Reissner-Nordstr$\ddot{o}$m black holes surrounded by perfect fluid dark
matter: testing the viability of weak gravity conjecture and weak cosmic
censorship conjecture simultaneously
Jafar Sadeghia111Email<EMAIL_ADDRESS>Saeed Noori Gashtia222Email:
<EMAIL_ADDRESS>
aDepartment of Physics, Faculty of Basic Sciences, University of Mazandaran
P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

A possible violation of the weak gravity conjecture (WGC) by cosmic censorship
is one of the major challenges in the field of general relativity. The cosmic
censorship is a fundamental principle that ensures the consistency of the
theory of gravity. According to this principle, a charged black hole in four
dimensions cannot violate the weak cosmic censorship conjecture (WCCC) under
normal conditions. However, in this paper, we explore the possibility of
reconciling the WGC and the WCCC by considering Reissner-Nordstr$\ddot{o}$m
(R-N) black holes embedded in perfect fluid dark matter (PFDM) in
asymptotically flat spacetimes. These two conjectures are seemingly unrelated,
but a recent proposal suggested that they are connected surprisingly. In
particular, We argue a promising class of valid counterexamples to the WCCC in
the four-dimensional Einstein-Maxwell theory, considering a charged black hole
when WGC is present. We demonstrate that by imposing certain constraints on
the parameters of the metric, the WGC and the WCCC can be compatible.
Furthermore, we investigate the properties of the charged black hole in the
presence of PFDM for $Q>M$ and present some intriguing figures to test the
validity of the WGC and the WCCC simultaneously. When PFDM is absent
($\gamma=0$), the RN black hole either has two event horizons if
$Q^{2}/M^{2}\leq 1$ or none if $Q^{2}/M^{2}>1$. The second scenario results in
a naked singularity, which contradicts the WCCC. But when PFDM is present
($\gamma\neq 0$), the RN black hole has event horizons with regard to Q and M.
This implies that the singularity is always covered, and the WGC and the WCCC
are fulfilled. Furthermore, we demonstrate that there is a critical value of
$\gamma$, called $\gamma_{ext}$, that makes the RN black hole extremal when
$\gamma=\gamma_{ext}$. In this situation, the black hole has an event horizon,
and the WGC and the WCCC are still fulfilled. We infer that PFDM can make the
WGC and the WCCC compatible with the RN black hole and that the WGC and the
WCCC agree with each other when PFDM is present.

Keywords: Reissner-Nordstr$\ddot{o}$m black hole, Perfect fluid dark matter,
Weak cosmic censorship conjecture, Weak gravity conjecture,

###### Contents

  1. 1 Introduction
  2. 2 The Model
  3. 3 Mild WGC $\&$ WCCC
  4. 4 Discussion and result
  5. 5 Conclusions
  6. 6 Acknowledgments

##  1 Introduction

Quantum gravity is a fascinating topic that has attracted the attention of
many researchers from different perspectives. The swampland program is a
research effort to find the universal principles that any consistent theory of
quantum gravity must obey. It is based on the idea that not all low-energy
effective field theories can be embedded in a quantum theory of gravity, such
as string theory. The swampland program tries to identify the criteria that
distinguish the theories that can be part of the string landscape from those
that belong to the swampland. The motivation for the swampland program comes
from various sources, such as black hole physics, AdS/CFT correspondence, and
string theory constructions. The swampland program aims to shed light on the
nature of quantum gravity and its implications for cosmology and particle
physics. The AdS/CFT correspondence is a duality that relates a gravitational
theory in anti-de Sitter (AdS) space to a conformal field theory (CFT) on its
boundary. It is a powerful tool to study quantum gravity and its holographic
nature. The swampland program is an attempt to find the universal constraints
that any consistent theory of quantum gravity must satisfy. One of the sources
of motivation for the swampland program is the AdS/CFT correspondence, as it
provides concrete examples of quantum gravity theories and their low-energy
limits. The AdS/CFT correspondence can also be used to test some of the
swampland conjectures, such as the bound on the number of massless modes, the
weak gravity conjecture, and the distance conjecture[1, 2, 3, 4, 5, 6].

A key criterion in the swampland program is the absence of global symmetries
in quantum gravity, while gauge symmetries are allowed [1, 2, 3, 4, 5, 6].
This criterion leads to the weak gravity conjecture (WGC), which states that
there must exist particles whose charge-to-mass ratio is greater than one,
i.e. $q/m>1$, in any quantum theory of gravity [4, 5, 6]. In this way, gravity
is the weakest force among all the interactions. The WGC is one of the several
conjectures in the swampland program that help to identify effective field
theories that are consistent with quantum gravity. For more details, see[1, 2,
3, 4, 5, 6, 7, 8]. Also for further study about the swampland program in
various cosmological concepts such as the physics of black holes.
thermodynamics, black brane, cosmological inflation, etc, you can see[9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69]. Another important concept in theoretical physics is the weak cosmic
censorship conjecture (WCCC), which was proposed by Penrose to avoid the
paradoxes caused by the existence of singularities in general relativity[70].
The WCCC asserts that the singularities formed by gravitational collapse are
always hidden behind event horizons, thus preserving the causal structure and
predictability of the theory. However, the WGC and the WCCC seem to be
incompatible with each other in the case of the Reissner–Nordström (RN) black
hole, which is a solution of the Einstein–Maxwell equations describing a
charged black hole. When $Q>M$, where Q is the charge and M is the mass of the
black hole, the RN solution violates the WCCC, as the singularity becomes
naked and visible to distant observers. On the other hand, when the RN black
hole decays to the extremal state, where Q = M, the energy conservation
implies that there are decay products whose charge-to-mass ratio is greater
than one. In this case, the WGC is satisfied, but the WCCC is violated, as
these products cannot be black holes, but rather particles[71]. This is one of
the main challenges for the WGC.

Recently, researchers have studied WCCC for the RN-dS black hole with PFDM.
They demonstrated that the presence of dark matter and a cosmological constant
prevents the overcharging of this black hole, a scenario not possible in a
vacuum. The only condition under which overcharging could occur is if there is
an exact equilibrium between the influences of dark matter and the
cosmological constant. Moreover, there is a specific limit where the influence
of the cosmological constant becomes dominant, ensuring the black hole cannot
be overcharged and thus, the WCCC is maintained. This conclusion is consistent
across both linear and non-linear accretion scenarios, indicating that these
cosmic factors act as safeguards against the overcharging of black holes[72].
In this paper, we address this challenge by considering the effects of perfect
fluid dark matter (PFDM) on the charged black holes. PFDM is a model of dark
matter that behaves as a perfect fluid with a constant equation of state[73,
78, 79, 74, 75, 76, 77]. We study the properties of the RN black holes
surrounded by PFDM in asymptotically flat spacetimes. We show that by imposing
certain constraints on the parameters of the metric, such as the mass, the
charge, and the PFDM density, we can achieve compatibility between the Mild
WGC and the WCCC. We also investigate the case of $Q>M$ and present some
interesting points to test the validity of the Mild WGC and the WCCC
simultaneously. Our paper is motivated by the recent observational evidence of
the accelerated expansion of the universe[78, 79], which suggests the
existence of a repulsive gravitational force due to a negative pressure on
cosmological scales. One possible explanation for this phenomenon is the
presence of a cosmological constant, which corresponds to the vacuum energy in
Einstein’s equations. However, the cosmological constant also affects the
properties of the black holes and their compatibility with the Mild WGC and
the WCCC. Therefore, we explore how the charged black holes with the PFDM
model can account for the consistency of the quantum gravity due to consistent
with Mild WGC and its effect on the validation of the Mild WGC and WCCC. On
the other hand, The motivation of the article is to explore the possibility of
resolving a theoretical conflict between two conjectures (Mild WGC and the
WCCC simultaneously), with respect to the effects of PFDM that has important
implications for quantum gravity and black hole physics. This is a challenging
and interesting research topic that could shed light on some fundamental
aspects of nature.

The paper is organized as follows. In section 2, we review the basic features
of the RN black hole surrounded by PFDM and the effects of the Mild WGC and
the WCCC. In section 3, we study the consistency between Mild WGC and WCCC for
the mentioned model viz the RN black hole surrounded by PFDM. In section 4, we
analyze the properties of the parameters and the constraints on them and
present some interesting figures to test the validity of the Mild WGC and the
WCCC simultaneously. In section 5, we summarize our results and conclude with
some remarks.

##  2 The Model

Reissner-Nordström black holes surrounded by perfect fluid dark matter are
electrically charged black holes that are influenced by the presence of a
hypothetical form of matter that accounts for most of the mass in the
universe. They have interesting thermodynamic and phase transition properties
that can be studied by extending the system’s phase space. Some researchers
have also tested the weak cosmic censorship conjecture for these black holes,
which states that the singularity inside a black hole cannot be exposed to an
external observer. The following action describes the gravitational theory
that is minimally coupled with a gauge field in the presence of perfect fluid
dark matter (PFDM)[80, 81, 82, 83],

$S=\int d^{4}x\sqrt{-g}\bigg{[}\frac{1}{16\pi
G}R+\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\mathcal{L}_{DM}\bigg{]},$ (1)

where $g$ as the determinant of the metric tensor $[g=det_{(}g_{ab})]$, $R$ as
the scalar curvature, $G$ as the gravitational constant, and $F_{\mu\nu}$ as
the electromagnetic tensor derived from the gauge potential $A_{\mu}$.
$L_{DM}$ is the Lagrangian density that describes the PFDM. By applying the
principle of least action, one can derive the Einstein field equations as
follows,

$\begin{split}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi
G(T^{M}_{\mu\nu}-T^{DM}_{\mu\nu})\equiv-8\pi GT_{\mu\nu},\end{split}$ (2)
$\begin{split}&F^{\mu\nu}_{\nu}=0,\\\
&F^{\mu\nu;\alpha}+F^{\nu\alpha;\mu}+F^{\alpha\mu;\nu}=0.\end{split}$ (3)

We denote $T^{M}_{\mu\nu}$ as the energy-momentum tensor of the ordinary
matter and $T^{DM}_{\mu\nu}$ as the energy-momentum tensor of the PFDM, as
defined in[80, 81, 82, 83]. So,

$\begin{split}&T^{\mu}_{\nu}=g^{\mu\nu}T_{\mu\nu},\\\ &T^{t}_{t}=-\rho,\hskip
14.22636ptT^{r}_{r}=T^{\theta}_{\theta}=T^{\phi}_{\phi}=P.\end{split}$ (4)

where one can assume that
$T^{r}_{r}=T^{\theta}_{\theta}=T^{\phi}_{\phi}=T^{t}_{t}(1-\delta)$, where
$\delta$ is a constant, based on[80, 81, 82, 83]. The expression for the
Reissner-Nordström metric in the context of PFDM is given below,

$\begin{split}ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\end{split}$ (5)

where,

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{\gamma}{r}\ln(\frac{r}{\gamma}).$
(6)

The parameters M, Q, and $\gamma$ represent the mass, charge, and PFDM
contribution of the black hole, respectively. If the PFDM is absent ($\gamma$
= 0), the above space-time metric reduces to the Reissner-Nordström black
hole. The values of $\gamma$ can be both positive and negative and here, we
consider both of them in this work.

##  3 Mild WGC $\&$ WCCC

We consider the metric of a R-N black hole surrounded by PFDM in asymptotic
flatness. By solving $f(r)=0$, and using the substitution $x=1/r$, we can find
the location of the event horizon for the black hole. Generally, we can find
the location of the event horizons for a charged black hole by solving f(r)=0
and get $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$, where M is the mass and Q is the
charge of the black hole. If $Q>M$, then the black hole does not have any
event horizon, and its singularity is exposed to the outside observers. This
is called a naked singularity, and it violates the weak cosmic censorship
conjecture (WCCC), which states that singularities should be hidden behind
event horizons. So, we rewrite the equation (6) as follows,

$1-2Mx+Q^{2}x^{2}=-\gamma x\ln(\frac{1}{x\gamma})$ (7)

Eq. (7) is a difficult equation to solve. We use some plots and analyze
various scenarios to find their solutions. We start with the case of
asymptotically flat space, where $\gamma<0$ or $\gamma>0$. Based on Eq. (7),
we sketch two curves $\Upsilon_{L}=1-2Mx+Q^{2}x^{2}$ and $\Upsilon_{R}=-\gamma
x\ln(\frac{1}{x\gamma})$ and label them as Fig. 1. We consider two situations:
$Q^{2}/M^{2}\leq 1$ and $Q^{2}/M^{2}>1$. In Fig. 1c, when $\Upsilon_{R}=0$,
the curve $\Upsilon_{L}$ crosses the $x$ axis at two points $(x_{+},x_{-})$.
These are the inner and outer event horizons of the standard RN black hole. We
use Eq. (7) and the condition that the curves of $\Upsilon_{R}$ and
$\Upsilon_{L}$ are tangent to each other at the point $x_{0}$ to find the
values of $x_{0}$ and $\gamma_{ext}$. This means that the slopes of the two
curves are equal at $x_{0}$, and we can use this to solve for $\gamma_{ext}$.
Then, we can calculate the value of $\gamma_{ext}$ and $x_{0}$ with the
following expressions,

$1-2Mx_{0}+Q^{2}x_{0}^{2}=-\gamma x_{0}\ln(\frac{1}{x_{0}\gamma}),$ (8)

and

$-\gamma-2M+2Q^{2}x_{0}=-\gamma\ln(\frac{1}{x_{0}\gamma}).$ (9)

Using the equations (8) and (9), we will have,

$x_{0}=\frac{\gamma\pm\sqrt{\gamma^{2}+4Q^{2}}}{2Q^{2}}$ (10)

Also, we have the $\gamma_{ext}$ by using the expansion of the logarithm up to
two terms and a series of simplifications,

$\gamma_{ext}=\frac{\bigg{[}2Q^{2}x_{0}^{3}\pm
2\sqrt{x_{0}^{2}\bigg{(}\big{(}-2Mx_{0}+Q^{2}x_{0}^{2}+3\big{)}{}^{2}-3\bigg{)}}\bigg{]}+2x_{0}\big{(}3-2Mx_{0}\big{)}}{6x_{0}^{2}}$
(11)

We can obtain the extremal state of the black hole by keeping (Q, M, $\gamma$)
constant and increasing $Q^{2}/M^{2}$ until it reaches the value of
$(Q^{2}/M^{2})_{ext}$. This is the condition for the black hole to have a
single event horizon. We can also use some approximation for the RN extremal
black hole with PFDM, by fixing $\gamma$, Q, M and using Eq. (6) and (7). This
gives us the relation for the extremal black hole with PFDM, which is,

$(\frac{M^{2}}{Q^{2}})_{ext}=\frac{Q^{2}\bigg{(}\frac{6\gamma\big{(}\gamma\pm\sqrt{\gamma^{2}+4Q^{2}}\big{)}}{Q^{2}}+\frac{\gamma\big{(}\gamma\pm\sqrt{\gamma^{2}+4Q^{2}}\big{)}^{3}}{4Q^{4}}-\frac{3\gamma^{2}\big{(}\gamma\pm\sqrt{\gamma^{2}+4Q^{2}}\big{)}^{2}}{4Q^{4}}-1\bigg{)}}{\gamma\bigg{(}\gamma\pm\sqrt{\gamma^{2}+4Q^{2}}\bigg{)}^{2}}$
(12)

With a little simplification and a series of straightforward calculations, the
relation $(\frac{Q^{2}}{M^{2}})_{ext}$ is rewritten in the following form,

$(\frac{Q^{2}}{M^{2}})_{ext_{1,2}}=\pm
4\gamma(1\mp\frac{\gamma^{2}}{Q^{2}}+...)$ (13)

So, finally for the second term (the negative part behind the parentheses)
with $\gamma\leq-1/4$ and $Q>0$, we will have,

$(\frac{Q^{2}}{M^{2}})_{ext}=1+\varepsilon,$ (14)

where $\varepsilon$ is a constant positive value. From the above concepts, we
want to investigate whether the weak gravity conjecture (WGC) and the weak
cosmic censorship conjecture (WCCC) are satisfied for the Reissner-Nordström
(RN) black hole with (PFDM) in both normal and extremal states.

##  4 Discussion and result

We examine the graph of $\Upsilon_{L}$ and find that it has a minimum value of
$(Q^{2}-M^{2})/Q^{2}$ at the point $x_{min}=M/Q^{2}$. We also look at Fig. 1c,
where we plot $\Upsilon_{R}$ and $\Upsilon_{L}$ for the case of
$Q^{2}/M^{2}\leq 1$. We observe that the two curves intersect at two points
$(x_{1},x_{2})$. This implies that the RN black hole with PFDM has an outer
event horizon $r_{1}=1/x_{1}$ that is larger than the outer event horizon of
the ordinary RN black hole $r_{+}=1/x_{+}$, and an inner event horizon
$r_{2}=1/x_{2}$ that is smaller than the inner event horizon of the ordinary
RN black hole $r_{-}=1/x_{-}$. On the other hand, in Fig. 1a, we plot
$\Upsilon_{R}$ and $\Upsilon_{L}$ for the case of $Q^{2}/M^{2}>1$. We notice
that the curve $\Upsilon_{L}$ does not cross the $x$-axis, which means that
the ordinary RN black hole has no event horizon and exposes its singularity to
the outside. This is called a naked singularity, and it violates the WCCC and
the Mild WGC is not valid. However, in the presence of PFDM, the curves
$\Upsilon_{R}$ and $\Upsilon_{L}$ still intersect at two points $x_{1}$ and
$x_{2}$, even when $Q^{2}/M^{2}>1$. This means that the RN black hole with
PFDM has two event horizons that hide its singularity from the observers.
Therefore, the WCCC and the Mild WGC are not violated, and they are consistent
with each other. In Fig. 1b, the curves $\Upsilon_{R}$ and $\Upsilon_{L}$
touch each other at a single point. In this case, we can define the extremal
state of the black hole at the point $x_{0}$, where $Q^{2}/M^{2}>1$ and
$\gamma=\gamma_{ext}$. When the parameters M and Q are fixed, if
$\gamma>\gamma_{ext}$, then the black hole becomes a naked singularity and
violates the WCCC. If $\gamma<\gamma_{ext}$, then the black hole has two event
horizons and satisfies the WCCC. If $\gamma=\gamma_{ext}$, then the black hole
has a single event horizon $x_{0}=1$ and is in the extremal state. This
situation is clearly shown in Fig. 2. We plot $\Upsilon_{L}=1-2Mx+Q^{2}x^{2}$
and $\Upsilon_{R}=-\gamma x\ln(\frac{1}{x\gamma})$ in Fig. 1, using Eq. (7).
We consider two cases: $Q^{2}/M^{2}\leq 1$ and $Q^{2}/M^{2}>1$. In Fig. 1c,
when $\Upsilon_{R}=0$, the curve $\Upsilon_{L}$ crosses the $x$-axis at two
points $(x_{+},x_{-})$. These are the inner and outer event horizons of the
standard RN black hole. We investigate the conditions for the compatibility of
the Mild WGC and the weak cosmic censorship conjecture (WCCC) for the
Reissner-Nordström (RN) black hole with (PFDM). We use Yes to indicate that
the two conjectures are consistent with each other, and No to indicate that
they are not. We summarize the results in Table 1. In summary, in studying R-N
black holes, it’s noted that without PFDM, black holes with charge $(Q)$
greater than mass $(M)$, $(Q>M)$, lack the event horizon, revealing
singularities. However, with PFDM, these black holes exhibit singularity and
event horizon. This condition supports the simultaneous compatibility of the
WGC and the WCCC, which is not possible in typical conditions. Also, at the
extremality limit, both conditions are met. For WCCC to hold, the PFDM
parameter must be within a certain range and also it must have a special range
for WGC. The intersection of these ranges indicates the conditions under which
both conjectures are satisfied for a charged black hole. Our research found a
common range for the PFDM parameter that ensures the coexistence of WGC and
WCCC, along with the presence of event horizon, satisfying the condition
$(Q>M)$. The compatibility is shown using parameters $(Y_{L})$ and $(Y_{R})$,
with constraints like those in Eq. (14). For instance, WGC is met for
$(\gamma\leq-0.25)$ with respect to $Q>M$ also it is compatible with WCCC.
Conversely, for $(\gamma>-0.25)$, WCCC holds, but not WGC (Eq. (14)), as seen
in the intersection of $(Y_{L})$ and $(Y_{R})$. We employed plots to visually
depict the parameter space where WGC and WCCC conditions align. These plots
clarify the regions of compatibility, offering a graphical representation of
our theoretical results. Furthermore, we utilized the Mild WGC, which posits
gravity as the weakest force, allowing for a wider range of scenarios than the
Strong WGC, which is more restrictive and particle-specific. Our studies
reveal that under certain conditions, such as the inclusion of PFDM, WGC, and
WCCC can coexist without contradiction, ensuring consistent physical
predictions for black holes and their thermodynamics.

Figure 1: The plot of $\Upsilon-x$. $\Upsilon_{L}=1-2Mx+Q^{2}x^{2}$ and
$\Upsilon_{R}=-\gamma x\ln(\frac{1}{x\gamma})$ with respect to $\gamma<0$
Figure 2: The plot of $\Upsilon-x$. $\Upsilon_{L}=1-2Mx+Q^{2}x^{2}$ and
$\Upsilon_{R}=-\gamma x\ln(\frac{1}{x\gamma})$ with respect to $Q>M$ and
$\gamma<0$

##  5 Conclusions

In this paper, the compatibility of the Mild weak gravity conjecture (WGC) and
the weak cosmic censorship conjecture (WCCC) was explored by considering
Reissner-Nordstr$\ddot{o}$m (R-N) black holes surrounded by perfect fluid dark
matter (PFDM) in asymptotically flat spacetimes. The Mild WGC is a conjecture
that imposes a lower bound on the charge-to-mass ratio in a quantum theory of
gravity. The WCCC is a conjecture that states that singularities formed by
gravitational collapse are hidden behind event horizons, preserving the
determinism and predictability of general relativity. These two conjectures
are seemingly unrelated, but a recent proposal suggested that they are
connected surprisingly. In particular, it was argued a promising class of
valid counterexamples to the WCCC in the four-dimensional Einstein-Maxwell
theory, considering charged black holes when Mild WGC was present. This
implies that the MILD WGC can protect the WCCC from violations, and that
cosmic censorship is a robust principle that can withstand the challenges
posed by the Mild WGC. In this paper, we extended this analysis to the case of
R-N black holes embedded in PFDM in asymptotically flat spacetimes. PFDM is a
model of dark matter that behaves as a perfect fluid with a constant equation
of state. We considered the effects of PFDM on the properties of the charged
black hole, such as the horizon radius with respect to Mild WGC and WCCC. We
imposed certain constraints on the parameters, such as the mass, the charge,
and the PFDM to demonstrate that by choosing appropriate values of these
parameters, the Mild WGC and the WCCC can be compatible. Moreover, we
investigated the properties of the charged black hole in the presence of PFDM
for $Q>M$, where Q is the charge and M is the mass of the black hole. We
presented some interesting figures to test the validity of the Mild WGC and
the WCCC simultaneously. We showed that the Mild WGC and WCCC can be satisfied
for a special range of values of Q, M, and PFDM. The main goal of this study
is to examine the effects of PFDM on the Reissner-Nordström (RN) black hole
and its implications for the weak gravity conjecture (WGC) and the weak cosmic
censorship conjecture (WCCC). We use a graphical method to plot the functions
$\Upsilon_{L}$ and $\Upsilon_{R}$, which represent the left-hand side and the
right-hand side of the equation of motion for the RN black hole with PFDM,
respectively. We consider two cases: $Q^{2}/M^{2}\leq 1$ and $Q^{2}/M^{2}>1$,
where Q and M are the charge and mass of the black hole, respectively. We also
vary the parameter $\gamma$, which affects of the PFDM on Mild WGC and WCCC.
We find that, in the absence of PFDM ($\gamma=0$), the RN black hole has two
event horizons for $Q^{2}/M^{2}\leq 1$, and no event horizon for
$Q^{2}/M^{2}>1$. The latter case leads to a naked singularity, which violates
the WCCC. However, in the presence of PFDM ($\gamma\neq 0$), the RN black hole
has event horizons, with regard to the values of Q and M. This means that the
singularity is always hidden, and the Mild WGC and the WCCC are satisfied.
Moreover, we show that there exists a critical value of $\gamma$, denoted by
$\gamma_{ext}$, such that the RN black hole becomes extremal when
$\gamma=\gamma_{ext}$. In this case, the black hole has an event horizon, and
the Mild WGC and the WCCC are still satisfied. We conclude that PFDM can
restore the compatibility of the Mild WGC and the WCCC for the RN black hole
and that the Mild WGC and the WCCC are consistent with each other in the
presence of PFDM. We also present a table that summarizes results for
different values $\gamma$. The results of this paper have important
implications for the field of general relativity and the quest for a quantum
theory of gravity. They also suggest that cosmic censorship is a robust
principle that can withstand the challenges.
The Weak Gravity Conjecture (WGC) is a new idea in theoretical physics. While
we have not yet obtained significant observational evidence for the WGC,
recently it plays a crucial role in understanding cosmic structure and
particle physics. This includes phenomena such as physics of black hole,
thermodynamics, dark energy, black brane and so on. Despite the scarcity of
directly observable data related to these concepts, the WGC has already made
some predictions. We anticipate that research in this area will accelerate in
the coming years, filling gaps and providing a deeper understanding of
fundamental physics. Of course, there are observational signatures for each of
the examined concepts, which we briefly mention below. We hope that more
observables related to combining concepts and proving the correctness of the
issue will be identified in the near future.
We face with some concepts related to WGC such as: 1\. Neutrinos: The WGC
implies that Standard Model neutrinos must be electrically neutral. 2\.
Quantization of electric charge: The electric charge in the Standard Model
must be quantized. 3\. Masslessness of the photon: The photon must remain
massless[71, 84].
To test the WCCC, scientists explore the following areas: 1\. Black hole
shadows: Investigating shadows cast by black holes, especially those with some
properties (e.g., charged black holes or modified gravity theories). 2\.
Photon rings and lensing rings: These features relate to dark matter presence
and black hole spacetime geometry.
Additionally, researchers study the Reissner-Nordstr$\ddot{o}$m, Kerr–Newman
black holes and so on: 1\. Shadow size: Increases with black hole charge. 2\.
Specific intensities: Charge affects observed intensities. 3\. Photon rings
and lensed rings: Additional observable features. 4\. Torsion charge: Alters
shadow size and intensity.
Regarding the (PFDM) model, assuming a perfect fluid: 1\. Background geometry:
Flat rotation curves hint at dark matter existence and background universe
geometry. 2\. Energy density and pressure: Expressions for dark matter energy
density and pressure can be derived[73, 74, 75, 76, 77]. Perfect fluid dark
matter is characterized by its uniform density and pressure, which can
significantly affect the spacetime around black holes. This influence can lead
to observable effects that might alter the expected signatures of the WGC and
WCCC. The presence of perfect fluid dark matter could cause the black hole
shadow to appear larger or more distorted than predicted in its absence[85,
86, 87, 88, 89, 90]. This is due to the additional gravitational lensing
effect that dark matter exerts on the light passing near the black hole. Dark
matter could also impact the emission spectra from the accretion disk of a
black hole. The interaction between dark matter particles and ordinary matter
could lead to unique spectral lines or shifts in the energy levels of emitted
photons[85, 86, 87, 88, 89, 90]. The merger of black holes embedded in a dark
matter halo might produce variations in the amplitude and frequency of the
emitted gravitational waves. These variations would be a result of the
additional gravitational potential contributed by the dark matter. By
analyzing these observational signatures, researchers can compare them with
theoretical models that include the effects of perfect fluid dark matter[85,
86, 87, 88, 89, 90]. Indeed, as mentioned, we initially selected a model that
assessed the impact of each observable signature, ensuring that the existence
of a black hole structure remains unchallenged. Following this verification,
we only investigated the Weak Gravity Conjecture (WGC) to identify a range
that preserves all possible conditions while also supporting the establishment
of the WCCC. It is crucial to note that the removal of the WGC would result in
only a minor alteration in the range related to WCCC compatibility. This
implies that, upon validating all the aforementioned cases, a new
compatibility—called to as the WGC-can be integrated into this framework. In
essence, the WGC may serve as a method to verify the existence and endorsement
of the visible signatures we have discussed. In summary, while maintaining all
conditions, we endeavored to incorporate a permissible and common interval of
WGC into the accptsable range with the WCCC, thereby establishing important
compatibility with preserving all structures, as elaborated in the text. These
concepts deepen our understanding of the fundamental nature of the universe.
Future work could extend this analysis to other types of black holes and dark
matter models, as well as explore the possible observational signatures of the
Mild WGC and the WCCC.

$\gamma<\gamma_{ext}$ |  $\gamma=\gamma_{ext}$ |  $\gamma>\gamma_{ext}$ |  Kind
---|---|---|---
No |  No |  No |  $\gamma>0$
Yes |  Yes |  No |  $\gamma<0$
Table 1: Summary of the results. Yes, indicates the consistency between Mild
WGC and WCCC, and No is vice versa.

##  6 Acknowledgments

The authors would like to express their gratitude to Mohammad Reza Alipour for
his insightful discussions. This research is supported by the research grant
of the University of Mazandaran (number: 33/26378)

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