and charge flux
is positive, indicating the particle’s entry into the black hole. But for
$\omega<q\Phi_{h}$, the energy and charge flux become negative, which means
the exit of the particle from the black hole, and this phenomenon is called
black hole superradiation. These fluxes can change the internal energy and
electric charge of black holes in infinitesimal time interval $dt$. Since we
consider the thermodynamics of the black hole in the RPS, the changes in the
internal energy of the black hole are equal to the changes in the mass of the
black hole. Also, we consider the changes in electric charge according to
equations (7) and (27), which has been re-scaled in this space. So in that
case, we have,

$dE=dM=\omega(\omega-\hat{q}\hat{\Phi}_{h})r_{h}^{2}dt$ (28)
$d\hat{Q}=\hat{q}(\omega-\hat{q}\hat{\Phi}_{h})r_{h}^{2}dt+\frac{\hat{Q}}{2C}dC$
(29)

where

$\hat{q}=q\sqrt{C},\qquad\hat{\Phi}_{h}=\frac{\Phi_{h}}{\sqrt{C}}$ (30)

Since the charge in RPS is rescaled, the variations of charge changes in this
space are different from the variations of charge in the normal and extended
space. Because it depends on the central charge differential in addition to
the time differential.

##  4 Charged Scalar Field and RPS Thermodynamics

In RPS thermodynamics, the radius AdS is fixed, and Newton’s constant is
considered as a variable. It can also be found that by attracting a scalar
field or a particle into the black hole, there are changes in the quantities
$(M,\hat{Q},C,r_{h})$ of the black hole, which, according to relations (6) and
(11), also leads to changes in the metric of the black hole. Also, WCCC, the
event horizon of the black hole plays an important role and is obtained from
the relation $h(M,\hat{Q},r_{h},C)=0$. Therefore, in the case where the black
hole attracts the scalar field or particle, this event horizon must exist, in
this case, the event horizon of the black hole changes and is obtained from
the following relationship,

$h(M+dM,\hat{Q}+d\hat{Q},r_{h}+dr_{h},C+dC)=0,$ (31)

and,

$\begin{split}&h(M+dM,\hat{Q}+d\hat{Q},r_{h}+dr_{h},C+dC)=h(M,\hat{Q},r_{h},C)\\\
&\qquad+\frac{\partial h}{\partial M}|_{r=r_{h}}dM+\frac{\partial
h}{\partial\hat{Q}}|_{r=r_{h}}d\hat{Q}+\frac{\partial h}{\partial
r}|_{r=r_{h}}dr+\frac{\partial h}{\partial C}|_{r=r_{h}}dC\end{split}$ (32)

where

$\begin{split}&h(M,\hat{Q},r_{h},C)=0,\qquad\frac{\partial h}{\partial
M}|_{r=r_{h}}=-\frac{2\ell^{2}}{Cr_{h}},\qquad\frac{\partial
h}{\partial\hat{Q}}|_{r=r_{h}}=\frac{2\ell^{2}\hat{Q}}{C^{2}r_{h}^{2}},\\\
&\frac{\partial h}{\partial
r}|_{r=r_{h}}=-\frac{2\ell^{2}\hat{Q}^{2}}{C^{2}r_{h}^{3}}+\frac{2\ell^{2}M}{Cr_{h}^{2}}+\frac{2r_{h}}{\ell^{2}}=4\pi
T,\qquad\frac{\partial h}{\partial
C}|_{r=r_{h}}=\frac{1}{C}\bigg{(}\frac{2\ell^{2}M}{Cr_{h}}-\frac{2\ell^{2}\hat{Q}^{2}}{C^{2}r_{h}^{2}}\bigg{)}\end{split}$
(33)

By using equations (28), (29), (30), (31), (31) and (33), one can obtain the
variations in the event horizon,

$\begin{split}dr_{h}=\frac{\ell^{2}r_{h}(\omega-\hat{q}\hat{\Phi})^{2}}{2\pi
CT}dt-\frac{1}{4\pi
CT}\bigg{(}1+\frac{r_{h}^{2}}{\ell^{2}}\bigg{)}dC\end{split}$ (34)

Using equation (12), we calculate the variations of entropy,

$\begin{split}dS=\frac{2\pi Cr_{h}}{\ell^{2}}dr_{h}+\frac{\pi
r_{h}^{2}}{\ell^{2}}dC\end{split}$ (35)

By combining equations (34) and (35), we rewrite the variations of entropy ,

$\begin{split}dS=\frac{r_{h}^{2}(\omega-\hat{q}\hat{\Phi})^{2}}{T}dt-\frac{r_{h}}{2\ell^{2}T}\bigg{(}1+\frac{r_{h}^{2}}{\ell^{2}}-2\pi
r_{h}T\bigg{)}dC\end{split}$ (36)

When we set the central charge changes to zero, we observe entropy changes in
normal space consistent with [32]. In addition, in the case that the
temperature of the black hole will be as,

$\begin{split}T=\frac{1}{2\pi
r_{h}}\bigg{(}1+\frac{r_{h}^{2}}{\ell^{2}}\bigg{)}\end{split}$ (37)

despite maintaining the conditions of the RPS, the entropy changes will still
reduce to the solution of the normal space. According to equation (36), we
find that entropy changes can be negative; in this case, Newton’s second law
can be violated. If the Hawking temperature of the black hole has a lower
bound, the entropy changes can always be positive,

$\begin{split}T\geq\frac{1}{2\pi
r_{h}}\bigg{(}1+\frac{r_{h}^{2}}{\ell^{2}}\bigg{)}.\end{split}$ (38)

Also, in the situation where $\omega=\hat{q}\hat{\Phi}$, and
$\bigg{(}T<\frac{1}{2\pi
r_{h}}\bigg{[}1+\frac{r_{h}^{2}}{\ell^{2}}\bigg{]}\bigg{)}$ the entropy
changes are negative.

##  5  WCCC in Near-Extremal and Extremal Charged AdS black hole

The WCCC suggests that a black hole’s singularity should be hidden by the
horizon from an observer at infinity, preventing a naked singularity within
the black hole’s structure if it has a stable horizon. Our current study
explores the stability of the outer horizon during the scattering of a charged
scalar field, considering RPS thermodynamic. By absorbing the scalar field,
the initial state $h(M,Q,r_{h},C)$ transitions to the final state
$h(M+dM,Q+dQ,r_{h}+dr_{h},C+dC)$ over a small time interval $dt$. By setting
the outer horizon as $h(M,Q,r,C)=0$, we can determine the presence of the
horizon by examining solutions in $h(M+dM,Q+dQ,r_{h}+dr_{h},C+dC)=0$. This
analysis simplifies to studying the change in the minimum value of $h$.

Initially, the minimum value of $h$ is negative or zero, indicating solutions
corresponding to their respective horizons. If the scalar field fluxes enter
the black hole, changes in the black hole’s mass and charge occur due to the
charged scalar field within $dt$, leading to variations in the minimum value
based on these changes. If the minimum value becomes positive during these
changes, no solution indicates a horizon in the final state, resulting in the
black hole becoming a naked singularity, contradicting the WCCC. On the other
hand, in other cases, the horizon consistently hides the interior of the black
hole in the final state, supporting the validity of the WCCC. Thus,
determining the sign of the minimum value in the final state is crucial for
validating the WCCC in the context of the charged scalar field scattering. Our
analysis focuses on determining this sign in the final state, which can be
inferred from the initial state as the final state differs slightly due to
conserved charges transferred through fluxes over $dt$.

This slight modification is more notable when the initial state pertains to a
near-extremal or extremal black hole as opposed to a nonextremal one. The
minimum value of a near-extremal black hole (including an extremal one)
approaches zero. It may shift to a positive value due to minor alterations in
the external scalar field. Therefore, our attention is directed towards a
near-extremal black hole as the initial state, distinguished by the near-
extremal circumstance at the minimum point $r_{min}$ with a negative constant
$\delta\ll 1$ signifying the minimum value of $h$,

$\begin{split}h_{min}\equiv
h(r_{min})=1-\frac{2\ell^{2}M}{Cr_{min}}+\frac{\ell^{2}\hat{Q}^{2}}{C^{2}r_{min}^{2}}+\frac{r_{min}^{2}}{\ell^{2}}\end{split}$
(39)

$\begin{split}\frac{\partial h_{min}}{\partial
r_{min}}=-\frac{2\ell^{2}\hat{Q}^{2}}{C^{2}r_{min}^{3}}+\frac{2\ell^{2}M}{Cr_{min}^{2}}+\frac{2r_{min}}{\ell^{2}}=0\end{split}$
(40)

$\begin{split}\frac{\partial^{2}h_{min}}{\partial
r_{min}^{2}}=\frac{6\ell^{2}\hat{Q}^{2}}{C^{2}r_{min}^{4}}-\frac{4\ell^{2}M}{Cr_{min}^{3}}+\frac{2}{\ell^{2}}>0.\end{split}$
(41)

Therefore, by considering the fluxes that change the mass and electric charge
of the black hole in infinitesimally small time changes $dt$, which lead to
the change of position, the minimum location as $r_{min}+dr_{min}$ we have,

$\begin{split}h(M+dM,\hat{Q}+d\hat{Q},C+dC,r_{min}+dr_{min})=h_{min}+\frac{\partial
h_{min}}{\partial M}dM+\frac{\partial
h_{min}}{\partial\hat{Q}}d\hat{Q}+\frac{\partial h_{min}}{\partial
C}dC+\frac{\partial h_{min}}{\partial r_{min}}dr_{min}\end{split}$ (42)

where

$\begin{split}&h_{min}=\delta\leq 0,\qquad\frac{\partial h_{min}}{\partial
M}=-\frac{2\ell^{2}}{Cr_{min}},\qquad\frac{\partial
h_{min}}{\partial\hat{Q}}=\frac{2\ell^{2}\hat{Q}}{C^{2}r_{min}^{2}},\\\
&\frac{\partial h_{min}}{\partial
C}=\frac{1}{C}\bigg{(}\frac{2\ell^{2}M}{Cr_{min}}-\frac{2\ell^{2}\hat{Q}^{2}}{C^{2}r_{min}^{2}}\bigg{)},\qquad\frac{\partial
h_{min}}{\partial
r_{min}}=\frac{C^{2}r_{h}r_{min}\left(\ell^{2}r_{h}+r_{h}^{3}+2r_{min}^{3}\right)-\ell^{4}\hat{Q}^{2}\left(2r_{h}-r_{min}\right)}{C^{2}\ell^{2}r_{h}r_{min}^{3}}=0\end{split}$
(43)

By putting relations (28), (29), (30) and (34) in relation (42), we get,

$\begin{split}&h(M+dM,\hat{Q}+d\hat{Q},C+dC,r_{min}+dr_{min})=\\\
&-\frac{2\ell^{2}r_{h}(\omega-\hat{q}\hat{\phi}_{h})\big{[}r_{h}(\omega-\hat{q}\hat{\phi}_{min})-r_{min}(\omega-\hat{q}\hat{\phi}_{h})\big{]}}{Cr_{min}}dt+\frac{\left(r_{h}-r_{min}\right)\left[C^{2}r_{h}r_{min}\left(\ell^{2}+r_{h}^{2}\right)-\ell^{4}\hat{Q}^{2}\right]}{C^{3}\ell^{2}r_{h}r_{min}^{2}}dC\end{split}$
(44)

First, we examine the black hole in the extremal state $r_{min}=r_{h}$ and
$h_{min}=\delta=0$, in this case, the above equation is rewritten as follows,

$\begin{split}h(M+dM,\hat{Q}+d\hat{Q},C+dC,r_{min}+dr_{min})=0\end{split}$
(45)

Thus, despite the scattering, the black hole remains in an extremal state, and
the WCCC is valid. Also, the result obtained in the RPS is the same as the EPS
and the normal space. To study the near-extremal black hole, we analyze the
event horizon close to the minimum point of $f(r_{min})$, i.e. at
$r_{h}=r_{min}+\epsilon$ and $0<\epsilon<<1$. In this scenario, equation (44)
can be expressed as follows,

$\begin{split}&h(M+dM,\hat{Q}+d\hat{Q},C+dC,r_{min}+dr_{min})=\delta-\epsilon\left[\frac{2\ell^{2}(\omega-\hat{q}\hat{\phi}_{h})(\omega-2\hat{q}\hat{\phi}_{h})}{C}+\mathcal{O}(\epsilon)\right]dt\\\
&+\epsilon\left[\frac{C^{2}r_{h}^{2}\left(\ell^{2}+r_{h}^{2}\right)-\ell^{4}\hat{Q}^{2}}{C^{3}\ell^{2}r_{h}^{3}}+\mathcal{O}(\epsilon)\right]dC\end{split}$
(46)

We also have,

$\begin{split}h_{min}=\delta=-\epsilon\bigg{[}\frac{C^{2}\ell^{2}r_{h}^{2}+3C^{2}r_{h}^{4}-\ell^{4}\hat{Q}^{2}}{C^{2}\ell^{2}r_{h}^{3}}+\mathcal{O}(\epsilon)\bigg{]}\end{split}$
(47)

Therefore, for small time changes that lead to small charge center changes,
i.e. $dt\sim\epsilon$ and $dC\sim\epsilon$, equation (46) is obtained as
follows,

$\begin{split}h(M+dM,\hat{Q}+d\hat{Q},C+dC,r_{min}+dr_{min})=\delta\pm\mathcal{O}(\epsilon^{2})<0\end{split}$
(48)

Hence, the adjustment to the minimum value can be considered negligible in the
initial order of d. Consequently, it can be inferred that there is no
alteration to the condition of the black hole: a nearly extremal black hole
remains nearly extremal even with variations in mass and electric charge.
Thus, the WCCC is valid for near-extremal black holes as well. However,
according to relations (46) and (47), we find that, in the condition that
$dC>>1$, there is a possibility that the conjecture of weak cosmic censorship
is violated. Also, when there are no changes in the central charge, the result
obtained is in agreement with the result obtained in the extended phase space
[36].

##  6 WGC with equivalence of mass and energy

Here we use the equivalence of mass and energy, and we have (with $\hbar=1$
and $c=1$),

$\begin{split}E=\mu_{s},\qquad
E=\omega\qquad\Rightarrow\qquad\mu_{s}=\omega\end{split}$ (49)

Also, according to relation (7), we can have,

$q\Phi_{h}=\hat{q}\hat{\Phi}_{h}$

Now, considering the second order of $\epsilon$, we get some interesting
results which is shown in Table 1.

|  fluxes |  WGC |  State
---|---|---|---
$\mu_{s}<q\Phi_{h}$ $\rightarrow$ $\frac{1}{\Phi_{h}}<\frac{q}{\mu_{s}}$ |  Superradiance |  $C>>\hat{Q}$ |  $N-E\rightarrow N-E$
$q\Phi_{h}<\mu_{s}<2q\Phi_{h}$ $\rightarrow$ $\frac{1}{2\Phi_{h}}<\frac{q}{\mu_{s}}<\frac{1}{\Phi_{h}}$ |  Absorption |  $C>>\hat{Q}$ , $r_{min}=\epsilon$ |  $N-E\rightarrow S-E$
$\mu_{s}>2q\Phi_{h}$ |  Absorption |  $\times$ |  $N-E\rightarrow N-E$
Table 1: near-extremal (N- E) , slightly extremal (S- E)

Using the equivalence of mass and energy and relation (46), we find that the
black hole emits particles in its radiation that obey the weak gravity
conjecture, which nevertheless holds the weak cosmic censorship conjecture, so
that the black hole from The extremity becomes more distant. But in the case
where the black hole attracts the particle, two situations occur, in the first
case, the black hole attracts the particle that follows the WGC. In this case,
the WCCC is valid, with the difference that the black hole is closer to the
extremity state. In the second case, the black hole attracts a particle that
does not follow the WGC, the WCCC will be valid, so that the black hole moves
away from the extremity state. Next, we obtain the electric potential of the
black hole near the extreme state. Given $r_{h}=r_{min}+\epsilon$, the
electric potential can be expressed as,

$\begin{split}\Phi_{h}=\frac{Q}{r_{h}}=\frac{Q}{r_{min}+\epsilon}=\frac{\hat{Q}}{r_{min}\sqrt{C}}(1-\frac{\epsilon}{r_{min}})\end{split}$
(50)

Also, by using $h(r)=0,T=\frac{h^{\prime}(r)}{4\pi}$ and 11, we can obtain
$r_{min}$ as follows,

$\begin{split}r_{min}=\frac{\sqrt{G}}{\sqrt{6}}\bigg{(}\sqrt{C^{2}+12\hat{Q}^{2}}-C\bigg{)}^{\frac{1}{2}}\end{split}$
(51)

By placing (50) and (51) into Superradiance conditions (
$\frac{1}{\Phi_{h}}<\frac{q}{\mu_{s}}$), we have,

$\begin{split}\frac{q}{\mu_{s}}>\sqrt{G}\big{(}1+\frac{\epsilon}{r_{min}}\big{)}\frac{\sqrt{C}\big{(}\sqrt{C^{2}+12\hat{Q}^{2}}-C\big{)}^{\frac{1}{2}}}{\sqrt{6}\hat{Q}}\end{split}$
(52)

According to relation (52), in the condition that $C>>\hat{Q}$, we have
$\frac{q}{\mu_{s}}>\sqrt{G}\big{(}1+\frac{\epsilon}{r_{min}}\big{)}$, in which
case the particles emitted from the black hole follow the WGC. But in the case
where the black hole absorbs the particle, even when $C>>\hat{Q}$, we have,

$\begin{split}\frac{\sqrt{G}}{2}\big{(}1+\frac{\epsilon}{r_{min}}\big{)}<\frac{q}{\mu_{s}}<\sqrt{G}\big{(}1+\frac{\epsilon}{r_{min}}\big{)}\end{split}$
(53)

Therefore, the WGC does not hold in this case. The black hole attracts
particles that do not obey the WGC and repels particles that obey the WGC. In
both cases the WCCC holds. Also, according to relation (53), we find that only
if the black hole has a very small event horizon and is in an extreme state
$(r_{min}=\epsilon)$ can absorb particles that follow the WGC. Therefore,
these small black holes get closer to their extreme limit, which increases the
probability of the black hole collapsing.

##  7 Conclusions

We employed the restricted phase space(RPS) thermodynamics and investigated
the weak cosmic censorship conjecture (WCCC). Also in this paper, we used the
equivalence of mass and energy principle and discussed the relationship
between two important conjectures as WCCC which is derived by general
relativity and WGC is derived from string theory. Here, we assumed that the
internal charge and energy of the black hole change with the superradiance and
absorption of the particle (or scalar field) in an infinitesimal time
interval. By using the reproduced first law of thermodynamics, we have shown
that the second law holds in conditions where the temperature has a lower
limit as $(T\geq\frac{1}{2\pi
r_{h}}\bigg{(}1+\frac{r_{h}^{2}}{\ell^{2}}\bigg{)})$. But the second law of
thermodynamics can be violated in conditions where the temperature has an
upper limit and the black hole is in equilibrium
$\omega=\hat{q}\hat{\Phi_{h}}$. Also, here we point out that when the central
charge does not change, the obtained results are the same as the results
obtained in normal space. In addition, we showed using the RPS thermodynamics
that the WCCC holds despite the input and output flux in the extremal state
and close to the extremal state. Therefore, the result obtained in the RPS is
consistent with the results obtained in the normal space and the extended
phase space(EPS). By applying the equivalence mass and energy principle for
the particle and second-order approximation in the state close to the extreme
black hole, we find that when we have superradiance , particles are emitted
that obey the WGC, of course, with the condition that the $C>>\hat{Q}$. In
this case, the black hole will be further away from its extreme state. Also, a
black hole will only attract a particle that obeys the WGC if the black hole
is very small and the central charge is larger than the scaled electric charge
$C>>\hat{Q}$. In this case, the black hole becomes closer to its extreme
state.

##  8 Acknowledgments

The authors would like to express their gratitude to Saeed Noori Gashti for
his insightful discussions.

## References

  * [1] Ooguri, Hirosi, and Cumrun Vafa. ”On the Geometry of the String Landscape and the Swampland.” Nuclear physics B 766.1-3 (2007): 21-33.
  * [2] Palti, Eran. ”The swampland: introduction and review.” Fortschritte der Physik 67.6 (2019): 1900037.
  * [3] van Beest, Marieke, et al. ”Lectures on the swampland program in string compactifications.” Physics Reports 989 (2022): 1-50.
  * [4] Arkani-Hamed, Nima, et al. ”The string landscape, black holes and gravity as the weakest force.” Journal of High Energy Physics 2007.06 (2007): 060.
  * [5] Harlow, Daniel, et al. ”Weak gravity conjecture.” Reviews of Modern Physics 95.3 (2023): 035003.
  * [6] Schöneberg, Nils, et al. ”News from the Swampland—constraining string theory with astrophysics and cosmology.” Journal of Cosmology and Astroparticle Physics 2023.10 (2023): 039.
  * [7] Kinney, William H. ”Eternal inflation and the refined swampland conjecture.” Physical review letters 122.8 (2019): 081302.
  * [8] Geng, Hao, Sebastian Grieninger, and Andreas Karch. ”Entropy, entanglement and swampland bounds in DS/dS.” Journal of High Energy Physics 2019.6 (2019): 1-16.
  * [9] JSadeghi, J., et al. ”The investigation of universal relation between corrections to entropy and extremality bounds with verification WGC.” Physics of the Dark Universe 30 (2020) : (100626).
  * [10] Sadeghi, J., et al. ”de Sitter Swampland Conjecture in String Field Inflation.” The European Physical Journal C 83 (2023) : (635).
  * [11] Oikonomou, V. K. ”Rescaled Einstein-Hilbert gravity from f (R) gravity: Inflation, dark energy, and the swampland criteria.” Physical Review D 103.12 (2021): 124028.
  * [12] Sadeghi, J., S. Noori Gashti, and F. Darabi. ”Swampland conjectures in hybrid metric-Palatini gravity.” Physics of the Dark Universe (2022): 101090.
  * [13] Crisford, Toby, Gary T. Horowitz, and Jorge E. Santos. ”Testing the weak gravity-cosmic censorship connection.” Physical Review D 97.6 (2018): 066005.
  * [14] Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. ”Strong cosmic censorship in light of weak gravity conjecture for charged black holes.” Journal of High Energy Physics 2023.2 (2023): 1-14.
  * [15] Das, Suratna. ”Distance, de Sitter and Trans-Planckian Censorship conjectures: the status quo of Warm Inflation.” Physics of the Dark Universe 27 (2020): 100432.
  * [16] Sadeghi, J., et al. ”de Sitter Swampland Conjecture in String Field Inflation.” The European Physical Journal C 83 (2023): (635).
  * [17] Yuennan, Jureeporn, and Phongpichit Channuie. ”Further Refining Swampland Conjecture on Inflation in General Scalar‐Tensor Theories of Gravity.” Fortschritte der Physik 70.6 (2022): 2200024.
  * [18] Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. ”Scalar Weak Gravity Conjecture in Super Yang-Mills Inflationary Model.” Universe 8.12 (2022): 621.
  * [19] Kallosh, Renata, et al. ”dS Vacua and the Swampland.” Journal of High Energy Physics 2019.3 (2019): 1-18.
  * [20] Sadeghi, J., and S. Noori Gashti. ”Anisotropic constant-roll inflation with noncommutative model and swampland conjectures.” The European Physical Journal C 81 (2021): 1-10.
  * [21] Gashti, S. Noori, J. Sadeghi, and B. Pourhassan. ”Pleasant behavior of swampland conjectures in the face of specific inflationary models.” Astroparticle Physics 139 (2022): 102703.
  * [22] Kinney, William H., Sunny Vagnozzi, and Luca Visinelli. ”The zoo plot meets the swampland: mutual (in) consistency of single-field inflation, string conjectures, and cosmological data.” Classical and quantum gravity 36.11 (2019): 117001.
  * [23] Sadeghi, J., S. Noori Gashti, and E. Naghd Mezerji. ”The investigation of universal relation between corrections to entropy and extremality bounds with verification WGC.” Physics of the Dark Universe 30 (2020): 100626.
  * [24] Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. ”Strong cosmic censorship in light of weak gravity conjecture for charged black holes.” Journal of High Energy Physics 2023.2 (2023): 1-14.
  * [25] Penrose, Roger. ”Gravitational collapse: the role of general relativity.” Riv. Nuovo Cim. 1 (1969): 252.
  * [26] Penrose, Roger. ”Gravitational collapse: the role of general relativity.” Gen. Relativ. Gravit. 34 (2002): 1141.
  * [27] Hod, Shahar. ”Weak cosmic censorship: as strong as ever.” Physical review letters 100.12 (2008): 121101.
  * [28] Yang, Si-Jiang, et al. ”Weak cosmic censorship conjecture for a Kerr-Taub-NUT black hole with a test scalar field and particle.” Physical Review D 101.6 (2020): 064048.
  * [29] Barausse, Enrico, Vitor Cardoso, and Gaurav Khanna. ”Test bodies and naked singularities: Is the self-force the cosmic censor?.” Physical Review Letters 105.26 (2010): 261102.
  * [30] Bai, Tianhao, et al. ”Weak cosmic censorship conjecture in the nonlinear electrodynamics black hole under the charged scalar field.” Commun. Theor. Phys. 72, (2020) 015401.
  * [31] Gwak, Bogeun. ”Weak cosmic censorship conjecture in Kerr-(anti-) de Sitter black hole with scalar field.” Journal of High Energy Physics 2018.9 (2018): 1-22.
  * [32] Tóth, Gábor Zsolt. ”Test of the weak cosmic censorship conjecture with a charged scalar field and dyonic Kerr–Newman black holes.” General Relativity and Gravitation 44 (2012): 2019-2035.
  * [33] Hong, Wei, Benrong Mu, and Jun Tao. ”Thermodynamics and weak cosmic censorship conjecture in the charged RN-AdS black hole surrounded by quintessence under the scalar field.” Nuclear Physics B 949 (2019): 114826.
  * [34] Chen, Deyou, Wei Yang, and Xiaoxiong Zeng. ”Thermodynamics and weak cosmic censorship conjecture in Reissner-Nordström anti-de Sitter black holes with scalar field.” Nuclear Physics B 946 (2019): 114722.
  * [35] Sorce, Jonathan, and Robert M. Wald. ”Gedanken experiments to destroy a black hole. II. Kerr-Newman black holes cannot be overcharged or overspun.” Physical Review D 96.10 (2017): 104014.
  * [36] Gwak, Bogeun. ”Weak cosmic censorship with pressure and volume in charged anti-de Sitter black hole under charged scalar field.” Journal of Cosmology and Astroparticle Physics 2019.08 (2019): 016.
  * [37] Han, Yi-Wen, Xiao-Xiong Zeng, and Yun Hong. ”Thermodynamics and weak cosmic censorship conjecture of the torus-like black hole.” The European Physical Journal C 79.3 (2019): 1-6.
  * [38] He, Yan-Li, and Jie Jiang. ”Weak cosmic censorship conjecture in Einstein-Born-Infeld black holes.” Physical Review D 100.12 (2019): 124060.
  * [39] He, Ke-Jian, Xin-Yun Hu, and Xiao-Xiong Zeng. ”Weak cosmic censorship conjecture and thermodynamics in quintessence AdS black hole under charged particle absorption.” Chinese Physics C 43.12 (2019): 125101.
  * [40] Kubizňák, David, and Robert B. Mann. ”P - V criticality of charged AdS black holes.” Journal of High Energy Physics 2012.7 (2012): 1-25.
  * [41] Caceres, Elena, Phuc H. Nguyen, and Juan F. Pedraza. ”Holographic entanglement entropy and the extended phase structure of STU black holes.” Journal of High Energy Physics 2015.9 (2015): 1-33.
  * [42] Dolan, Brian P. ”Pressure and volume in the first law of black hole thermodynamics.” Classical and Quantum Gravity 28.23 (2011): 235017.
  * [43] Visser, Manus R. ”Holographic thermodynamics requires a chemical potential for color.” Physical Review D 105.10 (2022): 106014.
  * [44] Gao, Zeyuan, and Liu Zhao. ”Restricted phase space thermodynamics for AdS black holes via holography.” Classical and Quantum Gravity 39.7 (2022): 075019.
  * [45] Gao, Zeyuan, Xiangqing Kong, and Liu Zhao. ”Thermodynamics of Kerr-AdS black holes in the restricted phase space.” The European Physical Journal C 82.2 (2022): 112.
  * [46] Sadeghi, Jafar, et al. ”RPS thermodynamics of Taub–NUT AdS black holes in the presence of central charge and the weak gravity conjecture.” General Relativity and Gravitation 54.10 (2022): 129.
  * [47] Kong, Xiangqing, Zhiqiang Zhang, and Liu Zhao. ”Restricted phase space thermodynamics of charged AdS black holes in conformal gravity.” Chinese Physics C 47.9 (2023): 095105.
  * [48] Ladghami, Yahya, et al. ”Barrow entropy and AdS black holes in RPS thermodynamics.” Physics of the Dark Universe 44 (2024): 101470.
  * [49] Sadeghi, Jafar, et al. ”Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives.” Annals of Physics 460 (2024): 169569.
  * [50] Ladghami, Y., et al. ”4D-EGB black holes in RPS thermodynamics.” Physics of the Dark Universe 41 (2023): 101261.
  * [51] Sadeghi, Jafar, et al. ”Bulk-boundary and RPS Thermodynamics from Topology perspective.” Chinese Physics C (2023).
  * [52] Sadeghi, Jafar, et al. ”Weak Cosmic Censorship and Weak Gravity Conjectures in CFT Thermodynamics.” arXiv preprint arXiv:2404.15981 (2024).
  * [53] Sadeghi, Jafar, and Saeed Noori Gashti. ”Reissner-Nordström black holes surrounded by perfect fluid dark matter: testing the viability of weak gravity conjecture and weak cosmic censorship conjecture simultaneously.” Physics Letters B (2024): 138651.
  * [54] Alipour, Mohammad Reza, Jafar Sadeghi, and Mehdi Shokri. ”WGC and WCCC of black holes with quintessence and cloud strings in RPS space.” Nuclear Physics B 990 (2023): 116184.
  * [55] Alipour, Mohammad Reza, Jafar Sadeghi, and Mehdi Shokri. ”WGC and WCC for charged black holes with quintessence and cloud of strings.” The European Physical Journal C 83.7 (2023): 1-7.
  * [56] Mezerji, E. Naghd, and J. Sadeghi. ”The correlation of WGC and hydrodynamics bound with R4 correction in the charged AdSd+ 2 black brane.” Nuclear Physics B 981 (2022): 115858.
  * [57] Dolan, Brian P., et al. ”Thermodynamic volumes and isoperimetric inequalities for de Sitter black holes.” Physical Review D 87.10 (2013): 104017.
  * [58] Liang, Jing, et al. ”Remarks on the weak cosmic censorship conjecture of RN-AdS black holes with cloud of strings and quintessence under the scalar field.” Nuclear Physics B 965 (2021): 115335.
  * [59] Sadeghi, Jafar, et al. ”Weak gravity conjecture from conformal field theory: a challenge from hyperscaling violating and Kerr-Newman-AdS black holes.” Chinese Physics C 47.1 (2023): 015103.