Weak Cosmic Censorship and Weak Gravity Conjectures
in CFT Thermodynamics
Jafar Sadeghi⋆111Email<EMAIL_ADDRESS>Saeed Noori Gashti ⋆,†222Email:
<EMAIL_ADDRESS>Mohammad Reza Alipour⋆333Email:
<EMAIL_ADDRESS>
Mohammad Ali S. Afshar ⋆444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences, University of Mazandaran
P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, we explore the intriguing interplay between fundamental
theoretical physics concepts within the context of charged black holes.
Specifically, we focus on the consistency of the weak gravity conjecture (WGC)
and weak cosmic censorship conjecture (WCCC) in the thermodynamics of
conformal field theory (CFT), and restricted phase space (RPS) for AdS
Reissner-Nordström black holes with a perfect fluid dark matter (RN-PFDM). The
WGC ensures that gravity remains the weakest force in the system. Meanwhile,
the WCCC addresses the cosmic censorship problem by preventing the violation
of fundamental physical laws near the black hole singularity. First, we
analyze the RN black hole’s free energy in both spaces, revealing a
distinctive swallowtail pattern indicative of a first-order phase transition
when certain free parameter conditions are met. We explore the WGC across
different phase spaces, emphasizing the need for certain conditions in EPST,
RPST, and CFT. We demonstrate that PFDM $\gamma$ and the radius of AdS $l$
have a vital role in proving the satisfaction of the WGC. Also, these values
have a linear relation with the range compatibility of WGC. The range of
compatibility for WGC in RPST and EPST is the same while for CFT, this range
is larger than EPST, and RPST. It means somehow the WGC and CFT are more
consistent. The WCCC was examined at the critical juncture, confirming its
validity with $f(M_{c},Q_{c},r_{c})=0$. We conclude that the WGC is supported
at the critical point of black holes, and the WCCC is also maintained,
demonstrating the robustness of these conjectures within the critical ranges
of black hole parameters.
Keywords: Weak Gravity Conjecture, Weak Cosmic Censorship Conjecture, Perfect
Fluid Dark Matter, Restricted Phase Space, Conformal Field Theory

###### Contents

  1. 1 Introduction
  2. 2 Weak gravity conjecture
  3. 3 The Model
  4. 4 RPS thermodynamics
  5. 5 CFT thermodynamics
  6. 6 WGC $\&$ WCCC
  7. 7 Conclusions

##  1 Introduction

The exploration of black hole thermodynamics, initially spurred by the work of
Hawking and Bekenstein, delves into the thermal characteristics of these
cosmic entities from a thermodynamic physics perspective[1, 2, 3]. This field
posits a correlation between surface gravity and the event horizon’s area with
temperature and entropy, respectively, while often treating a black hole’s
mass as its internal energy. Despite the progress achieved, questions remain,
particularly regarding the microscopic contributors to black hole entropy. The
principles of holography and the AdS/CFT duality are instrumental in examining
various black hole properties, where a state of an AdS black hole in the bulk
corresponds to a thermal state in the dual field theory[4, 5]. Additionally,
the thermodynamics of black holes can be analyzed through the lens of
different symmetries within a gravitational model[6]. Hawking and Page’s
advanced research on AdS black holes revealed a phase transition between such
black holes accompanied by radiation and their thermal AdS counterparts[7].
The thermodynamic properties of black holes, especially within the framework
of extended phase space thermodynamics, have been extensively discussed, where
the negative cosmological constant is equated with pressure[8, 9, 10, 11, 12].
This perspective has led to the inclusion of other model parameters as novel
thermodynamic variables[13, 14].

Furthermore, the use of P-V variables suggests that AdS black holes could
function as heat engines[15, 16]. A recent theoretical development has
introduced central charge and chemical potential as a new pair of conjugate
thermodynamic variables, enhancing our understanding of black hole
thermodynamics. In the corresponding dual theory, the central charge is linked
to the square of the number of colors[17, 18, 19]. In the realm of AdS/CFT
correspondence, a novel thermodynamic framework known as restricted phase
space thermodynamics (RPST) has emerged, as introduced by Gao et al. This
paradigm redefines black hole masses as a form of internal energy. Unlike the
extended phase space thermodynamics (EPST) that utilizes pressure and volume,
RPST employs a distinct set of conjugate thermodynamic variables: the chemical
potential, denoted as µ, and the central charge, C. These variables represent
color susceptibility and the degrees of freedom within the conformal field
theory (CFT), respectively[18, 19, 20]. The central charge, C, plays a role
analogous to the particle count in the statistical physics of conventional
matter. It is mathematically expressed in relation to Newton’s constant, G,
and the anti-de Sitter (AdS) curvature radius, through the formula
$C=\frac{l^{d-2}}{G}$ Here, $d$ signifies the dimensionality of the AdS
spacetime. The RPST framework has been applied to explore the homogeneity of
the Smarr relation, various thermodynamic processes, and phase transitions
across different black hole configurations such as Reissner-Nordström-AdS (RN-
AdS), Kerr-AdS, Bañados-Teitelboim-Zanelli (BTZ), and Taub-NUT within the AdS
context[20, 21, 22, 23, 24, 25].

The holographic principle has revolutionized our understanding of space and
information, suggesting that the information contained within a volume of
space can be represented on a lower-dimensional boundary. This concept has
given rise to the field of holographic black hole chemistry. It posits a
duality between gravitational dynamics in higher-dimensional Anti-de Sitter
(AdS) space and quantum field theory on its boundary, a relationship known as
the AdS/CFT correspondence. Through this lens, we can explore black hole
thermodynamics by treating the cosmological constant as a variable akin to
thermodynamic pressure in AdS space or a scaling factor on the boundary. This
innovative approach has led to the development of black hole chemistry, which
can define thermodynamic quantities like enthalpy, Gibbs free energy, and
chemical potential for black holes[26, 27, 28, 29]. It allows probing of the
properties and phase transitions of black holes in AdS space through
holographic duality. Notably, it provides insights into phenomena such as the
Hawking-Page transition, which reflects a phase shift from thermal AdS space
to a large AdS black hole, analogous to the confinement/deconfinement
transition in conformal field theory. One of the intriguing outcomes is that
AdS black holes adhere to a generalized Smarr relation, linking their mass,
area, charge, angular momentum, and pressure. However, integrating the first
law of black hole thermodynamics with the thermodynamics of its holographic
dual presents challenges. This research is particularly focused on how the
variation of the cosmological constant ($\Lambda$) correlates with changes in
the central charge (C) and the volume (V) of the boundary CFT. This
relationship necessitates a revision of the thermodynamic laws to account for
the volume and its associated thermodynamic pressure[26, 27, 28, 29]. The
possibility of varying fundamental constants, like Newton’s constant (G) and
the cosmological constant, hints at more profound underlying theories.
researchers addressed this by demonstrating that altering G alongside
$\Lambda$ maintains the constancy of the central charge C in the dual CFT.
Consequently, the first law can be reformulated to include a new thermodynamic
volume and chemical potential, bridging holographic and bulk thermodynamics
through the central charge duality. Their work also uncovered phase structures
with a critical central charge value, beyond which the phase diagram shows a
swallowtail pattern indicative of a first-order phase transition. By treating
G as a variable, we can incorporate the central charge into the first law and
extend the thermodynamic discussion to various black holes. In essence, black
hole chemistry reveals that AdS black holes experience phase transitions
mirroring those found in conventional thermodynamic systems. Yet, the extended
thermodynamics’ holographic interpretation remained elusive for some time.
Initial theories proposed that the volume-pressure term in AdS should
correspond to a chemical potential-central charge term in the dual CFT, where
the central charge is C and the chemical potential is its thermodynamic
conjugate[26, 27, 28, 29].

The Swampland Program is an initiative in theoretical physics that seeks to
identify effective low-energy theories which are not compatible with quantum
gravity. This contrasts with the ”string theory landscape,” which includes
theories known to be compatible with string theory, a consistent quantum
theory of gravity. The Swampland encompasses theories that appear consistent
but lack a consistent ultraviolet completion when gravity is added. The
program was initiated by Cumrun Vafa and aims to delineate the boundaries of
the quantum gravity landscape by identifying universal principles shared among
all theories compatible with gravitational UV completion[30, 31, 32, 33, 34,
35, 36, 37, 38]. The WGC posits that in any theory of quantum gravity, gauge
forces must mediate interactions that are stronger than gravity for some
particles. This conjecture has deep connections across various areas of
physics and mathematics, suggesting that gravity should be the weakest force
in any consistent theory of quantum gravity. It has been reviewed extensively
and is supported by evidence from string theory examples. The Swampland
Program or the WGC is significant in the ongoing effort to understand the
fundamental principles that govern our universe at the quantum level. They
have implications for particle physics, cosmology, general relativity, and
mathematics, guiding researchers in exploring the limits and possibilities of
theoretical physics. To study an example of applications of the swampland
program in inflation, black holes, thermodynamics, and black brane, you can
see in[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102].

In this exploration, we examine the nuanced interplay between the WGC and the
WCCC in the realm of charged black holes. Our focus is on the (RN) black holes
that are encased within PFDM, considering diverse spatial frameworks like RPS
and CFT. This analysis aims to deepen our understanding of the theoretical
underpinnings that govern these cosmic entities. The WCCC, introduced by Roger
Penrose, is a pivotal concept in theoretical physics that addresses the
potential issues arising from singularities within general relativity. The
WCCC posits that singularities resulting from gravitational collapse are
invariably concealed behind event horizons, thereby maintaining the causal
structure and predictability of spacetime. This conjecture ensures that the
”naked” singularities, which could expose the inner workings of black holes,
do not disrupt the observable universe. In contrast, the WGC suggests that in
any consistent theory of quantum gravity, there must exist particles for which
the electromagnetic force is stronger than their gravitational pull. This
implies that black holes should be able to decay into these particles,
preventing the formation of stable remnants and ensuring the discharge of
black holes. The Reissner–Nordström (RN) black hole, a solution to the
Einstein–Maxwell equations for a charged black hole, presents a unique
challenge to these conjectures. If the charge $Q$ of the black hole exceeds
its mass $M$ ($Q>M$), the RN solution would contravene the WCCC by revealing a
naked singularity to distant observers. Conversely, when the RN black hole
approaches an extremal state ($Q=M$), it suggests the existence of decay
products with a charge-to-mass ratio exceeding unity. While this satisfies the
WGC, it simultaneously breaches the WCCC, as these decay products cannot form
black holes but are instead predicted to be elementary particles. This
apparent contradiction between the WGC and the WCCC in the context of RN black
holes is one of the intriguing puzzles at the intersection of quantum theory
and gravitational physics[103, 104, 105].

The WGC and the WCCC are two distinct ideas in theoretical physics, each
addressing different aspects of fundamental physics. While both conjectures
involve the term ”weak” and pertain to fundamental aspects of gravity and
spacetime, they are distinct in their focus and implications. The WGC
primarily concerns the relative strengths of forces in a quantum gravity
framework, while the WCCC deals with the behavior of singularities and event
horizons in gravitational collapse scenarios within classical general
relativity. Combining these concepts, one might speculate about potential
connections between the WGC and cosmic censorship, such as whether the WGC
could have implications for cosmic censorship scenarios. Building on the
concepts previously discussed, we delve into the intricate relationship
between the WGC and the WCCC within the framework of charged black holes,
specifically the RN AdS black hole surrounded by PFDM. We examine this
relationship from a thermodynamic standpoint, considering various theoretical
spaces such as RPS and CFT by mentioning an important point that means. The
Mild WGC posits that gravity is the weakest force, yet it accommodates a wider
spectrum of theoretical possibilities, making it less stringent than its
counterpart, the Strong WGC. The latter is more narrowly focused on particles,
asserting not just that gravity must be weaker but also stipulating the
existence of particles that strictly adhere to this inequality. When
considering black holes, the WCCC maintains that singularities should remain
veiled behind event horizons, safeguarding the causal structure of spacetime.
The interplay between the WGC and WCCC is subtle and complex. Our
investigations have revealed that under specific conditions, such as the
incorporation of PFDM, these conjectures can indeed coexist harmoniously. This
concord is vital as it ensures that our theoretical models remain consistent
and do not yield contradictory predictions about the thermodynamic behavior of
black holes. By demonstrating compatibility under these conditions, we
contribute to a more coherent understanding of the fundamental principles
governing black hole physics. The paper is organized as follows.
In Section 2, we conduct a comprehensive study of the RN AdS black hole in the
presence of PFDM. Sections 3 and 4 are dedicated to examining the
thermodynamics of the RN AdS black hole surrounded by PFDM within the
frameworks of RPST and CFT, respectively. We also analyze the thermodynamic
parameters in these two spaces. Section 5 explores the compatibility of the
WGC and the WCCC in both spaces, delving into the areas where these
conjectures align. Finally, in Section 6, we briefly summarize our findings.

##  2 Weak gravity conjecture

The Weak Gravity Conjecture is a theoretical framework in the field of quantum
gravity that suggests gravity should be the weakest force in any consistent
theory of quantum gravity. It posits that for any gauge force, there must
exist some particles for which the gauge force mediates interactions stronger
than gravity. This conjecture has profound implications across various domains
of physics and mathematics, and it has been a subject of significant interest
and debate within the theoretical physics community. The WGC is not without
its criticisms and potential weak points. Some of the criticisms include that
the conjecture is based on qualitative arguments rather than a rigorous
mathematical proof [51]. There are several variations of the conjecture, and
it’s not clear which, if any, is the correct one. The conjecture has been
challenged by certain string theory examples and other theoretical models that
do not seem to satisfy its conditions. The WGC has made several predictions,
including: The existence of particles with specific mass-to-charge ratios
ensures gravity remains the weakest force. Constraints on the properties of
black holes, particularly regarding their charge and mass. Implications for
the stability of cosmic structures and the behavior of fundamental particles.
The WGC has interesting connections with thermodynamics, particularly in the
context of black hole physics. It has been suggested that the conjecture could
be derived from thermodynamic principles applied to black holes, such as
entropy considerations and the behavior of quasinormal modes [72, 75]. The
implications of the WGC can extend to various spacetime geometries, including
extended, restricted, and CFT spaces: The conjecture has been explored in the
context of theories with extra dimensions, such as string theory, where it
places constraints on the geometry of the compactified dimensions. In more
constrained spacetime geometries, like those with a limited number of
dimensions or symmetries, the WGC still implies the existence of particles
that satisfy its mass-to-charge ratio condition. In the AdS/CFT
correspondence, the WGC translates into inequalities involving the dimension
and charge of operators, as well as central charges in the dual CFTs. However,
these translations are still speculative and may not apply universally across
all CFTs [52, 53]. In summary, the WGC is a conjecture that aims to provide
insights into the nature of gravity in quantum gravity theories. It has
sparked a wide range of research, leading to discussions about its validity,
implications, and connections to other physical principles. While it has faced
criticism and challenges, it continues to be a pivotal concept in theoretical
physics, driving exploration into the fundamental forces of nature. The WGC
has far-reaching implications that extend beyond the realm of theoretical
physics into cosmology, black hole thermodynamics, and the broader landscape
of string theory. In cosmology, the WGC can influence our understanding of the
early universe, particularly during the inflationary period. The conjecture
suggests that inflationary models involving scalar fields should be coupled to
gauge fields in a way that satisfies the WGC criteria [54]. This can lead to
predictions about the spectrum of primordial fluctuations, potentially
observable in the Cosmic Microwave Background (CMB) data. String theory
provides a fertile ground for testing the WGC. The conjecture has been used to
argue for the existence of certain particles in the string spectrum that have
implications for the stability of extra dimensions and the nature of
supersymmetry breaking. A proof of the WGC within the context of perturbative
string theory has been proposed, which relates the black hole extremality
bound to long-range forces calculated on the worldsheet [55]. The
thermodynamics of black holes offers a unique testing ground for the WGC. The
conjecture has been studied in the context of 4D Einstein-Maxwell-dilaton
theory, where it places constraints on the Wilson coefficients of the theory,
which are satisfied under certain assumption about the UV completion. This
study bridges the gap between quantum field theory and gravitational
phenomena, providing insights into the entropy and stability of black holes
[56]. The WGC may serve as a bridge between quantum mechanics and cosmic-scale
physics. By examining the thermodynamics of black holes and their connection
with the WGC, we can explore how quantum-level interactions manifest at cosmic
scales. This could lead to a better understanding of the holographic principle
and the nature of spacetime itself. The phenomenological implications of the
WGC are vast. It can affect predictions for particle accelerators, inform the
search for dark matter candidates, and influence models of dark energy. The
conjecture’s requirement for the existence of certain particles could lead to
observable signatures in high-energy physics experiments. In summary, the WGC
is not just a theoretical curiosity but a principle that can potentially offer
solutions to existing problems in cosmology and string theory, provide a
framework for understanding black hole thermodynamics, and open new avenues
for research in the phenomenology of quantum gravity theories. The ongoing
dialogue between theory and observation will continue to refine our
understanding of the WGC and its implications across the cosmos. Considering
all the mentioned concepts, we must now explore other applications of the WGC
within various cosmological frameworks. This exploration aims to address
potential issues and provide evidence supporting both the conjecture itself
and string theory. By examining black hole thermodynamics and its relationship
with the WGC, we can establish a link from quantum to cosmic scales. This not
only allows for a review of existing concepts but also enables the results
obtained to unveil new perspectives on the implications and phenomenology
associated with the WGC.

##  3 The Model

Reissner-Nordström AdS black holes, which are enveloped by PFDM, carry an
electrical charge and are affected by a theoretical type of matter that
constitutes the majority of the universe’s mass. These black holes exhibit
intriguing thermodynamic behaviors and phase transition characteristics, which
can be explored by broadening the phase space of the system. Additionally,
some studies have investigated the WCCC in the context of these black holes.
This conjecture posits that a black hole’s singularity remains hidden from any
external observer. The action under consideration pertains to a gravitational
theory that interacts minimally with a gauge field amidst the presence of
PFDM[105, 106, 107, 108, 109],

$S=\int d^{4}x\sqrt{-g}\bigg{[}\frac{1}{16\pi
G}R+\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{\Lambda}{8\pi
G}+\mathcal{L}_{DM}\bigg{]},$ (1)

in the given context, $g$ represents the determinant of the metric tensor,
expressed as $g=\det(g_{ab})$, where $g_{ab}$ are the components of the metric
tensor. The scalar curvature is denoted by $R$, while $G$ stands for the
gravitational constant. The electromagnetic tensor $F_{\mu\nu}$ is obtained
from the gauge potential $A_{\mu}$. The Lagrangian density $L_{DM}$
characterizes the PFDM. Utilizing the principle of least action allows for the
derivation of the Einstein field equations, which can be articulated 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)

The energy-momentum tensor for ordinary matter is symbolized by
$T^{M}_{\mu\nu}$, and the energy-momentum tensor for PFDM is represented by
$T^{DM}_{\mu\nu}$. These tensors are detailed in[105, 106, 107, 108, 109].
Consequently, the equations can be expressed as follows,

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

In the framework of the RN AdS metric influenced by PFDM, it is posited that
the components of the energy-momentum tensor satisfy
$T^{r}_{r}=T^{\theta}_{\theta}=T^{\phi}_{\phi}=T^{t}_{t}(1-\delta)$, with
$\delta$ being a constant. This assumption is substantiated by [105, 106, 107,
108, 109]. The formulation of the RN AdS metric within the PFDM context is
delineated as follows,

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

where,

$f(r)=1-\frac{2MG}{r}+\frac{GQ^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}+\frac{\gamma}{r}\ln(\frac{r}{\gamma}).$
(6)

In this analysis, the parameters $M$, $Q$, and $\gamma$ signify the mass,
charge and the influence of (PFDM) on the black hole, respectively. Also,
$\Lambda=-3/\ell^{2}$ is the cosmological constant with $\ell$ being the AdS
radius. In the absence of PFDM ($\gamma=0$), the space-time metric simplifies
to that of a standard RN AdS black hole. The parameter $\gamma$ can assume
both positive and negative values, reflecting different PFDM scenarios. This
study contemplates the implications of both positive and negative $\gamma$ on
the black hole’s characteristics.

##  4 RPS thermodynamics

In the RPS formalism, the traditional variables of pressure $P$ and volume $V$
are eschewed due to the altered connotations they acquire within the
holographic framework. Instead, two pivotal quantities, $C$ and $\mu$, assume
significant roles within the realm of CFT. The central charge $C$ is
instrumental in ascertaining the microscopic degrees of freedom inherent in
CFT, while its conjugate $\mu$ is regarded as a chemical potential, as
referenced in literature[20]. Within the domain of extended phase space
thermodynamics, the exploration of black hole thermodynamics necessitates the
employment of a dynamic cosmological constant. This adjustment invariably
leads to a transformation in the associated dynamical equations. To circumvent
this modification, the thermodynamics of a constrained phase space is
proposed, wherein Newton’s constant is treated as a variable, supplanting the
cosmological constant. This substitution preserves the original form of the
field equations. Moreover, a correlation is established between the central
charge and the cosmological and Newton’s constants, expressed as
$C=\frac{\ell^{2}}{G}$, where $\ell^{2}=-\frac{3}{\Lambda}$ [20]. The first
law of thermodynamics, as applied to a RN-AdS black hole surrounded by PFDM
within the RPS framework, is articulated as follows,

$dM=TdS+\hat{\phi}d\hat{Q}+\mu dC+\Pi d\gamma.$ (7)

The observation aligns with an Euler-like relation, which is a fundamental
principle in thermodynamics and fluid dynamics, often used to describe the
relationship between various state functions. In the context of the RPS
formalism, such relations are crucial for understanding the thermodynamic
properties of black holes and their behavior in different spacetime
geometries. By incorporating this Euler-like relation into the RPS framework,
researchers can further elucidate the intricate dynamics of black holes within
the holographic paradigm, offering deeper insights into the nature of
spacetime and the fundamental forces that govern it,

$M=TS+\hat{\phi}\hat{Q}+\mu C+\Pi\gamma,$ (8)

in the context of our black hole, the parameters $M$, $T$, and $S$ represent
the mass, temperature, and entropy, respectively. Moreover, these parameters
can generally be characterized by corresponding quantities in the dual CFT,

$\hat{Q}=\frac{Q\ell}{\sqrt{G}}\hskip 28.45274pt\hskip
28.45274pt\hat{\phi}=\frac{\phi\sqrt{G}}{\ell}.$ (9)

We will also have some parameters in RPS as follows,

$C=\frac{\ell^{2}}{G},\qquad\hat{Q}=Q\sqrt{C},\qquad
r_{+}=\ell\sqrt{\frac{S}{\pi C}}.$ (10)

Setting $(f(r_{+})=0)$ and utilizing equations (6), (7), (9), and (10), we can
derive some variables such as M, T, $\hat{\phi}$, $\Pi$, and $\mu$ of the
black hole as follows,

$M=\frac{\pi
CS+\pi^{2}\hat{Q}^{2}+S^{2}}{2\pi^{3/2}\ell\sqrt{CS}}+\frac{C\gamma}{2\ell^{2}}\ln\bigg{(}\sqrt{\frac{S}{\pi
C}}\frac{\ell}{\gamma}\bigg{)},$ (11)

$T=\bigg{(}\frac{\partial M}{\partial S}\bigg{)}_{\hat{Q},C,\gamma}=\frac{\pi
CS-\pi^{2}\hat{Q}^{2}+3S^{2}}{4\pi^{3/2}\ell
S\sqrt{CS}}+\frac{C\gamma}{4\ell^{2}S}$ (12)

$\hat{\phi}=\bigg{(}\frac{\partial
M}{\partial\hat{Q}}\bigg{)}_{S,C,\gamma}=\sqrt{\frac{\pi}{SC}}\frac{\hat{Q}}{\ell},$
(13)

$\Pi=\bigg{(}\frac{\partial
M}{\partial\gamma}\bigg{)}_{S,\hat{Q},C}=\frac{C}{2\ell^{2}}\bigg{[}-1+\ln\bigg{(}\sqrt{\frac{S}{\pi
C}}\frac{\ell}{\gamma}\bigg{)}\bigg{]},$ (14)

$\mu=\bigg{(}\frac{\partial
M}{\partial\gamma}\bigg{)}_{S,\hat{Q},\gamma}=\frac{\pi
CS-\pi^{2}\hat{Q}^{2}-S^{2}}{4\pi^{3/2}C\ell\sqrt{CS}}+\frac{\gamma}{4\ell^{2}}\bigg{[}-1+2\ln\bigg{(}\sqrt{\frac{S}{\pi
C}}\frac{\ell}{\gamma}\bigg{)}\bigg{]}.$ (15)

The $T-S$ curve at a fixed value of $\hat{Q}$ and $C$ is a fundamental feature
in thermodynamics. It captures the behavior of a system as temperature ($T$)
and entropy ($S$) vary. Notably, this curve exhibits a first-order phase
transition, which, intriguingly, becomes second-order at the critical point.
To identify the critical point on the $T-S$ curve (often referred to as the
inflection point), we employ the following equation,

$\bigg{(}\frac{\partial T}{\partial
S}\bigg{)}_{\hat{Q},C,\gamma}=0,\qquad\bigg{(}\frac{\partial^{2}T}{\partial
S^{2}}\bigg{)}_{\hat{Q},C,\gamma}=0.$ (16)

Using Eqs. (12) and (16) for the critical parameters, we have,

$Y:=C\ell^{\frac{4}{3}}\big{(}81\gamma^{2}-\ell^{2}+9\ell\gamma\sqrt{81\gamma^{2}-2\ell^{2}}\big{)}^{\frac{1}{3}},\qquad
S_{c}=\frac{\pi(C\ell^{2}+Y)^{2}}{18\ell^{2}Y},$ (17)

$\hat{Q}_{c}^{2}=-\frac{\left(C\ell^{2}+Y\right)^{2}\left(C^{2}\ell^{4}-4C\ell^{2}Y+Y^{2}\right)}{324\ell^{4}Y^{2}}+\frac{\sqrt{2}\gamma
C^{3/2}\left(C\ell^{2}+Y\right)}{9\sqrt{Y}\ell^{2}}.$ (18)

The critical values of $T$ and $M$ are calculated by,

$T_{c}=\frac{\left(C\ell^{2}+Y\right)\left(-9\sqrt{2}C^{3/2}\gamma\ell^{2}Y^{3/2}+C^{3}\ell^{6}+6C^{2}\ell^{4}Y+6C\ell^{2}Y^{2}+Y^{3}\right)}{3\pi\sqrt{2CY}\ell^{2}\left(C\ell^{2}+Y\right)^{3}}+\frac{9\gamma
CY}{2\pi\left(C\ell^{2}+Y\right)^{2}},$ (19)

$M_{c}=\frac{\sqrt{C}\left(C\ell^{2}+Y\right)\left(2C\ell^{2}+3\gamma\sqrt{2CY}+2Y\right)}{9\ell^{2}\sqrt{2Y}\left(C\ell^{2}+Y\right)}+\frac{\gamma
C}{2\ell^{2}}\ln\left(\frac{C\ell^{2}+Y}{\gamma\sqrt{18CY}}\right).$ (20)

When studying the thermodynamics of black holes using the Euclidean action, we
often encounter a challenge known as redshift degeneracy in the action. To
address this issue, we propose considering an appropriate counterterm derived
from the AdS/CFT correspondence within the Euclidean action. By doing so, we
can resolve this limitation and proceed to find the thermodynamic potential
associated with both the canonical and grand canonical ensembles. As an
illustrative example, let’s refer to the Gibbs energy potential, denoted as G:
$G=i\beta$ where i represent the Euclidean action, and $\beta$ is the inverse
of the Hawking temperature. From this Gibbs energy potential, we can derive
other essential thermodynamic quantities: Entropy (S): Given by
$S=-\frac{dG}{dt}$, angular momentum (J): Given by $J=-\frac{dG}{d\Omega}$,
electric charge (Q): Given by $Q=-\frac{dG}{d\phi}$. This approach is
equivalent to using the first law of thermodynamics to determine these
quantities. Additionally, we can express these thermodynamic quantities in
terms of the Helmholtz free energy (F) and its conjugates: Entropy (S):
$S=-\frac{dF}{dt}$, angular momentum (J): $J=-\frac{dF}{d\Omega}$, electric
charge (Q): $Q=-\frac{dF}{d\phi}$. The connection between the Helmholtz free
energy and the Euclidean action is given by $F=i\beta$. Remarkably, the
results obtained from the Helmholtz free energy remain consistent with those
derived from the first law of thermodynamics. In the RPS formalism, Newton’s
constant is treated as a variable external to the Euclidean action.
Consequently, it does not introduce any additional terms to the counterterm.
By adjusting the counterterm appropriately, we can eliminate the redshift
degeneracy for different types of black holes. Now, we delve into the
Helmholtz free energy,

$F=M-TS=\frac{\pi
CS+3\pi^{2}\hat{Q}^{2}-S^{2}}{4\pi^{3/2}\ell\sqrt{CS}}+\frac{\gamma
C}{4\ell^{2}}\bigg{[}-1+2\ln\bigg{(}\frac{\ell}{\gamma}\sqrt{\frac{S}{\pi
C}}\bigg{)}\bigg{]}.$ (21)

Figures 1 and 2 depict the evolution of free energy as a function of
temperature for various free parameter values of the RN AdS black hole,
analyzed within the framework of PFDM in RPS. When $\hat{Q}_{c}>\hat{Q}=0.5$,
a distinctive swallowtail pattern emerges, indicative of a first-order phase
transition. Also, for ($\hat{Q}_{c}=\hat{Q}$ ), we have a second-order phase
transition. To enhance clarity, we have separated the figures. The specific
free parameters pertinent to each figure are duly annotated. The separation of
the figures isolates and emphasizes each parameter set’s unique
characteristics, facilitating a more detailed examination of the system’s
behavior under varying conditions. It is essential to recognize that the free
parameters play a crucial role in shaping the physical properties of the black
hole.

Figure 1: With the $\gamma=2,C=\ell=1$ and $\hat{Q}_{c}=0.93$ . (a)
$\hat{Q}_{c}>\hat{Q}=0.5$ (b) $\hat{Q}_{c}=\hat{Q}=0.93$ (c)
$\hat{Q}_{c}<\hat{Q}=1.5$

.

Figure 2: With the $\gamma=1.5,C=\ell=1$ and $\hat{Q}_{c}=0.79$ . (a)
$\hat{Q}_{c}>\hat{Q}=0.5$ (b) $\hat{Q}_{c}=\hat{Q}=0.79$ (c)
$\hat{Q}_{c}<\hat{Q}=1.5$

.

Figure 3: spinodal curve ($T-S$) for charged black hole in presence perfect
fluid. We have set $C=\ell=1$ and (a) $\gamma=2$ and $\hat{Q}_{c}=0.93$, (b)
$\gamma=1.5$ and $\hat{Q}_{c}=0.79$

.

Figure 3 presents the Hawking temperature as a function of entropy for the AdS
(RN) black hole amidst the presence of (PFDM) in the (RPS). We have set the
constants $C$ and $\ell$ to 1. For Figure 3a, the parameters are set to
$\gamma=2$ and $\hat{Q}_{c}$= 0.93 , while for Figure 3b, $\gamma=1.5$ and
$\hat{Q}_{c}$= 0.79 . Each subplot within the figure reveals certain zero
points for varying free parameters within the range $0<S<1$. Upon examining
the entropy alongside the black hole’s free parameter values, it is observed
that for small values of $S$, the structural behavior of the black hole is
markedly unique. Conversely, as the value of $S$ increases, the figures tend
to converge, indicating a unification in behavior for larger entropy values.
This convergence suggests that the influence of the free parameters diminishes
as the entropy grows, leading to a more predictable and uniform structure for
the black hole.

##  5 CFT thermodynamics

To explore the thermodynamics of the charged AdS black hole with PFDM within
the CFT framework, it’s essential to understand the holographic links that
connect the bulk and boundary quantities. We denote the boundary’s curvature
radius by $R$, which is distinct from the AdS radius $\ell$ in the bulk. The
CFT metric, showcasing conformal scaling invariance, is given by,

$\begin{split}ds^{2}=\omega^{2}(-dt^{2}+\ell^{2}d\Omega_{d-2}^{2}).\end{split}$
(22)

Here, $\omega$ is the dimensionless conformal factor that varies freely,
signifying the boundary theory’s conformal symmetry. In a spherical scenario,
$d\Omega_{d-2}^{2}$ is the metric on a $(d-2)$-dimensional sphere. We assume
$\omega$ is constant across the

$\begin{split}\mathcal{V}=\Omega_{d-2}R^{d-2},\end{split}$ (23)

where $R=\omega\ell$ signifies the manifold’s variable curvature radius where
the CFT resides. Notably, variations in the central charge $C$ do not affect
$\mathcal{V}$. In Einstein’s gravity context, $C$ and $\ell$ exhibit a dual
relationship, as indicated in a previous equation. We maintain a constant
Newton’s constant $G$ while allowing $\ell$ to vary, which in turn alters $C$.
The holographic dictionary facilitates the correlation of bulk parameters
$M,T,S,\Phi,Q$ with their CFT counterparts
$E,\tilde{T},\tilde{S},\tilde{\Phi},\tilde{Q}$, as follows,

$\begin{split}E=\frac{M}{\omega},\qquad\tilde{T}=\frac{T}{\omega},\qquad\tilde{S}=S,\qquad\tilde{Q}=\frac{Q\ell}{\sqrt{G}},\qquad\tilde{\Phi}=\frac{\Phi\sqrt{G}}{\omega\ell}.\end{split}$
(24)

Employing these relationships, we can reformulate the first law of
thermodynamics for the charged AdS black hole in the CFT context:

$\begin{split}\delta E=\tilde{\Phi}\delta\tilde{Q}+\mu\delta
C-p\delta\mathcal{V}+\tilde{T}\delta S,\end{split}$ (25)

where the chemical potential $\mu$ and pressure $p$ are defined as,

$\begin{split}\mu=\frac{E-\tilde{T}S-\tilde{\Phi}\tilde{Q}}{C},\end{split}$
(26) $\begin{split}p=\frac{E}{(d-2)\mathcal{V}}.\end{split}$ (27)

Furthermore, it’s evident from the above equation that $\mathcal{V}$ and $C$
vary independently. This paper aims to delve into the consequences of this
theoretical framework on the thermodynamics of RN AdS black holes in the
presence of PFDM within the CFT paradigm. Our objective is to reformulate the
bulk thermodynamic first law in a manner that incorporates the boundary
central charge. To achieve this, we turn to the holographic dual relationship
between key quantities in Einstein’s gravity and the AdS scale, the central
charge C, and Newton’s constant G. In this context, we consider a setting
where both the cosmological constant ($\Lambda$) and gravitational Newton
constant ($G$) are allowed to vary within the bulk. By doing so, we can
express the first law in a new form that involves both $\Lambda$ (associated
with thermodynamic pressure) and the central charge $C$ from the dual CFT.
These variables and their conjugates play a crucial role in understanding the
thermodynamics of AdS black holes. This approach allows us to explore novel
aspects of black hole behavior, including phase transitions, critical points,
and universal scaling behavior, all while maintaining a fixed boundary central
charge. The interplay between holography and bulk thermodynamics provides
valuable insights into the nature of quantum gravity,

$C=\frac{\Omega_{d-2}\ell^{d-2}}{16\pi G}.$ (28)

Here, we aim to derive the first law of thermodynamics within the context of
the CFT mode for the RN-AdS black hole with PFDM. Our goal is to establish a
meaningful correspondence between quantities in the bulk (associated with the
black hole) and those on the boundary (related to the CFT). To achieve this,
we introduce a scale transformation given by:$E=\frac{M}{\omega}$ where
$\omega=\frac{R}{\ell}$ represents the ratio of the AdS radius ($R$) to the
characteristic length scale ($\ell$). We focus on the case where the spacetime
dimension is $d=4$. By analyzing the modified equations, we explore the
interplay between the RN-AdS black hole, its thermodynamic properties, and the
dual CFT description. This investigation sheds light on the intricate
relationship between the bulk geometry and the boundary theory, providing
valuable insights into the underlying physics,

$\Omega_{2}=4\pi,\qquad C=\frac{\ell^{2}}{4G},$ (29)

and

$\begin{split}&\delta\bigg{(}\frac{M}{\omega}\bigg{)}=\frac{T}{\omega}\delta\bigg{(}\frac{A}{4G}\bigg{)}+\bigg{(}\frac{M}{\omega}-\frac{TS}{\omega}-\frac{Q\Phi}{\omega}-\frac{\Pi\gamma}{\omega}\bigg{)}\frac{\delta(\ell^{2}/4G)}{\ell^{2}/4G}\\\
&-\frac{M}{2\omega}\frac{\delta(\Omega_{2}R^{2})}{(\Omega_{2}R^{2})}+\frac{\Phi\sqrt{G}}{\omega\ell}\delta\bigg{(}\frac{Q\ell}{\sqrt{G}}\bigg{)}+\frac{\Pi\sqrt{G}}{\omega\ell}\delta\bigg{(}\frac{\gamma\ell}{\sqrt{G}}\bigg{)}.\end{split}$
(30)

Based on the equation provided above, the correspondence between bulk and
boundary quantities for the black hole can be established as follows,

$\begin{split}\delta E=\tilde{T}\delta\tilde{S}+\mu\delta
C-p\delta\mathcal{V}+\tilde{\Phi}\delta\tilde{Q}+\tilde{\Pi}\delta\tilde{\gamma},\end{split}$
(31)

where

$\begin{split}E=\frac{M}{\omega},\quad\tilde{S}=S,\quad\tilde{T}=\frac{T}{\omega},\quad\tilde{\Phi}=\frac{\Phi\sqrt{G}}{\omega\ell},\quad\tilde{Q}=\frac{Q\ell}{\sqrt{G}},\quad\tilde{\Pi}=\frac{\Pi\sqrt{G}}{\omega\ell},\quad\tilde{\gamma}=\frac{\gamma\ell}{\sqrt{G}},\end{split}$
(32)

if $\beta=\frac{\tilde{\gamma}}{\ell}$, we also calculate thermodynamic
quantities for the CFT as follows,

$\begin{split}E=\frac{4\pi
CS+\pi^{2}\tilde{Q}^{2}+S^{2}}{2\pi\sqrt{CS\mathcal{V}}}+\frac{2\beta\sqrt{\pi
C}}{\sqrt{\mathcal{V}}}\ln\left(\frac{\sqrt{S}}{\sqrt{\pi}\beta}\right),\end{split}$
(33)

$\tilde{T}=\bigg{(}\frac{\partial E}{\partial
S}\bigg{)}_{\tilde{Q},C,\beta,\mathcal{V}}=\frac{4\pi
CS-\pi^{2}\tilde{Q}^{2}+3S^{2}}{4\pi
S^{3/2}\sqrt{C\mathcal{V}}}+\frac{\beta\sqrt{\pi C}}{S\sqrt{\mathcal{V}}},$
(34)

$\tilde{\Phi}=\bigg{(}\frac{\partial
E}{\partial\tilde{Q}}\bigg{)}_{C,S,\beta,\mathcal{V}}=\frac{\pi\tilde{Q}}{\sqrt{CS\mathcal{V}}},$
(35)

$\mu=\bigg{(}\frac{\partial E}{\partial
C}\bigg{)}_{\tilde{Q},S,\beta,\mathcal{V}}=\frac{4\pi
CS-\pi^{2}\tilde{Q}^{2}-S^{2}}{4\pi
C\sqrt{CS\mathcal{V}}}+\frac{\sqrt{\pi}\beta}{\sqrt{C\mathcal{V}}}\ln\bigg{(}\frac{\sqrt{S}}{\sqrt{\pi}\beta}\bigg{)},$
(36)

$\tilde{\Pi}=\bigg{(}\frac{\partial
E}{\partial\tilde{\gamma}}\bigg{)}_{\tilde{Q},S,C,\mathcal{V}}=\frac{\beta\sqrt{\pi
C}}{\tilde{\gamma}S\sqrt{\mathcal{V}}},$ (37)

$-p=\bigg{(}\frac{\partial
E}{\partial\mathcal{V}}\bigg{)}_{\tilde{Q},S,C,\beta}=-\frac{E}{2\mathcal{V}}.$
(38)

Additionally, we can utilize the following relationships to identify the
critical points,

$\begin{split}\bigg{(}\frac{\partial\tilde{T}}{\partial
S}\bigg{)}_{\tilde{Q},\mathcal{V},C,\beta}=0,\qquad\bigg{(}\frac{\partial^{2}\tilde{T}}{\partial
S^{2}}\bigg{)}_{\tilde{Q},\mathcal{V},C,\beta}=0.\end{split}$ (39)

By applying the connections between equations $\eqref{eq25}$ and
$\eqref{eq30'}$, we can determine the critical points as outlined below

$X:=(-4C^{3}+81\beta^{2}C^{2}+9\beta
C^{2}\sqrt{81\beta^{2}-8C})^{\frac{1}{3}},\qquad
S_{c}=\frac{1}{9}\left(\frac{2^{5/3}\pi C^{2}}{X}+4\pi C+2^{\frac{1}{3}}\pi
X\right),$ (40)

$\tilde{Q}_{c}^{2}=\frac{-2^{10/3}C^{4}+2^{11/3}C^{3}X+24C^{2}X^{2}+2^{7/3}CX^{3}-2^{2/3}X^{4}}{81X^{2}}+\frac{8}{9}\beta
C\sqrt{\frac{2^{5/3}C^{2}}{X}+4C+2^{\frac{1}{3}}X},$ (41)

$\tilde{T}_{c}=\frac{4\pi CS_{c}-\pi^{2}\tilde{Q}_{c}^{2}+3S_{c}^{2}}{4\pi
S_{c}^{3/2}\sqrt{C\mathcal{V}}}+\frac{\beta\sqrt{\pi
C}}{S_{c}\sqrt{\mathcal{V}}},\qquad E_{c}=\frac{4\pi
CS_{c}+\pi^{2}\tilde{Q}_{c}^{2}+S_{c}^{2}}{2\pi\sqrt{CS_{c}\mathcal{V}}}+\frac{2\beta\sqrt{\pi
C}}{\sqrt{\mathcal{V}}}\ln\bigg{(}\frac{\sqrt{S_{c}}}{\sqrt{\pi}\beta}\bigg{)}.$
(42)

We utilize equations $\eqref{eq24}$ and $\eqref{eq25}$ to determine the
Helmholtz energy.

$F=E-\tilde{T}\tilde{S}=\frac{12\pi
CS+\pi^{2}\tilde{Q}^{2}+5S^{2}}{4\pi\sqrt{CS\mathcal{V}}}+\frac{\beta\sqrt{\pi
C}}{\sqrt{\mathcal{V}}}\bigg{[}-1+2\ln\bigg{(}\frac{\sqrt{S}}{\sqrt{\pi}\beta}\bigg{)}\bigg{]}.$
(43)

Figures 4 and 5 showcase the evolution of free energy as a function of
temperature for diverse free parameter values of the RN AdS black hole,
situated within the theoretical framework of PFDM in Conformal Field Theory. A
notable swallowtail pattern is observed when $\tilde{Q}_{c}>\tilde{Q}=1.5$,
signaling a first-order phase transition. Also, in $\tilde{Q}_{c}=\tilde{Q}$,
we face with second phase transition. For enhanced visual clarity, we have
separated the figures. Each figure is meticulously labeled with the relevant
free parameters. By separating the figures, we aim to highlight and scrutinize
the distinct features of each parameter set, thereby enabling a more granular
analysis of the system’s dynamics under different conditions. The role of free
parameters is pivotal in determining the physical attributes of the black
hole.

Figure 4: With the $\beta=2,C=\mathcal{V}=1$ and $\tilde{Q}_{c}=2.52$ . (a)
$\tilde{Q}_{c}>\tilde{Q}=1.5$ (b) $\tilde{Q}_{c}=\tilde{Q}=2.52$ (c)
$\tilde{Q}_{c}<\tilde{Q}=3$

.

Figure 5: With the $\beta=1.5,C=\mathcal{V}=1$ and $\tilde{Q}_{c}=2.15$ . (a)
$\tilde{Q}_{c}>\tilde{Q}=1.5$ (b) $\tilde{Q}_{c}=\tilde{Q}=2.15$ (c)
$\tilde{Q}_{c}<\tilde{Q}=3$

.

Figure 6: spinodal curve ($\tilde{T}-S$) for charged black hole in presence
perfect fluid. We have set $C=\mathcal{V}=1$ and (a) $\beta=2$ and
$\tilde{Q}_{c}=2.52$, (b) $\beta=1.5$ and $\tilde{Q}_{c}=2.15$

.

Figure 6 illustrates the Hawking temperature as a function of entropy for the
RN AdS black hole in the context of PFDM within CFT. We set $C$
=$\mathcal{V}$=1. For Figure 6a, the parameters are chosen as $\beta=2$ and
$\tilde{Q}_{c}=2.52$, whereas for Figure 6b, $\beta=1.5$ and
$\tilde{Q}_{c}=2.15$ are selected. Each subplot in the figure discloses
distinct zero points for varying free parameters across the entropy range
$0<S<1$. A detailed analysis of the entropy in relation to the black hole’s
free parameter values reveals a pronounced uniqueness in structural behavior
for lower entropy values, denoted by $S$. This uniqueness is characterized by
significant deviations in the thermal properties of the black hole. In
contrast, as the entropy value $S$ ascends, the subplots exhibit a tendency to
align, signifying a behavioral unification at higher entropy levels.

##  6 WGC $\&$ WCCC

The primary objective of this work is to challenge the WGC and WCCC in
different spaces. So first we investigate the thermodynamic properties of
Reissner-Nordström black holes in the presence PFDM, utilizing the principles
of RPS and CFT thermodynamics. Then we explore the special ranges where these
conjectures are compatible with each other in the $Q/M>\sqrt{G}$. We aim to
explore the following key aspects: This comprehensive study aims to deepen our
understanding of black hole physics in the presence of dark matter and energy,
contributing to the broader field of quantum gravity and cosmology. This
research objective is designed to be broad yet focused, allowing for a
detailed exploration of the interconnected theories and their implications on
the thermodynamics and stability of black holes in astrophysical scenarios
influenced by dark matter.

Figure 7:  (a) $\ell=0.5$, (b) $\ell=1.5$, (c) $\ell=2$

.

To substantiate the WGC within various phase spaces, we must establish certain
specific relations. In the EPST, the condition is $H=\frac{Q^{2}}{M^{2}}-1>0$,
and for the RPST and CFT, the criterion is $H=\frac{Q^{2}}{M^{2}}-G>0$.
Consequently, we can pursue several approaches, among which thermodynamics and
the critical values of black hole parameters are particularly significant. In
light of this, we integrate the critical points delineated in prior sections
into each respective space under the conditions mentioned above. Given that
the critical values related with two parameters, namely $\gamma$ and $l$, it
becomes evident that these parameters play pivotal roles in validating the
WGC. As depicted in Figure 7, the ranges within which the WGC is
satisfied—specifically ($H>0$) for EPST and RPST are the same. However, for
CFT, this range is different and larger than other spaces.

$RPST=EPST\rightarrow\sqrt{\frac{2}{3}}\ell<\gamma<5\ell$

,

$CFT\rightarrow\sqrt{3}\ell<\gamma<10\ell$

.

Furthermore, through computational methods, we have approximated these
intervals. Additionally, upon scrutinizing the WCCC at the critical juncture,
we ascertain that it remains valid viz $f(M_{c},Q_{c},r_{c})=0$. Thus, it can
be concluded that even though the WGC is upheld for the black hole at the
critical point, the WCCC is also concurrently sustained. This dual validity
underscores the robustness of the conjectures within the critical framework of
black hole parameters.

The WGC in CFT thermodynamic space is considered to yield a better result and
is within a more acceptable range for several reasons and the coordination of
the WGC and CFT can be checked in different works[110, 111, 112, 113]

In summary, the WGC in CFT thermodynamic space is favored because it
integrates well with established principles of quantum field theory and
quantum gravity, offers a stable thermodynamic interpretation, and avoids
theoretical inconsistencies, making it a more reliable and acceptable result
within the theoretical physics community.

##  7 Conclusions

In this paper, we explore the intriguing interplay between fundamental
theoretical physics concepts within the context of charged black holes.
Specifically, we focus on the consistency of the WGC and WCCC in CFT, and RPS
for RN AdS-PFDM. First, we have meticulously explored the thermodynamic
properties of RN AdS black holes in the presence of PFDM within the realm of
RPS and CFT. Our research has been underpinned by a series of detailed figures
that depict the evolution of free energy and Hawking temperature as functions
of temperature and entropy, respectively. Our findings reveal a remarkable
swallowtail pattern in the free energy of RN AdS black holes, which emerges
under specific free parameter values and signifies a first-order phase
transition. We found that despite the differences in EPST, RPST and CFT
spaces, similar phase behavior is observed in all these spaces. In our study,
we delve into the intricacies of the ” WGC”, which is a theoretical framework
within the field of quantum gravity. The WGC posits that for any gauge force
to be consistent with quantum gravity, it must exert forces that are stronger
than gravitational forces on some particles. This conjecture has profound
implications across various phase spaces, and we specifically focus on its
application in ”EPST”, ” RPST, and CFT” spaces. Our investigation highlights
the critical role that thermodynamic principles and black hole parameters play
in substantiating the WGC. Through rigorous exploration, we delineate the
regions within these phase spaces where the WGC holds met. Interestingly, we
observe that the parameter space accommodating the WGC is more expansive in
the context of CFT, surpassing the confines observed in EPST and RPST spaces.
Moreover, we pinpoint specific intervals for the parameters $\gamma$ and $l$
that lend credence to the WGC. By employing computational techniques, we not
only bolster the WGC but also affirm the ”WCCC”. The WCCC, which is a
fundamental principle in the study of black holes, asserts that singularities
arising within black holes should be concealed from external observers,
thereby preserving the predictability of physical laws. Hence, for a charged
black hole, when $(\frac{Q}{M}>\sqrt{G})$, we will face a naked singularity,
which violates WCCC, and therefore is not compatible with WGC, but in this
case, adding PFDM will solve the mentioned problem and the compatibility of
these two conjectures appears. In a pivotal examination at the critical
juncture of black hole parameters, we scrutinize the WCCC and establish its
validity. The critical condition $f(M_{c},Q_{c},r_{c})=0$ serves as a
testament to the conjecture’s soundness. Our findings lead us to conclude that
both the WGC and WCCC receive substantial support at the critical points of
black holes. This not only demonstrates the robustness of these conjectures
but also underscores their significance within the critical areas of black
hole physics.

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