Ke-Jian He ^1,
, Xin-Yun Hu ^2,,
, Xiao-Xiong Zeng ^3,4,
,

Corresponding author: Xin-Yun Hu, huxinyun@126.com

1.Physics and Space College, China West Normal University, Nanchong 637000, China
2.College of Economic and Management, Chongqing Jiaotong University, Chongqing 400074, China
3.State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, China
4.Department of Mechanics, Chongqing Jiaotong University, Chongqing 400074, China
Received Date:2019-08-07
Available Online:2019-12-01
Abstract:Considering the cosmological constant as the pressure, this study addresses the laws of thermodynamics and weak cosmic censorship conjecture in the Reissner-Nordstr?m-AdS black hole surrounded by quintessence dark energy under charged particle absorption. The first law of thermodynamics is found to be valid as a particle is absorbed by the black hole. The second law, however, is violated for the extremal and near-extremal black holes, because the entropy of these black hole decrease. Moreover, we find that the extremal black hole does not change its configuration in the extended phase space, implying that the weak cosmic censorship conjecture is valid. Remarkably, the near-extremal black hole can be overcharged beyond the extremal condition under charged particle absorption. Hence, the cosmic censorship conjecture could be violated for the near-extremal black hole in the extended phase space. For comparison, we also discuss the first law, second law, and the weak cosmic censorship conjecture in normal phase space, and find that all of them are valid in this case.

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Black holes are important not only in astronomy, but also in gravitation theory. There is a geometric singularity located at the center of the black hole. If the event horizon of the black hole is unstable, the singularity of the black hole will be exposed. The exposed singularity will send uncertain information, leading to the break of the causal relation between space and time. To avoid this situation, Penrose conjectured that the singularity should be wrapped in the center of the event horizon of the black hole, which is the so-called weak cosmic censorship conjecture [1,2]. This conjecture also describes that the gravitational collapse of dust will eventually form a black hole with a singularity [2], which was proven by Penrose and Hawking [3].
Because there is no general method to prove the weak cosmic censorship conjecture in black holes, we need to check its validity in various spacetimes. One interesting method is to verify whether the final states of black holes are still black holes after the absorption. In four-dimensional space time, Wald proved that particles making an extremal Kerr-Newman black hole overcharge or overspin would not be absorbed through the Gedanken experiment [4]. This result is also applicable to the scalar field [5,6]. Subsequently, Hubeny found that for a near-extremal Reissner-Nordstr?m black hole, the weak cosmic censorship conjecture was violated and the charge can exceed its extremum boundary [7]. This behavior was also observed in the near-extremal Kerr and Kerr-Newman black hole [8,9]. However, as the self-force effect was considered in the Kerr-Newman black hole, the weak cosmic censorship conjecture was found to remain valid when the event horizon was stable [10-12]. Furthermore, the near-extremal Reissner-Nordstr?m black hole can also apply this conjecture with consideration of the back-reaction effect [7, 13]. Many studies have been performed on the validity of the weak cosmic censorship conjecture under different black hole backgrounds [14-39], such as black holes in Einstein’s gravity theory, modified gravitational theory, black holes with low or high dimensions, and black holes with electric charge.
The cosmological constant is initially fixed. However, with some later studies, the cosmological constant was considered as a variable quantity [40,41] and the extended phase space is constructed, where the pressure of the black hole is related to the cosmological constant [42,43], and its thermal conjugate is the thermodynamic volume of the black hole [44,45]. In this case, the mass of the black hole corresponds to the enthalpy, not the internal energy of the black hole thermodynamic system [46,47].
In the extended phase space, the first law of thermodynamics has been extensively investigated. However, few studies investigate the second law and the weak cosmic censorship conjecture. Validity of the first law does not imply that the second law and the weak cosmic censorship conjecture are valid. It thus important and necessary to study the second law and weak cosmic censorship in the extended phase space. Recently, Ref. [48] investigated the first law, second law, as well as the weak cosmic censorship conjecture in Reissner Nordstr?m-AdS black holes. The authors found that the first law and the weak cosmic censorship conjecture were valid, the second law was violated for the extremal and near-extremal black holes. In this study, we will extend the idea in Ref. [48] to black holes with quintessence dark energy. To this end, we explore the validity of the second law. We find that the second law is violated for the extremal and near-extremal black holes. In contrast to previous research, we investigate the change of the minimum value of the metric function at the higher order in the process of particle absorption. In Ref. [48], the authors assume that $ \mathcal O(\epsilon^2) $ is the higher order terms of $ \epsilon $, such that it can be considered as zero. However, we can not easily ignore the value of the higher order term, as we cannot determine the magnitude of $ \delta_\epsilon $ and $ \mathcal O(\epsilon^2) $. Hence, we have extended the investigation of the changes in the minimum value of the function to the higher order. Moreover, our results show that the weak cosmic censorship conjecture could be violated for the near-extremal black hole in the extended phase space. This result did not appear in previous studies.
The remainder of this article is organized as follows. In Section 2, we briefly review the thermodynamics of the Reissner-Nordstr?m-AdS black hole surrounded by quintessence dark energy. In Section 3, we first establish the first law of thermodynamics in the extended phase space and subsequently discuss the second law and weak cosmic censorship conjecture in this framework. In Section 4, we investigate the first law, second law, and the weak cosmic censorship conjecture in normal phase space. Section 5 presents our conclusions.

The metric of the spherically symmetric charged-AdS black hole surrounded by quintessence dark energy can be written as [49]

$ {\rm d}s^{2} = -f(r){\rm d}t^{2}+f^{-1}(r){\rm d}r^{2}+r^{2}({\rm d}\theta^2+\sin^2\theta {\rm d}\phi^2), $

(1)

with

$ f(r) = 1-\frac{a}{r^{3 \omega+1}}+\frac{r^2}{l^2}-\frac{2 M}{r}+\frac{Q^2}{r^2}, $

(2)

the electric potential of the black hole is

$ \begin{align} A_\mu = \left (-\frac{Q}{r}, 0, 0, 0\right ). \end{align} $

(3)

In the above equation, $ M $ and $ Q $ are the ADM mass and charge of the black hole, $ l $ is the radius of the $ {\rm AdS} $ spacetime, which is related to the cosmological constant, and $ a $ is the normalization factor closely related to quintessence density that should be greater than zero. Several scenarios exist for the value of $ \omega $, namely for $ -1 <\omega<-1/3 $, it is quintessence dark energy, and for $ \omega<-1 $, it is phantom dark energy. The values of state parameter $ \omega $ affect the structure of spacetime. When $ \omega = -1 $, it affects the AdS radius, while when $ \omega = -1/3 $, it affects the curvature $ \kappa $ of space-time [50].
At the event horizon $ r_h $, the Hawking-temperature, Bekenstein-Hawking entropy and electric potential can be expressed as

$ \begin{align} T_h = \frac{f'(r_h)}{4 \pi} = \frac{ r_h\left(3a \omega {\rm{r}}_h{}^{-3 \omega }+3r_h{}^3/l^2+r_h\right)-Q^2}{4\pi {\rm{r}}_h{}^{3 }}, \end{align} $

(4)

$ \begin{align} S_{h} = \pi r_h^2, \end{align} $

(5)

$ \begin{align} \Phi _h = -A_t(r_h) = \frac{ Q}{r_h}. \end{align} $

(6)

In the extended thermodynamic phase space, the cosmological constant plays the role of pressure $ P $ [42,43], and its thermal conjugate variable is the thermodynamic volume $ V $ of the black hole [44,45]. The pressure and volume can be expressed as

$ \begin{align} P = -\frac{\Lambda}{8\pi} = \frac{3}{8\pi l^2},\quad V_h = \left(\frac{\partial M}{\partial P}\right)_{S,Q} = \frac{4\pi {r_h}^3}{3}. \end{align} $

(7)

The first law of thermodynamics thus should be written as [51,52]

$ \begin{align} {\rm d}M = T_{h}{\rm d}S_{h}+\Phi_{h}{\rm d}Q+V_{h}{\rm d}P. \end{align} $

(8)

In this case, the mass is defined as enthalpy. Relations among the enthalpy, internal energy, and pressure are

$ \begin{align} M = U_h+P V_h. \end{align} $

(9)

In the extended phase space, the change of mass will affect not only the event horizon and the electric charge, but also the $ {\rm AdS} $ radius. Therefore, we will investigate the change in the black hole by the charged particle absorption.

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3.1.Energy-momentum relation of the absorbed particle

To obtain the relationship between the conserved quantities of particles in the electric field $ A_\mu $, we will employ the following Hamilton-Jacobi equation to study dynamics of the particles [49]

$ \begin{align} g^{\mu \nu }\left(p_{\mu }-{{eA}}_{\mu }\right)\left(p_{\nu }-{{eA}}_{\nu }\right)+\mu_b ^2 = 0, \end{align} $

(10)

where

$ \begin{align} p_\mu = \partial_\mu S. \end{align} $

(11)

In the above equation, $ \mu_b $ is the mass of the particle, $ p_\mu $ is the momentum, and $ S $ is the Hamilton action of the particle. In the spherically symmetric spacetime, the Hamilton action of a moving particle can be separated into

$ \begin{align} S = -Et+R(r)+H(\theta )+L\phi, \end{align} $

(12)

where $ E $ and $ L $ are the energy and angular momentum. To solve the Hamilton-Jacobi equation, we will use the inverse metric of the black hole as follows

$ \begin{align} g^{\mu \nu } \partial _{\mu }\partial _{\nu } = -\frac{1}{f(r)}\left(\partial _t\right){}^2+f(r)\left(\partial _r\right){}^2+\frac{1}{r^2}\left(\partial _{\theta }\right){}^2+\frac{1}{r^2\sin ^2\theta }\left(\partial _{\phi }\right){}^2. \end{align} $

(13)

Hence, the Hamilton-Jacobi equation changes into

$\begin{split} & - \frac{1}{{f(r)}}{\left( { - E - eA_t} \right)^2} +f(r){\left( {{\partial _r}S(r)} \right)^2} \\ & \quad +\frac{1}{{{r^2}}}{\left( {{\partial _\theta }H(\theta )} \right)^2} + \frac{1}{{{r^2}{{\sin }^2}\theta }}{L^2} + \mu _b^2 = 0 {\text{.}}\\ \end{split} $

(14)

Substituting Equation (12) into Equation (14), we can obtain the radial and angular equations

$ \begin{align} -\frac{r^2}{f (r)} \left(-E -{{eA}}_t\right){}^2+r^2f (r) \left(\partial _rS(r)\right){}^2+r^2\mu_b ^2 = -\mathcal{K}, \end{align} $

(15)

$ \begin{align} \left(\partial _{\theta }H(\theta )\right){}^2+\frac{1}{\sin ^2\theta }L^2 = \mathcal{K}. \end{align} $

(16)

Correspondingly, the radial momentum $ p^r $ and angular momentum $ p^{\theta } $ of the particle can be written as

$ \begin{align} p^r = f(r)\sqrt{\frac{-\mu_b ^2r^2-\mathcal{K}}{r^2f(r)}+\frac{1}{f(r)^2}\left(-E -{eA}_t\right){}^2}, \end{align} $

(17)

$ \begin{align} p^{\theta } = \frac{1}{r^2}\sqrt{\mathcal{\mathcal{K}}-\frac{1}{\sin ^2\theta }L^2}. \end{align} $

(18)

When the black hole completely absorbs a charged particle, the conserved quantity of the particle and the conserved quantity of the black hole are indistinguishable to an observer outside the horizon. By removing the separate variable $ \mathcal{K} $ in Equation (17), we can obtain the relationship between the energy and momentum at any radial location. Near the event horizon, we can obtain

$ \begin{align} E = \frac{Q}{r_h}e+p^r. \end{align} $

(19)

For the $ p^r $ term, we choose the positive sign thereafter, as done in Ref. [53], to ensure a positive time direction.
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3.2.Thermodynamics in extended phase space

In the process of absorption, the energy and electric charge of the particle equal to the change of the internal energy and charge of the black hole, that is

$ \begin{align} E = {\rm d}U_h = {\rm d}(M-P V_h),\quad e = {\rm d}Q. \end{align} $

(20)

In this case, the energy relation of Equation (19) becomes

$ \begin{align} {\rm d}U_h = \frac{Q}{r_h} {\rm d}Q+p^r. \end{align} $

(21)

Moreover, to rewrite Equation (21) to the first law of thermodynamics, we have to find the entropy variation. From Equation (5), we have

$ \begin{align} {\rm d}S_h = 2\pi r_h {\rm d}r_h, \end{align} $

(22)

where $ {\rm d}r_h $ is the variation of the event horizon of the black hole. The absorbed particles also change the function $ f(r) $, the shift of function $ f(r) $ satisfies

$ \begin{split} {\rm d}{f_h} &= \frac{\partial f_h}{\partial M}{{\rm d}M}+\frac{\partial f_h}{\partial Q}{{\rm d}Q}+\frac{\partial f_h}{\partial l}{{\rm d}l}+\frac{\partial f_h}{\partial r_h}{{\rm d}r}_h = 0,\\f_h& = f\left(M,Q,l,r_h\right), \end{split} $

(23)

where

$ \begin{align} &\frac{\partial f_h}{\partial M} = -\frac{2}{r_h}, \quad \frac{\partial f_h}{\partial Q} = \frac{2 Q}{r_h{}^2},\\ &\frac{\partial f_h}{\partial l} = -\frac{2 r_h{}^2}{l^3},\; \frac{\partial f_h}{\partial r_h} = -\frac{2 Q^2}{r_h{}^3}\!+\frac{2 M}{r_h{}^2}\!+\frac{2 r_h}{l^2}-a r_h{}^{-2-3 w} (-1-3 w). \end{align} $

(24)

Combining Equations (21) and (20), the energy relation in the Equation (21) can be rewritten as

$ \begin{align} {\rm d}M-{\rm d}(PV_h) = \frac{Q}{r_h} {\rm d}Q+p^r. \end{align} $

(25)

Combining Equations (23) and (25), we find that all variables except for $ {\rm d}r_h $ and $ p^r $are eliminated, hence we can obtain

$ \begin{align} {{\rm d}r}_h = \frac{2 l^2 p^r r_h{}^{1+3 w}}{r_h{}^{3 w} \left(r_h{}^3-2 l^2 \left(M-r_h\right)\right)+a l^2 (3 w-1)}. \end{align} $

(26)

Based on Equation (26), we can obtain the variation of entropy and volume

$ \begin{align} {{\rm d}S}_h = \frac{4 l^2 \pi p^r r_h{}^{2+3 w}}{r_h{}^{3 w} \left(r_h{}^3-2 l^2 \left(M-r_h\right)\right)+a l^2 (3 w-1)}, \end{align} $

(27)

$ \begin{align} {{\rm d}V}_h = \frac{8 l^2 \pi p^r r_h{}^{3+3 w}}{r_h{}^{3 w} \left(r_h{}^3-2 l^2 \left(M-r_h\right)\right)+a l^2 (3 w-1)}. \end{align} $

(28)

Incorporating Equations (4), (27), (7) and (28), we lastly obtain

$ \begin{align} T_h {{\rm d}S}_h-{P{\rm d}V}_h = p^r. \end{align} $

(29)

Moreover, incorporating Equations (4), (5), (6), (7), (27) and (28), the energy relation of Equation (21) can be expressed as

$ \begin{align} {\rm d}U_h = \Phi_h {\rm d}Q + T_h {\rm d}S_h-P {\rm d}V_h. \end{align} $

(30)

Because the mass of the black hole has been defined as enthalpy, the relation between the internal energy and enthalpy can be rewritten in the extended phase space as

$ \begin{align} {\rm d}M = T_h {\rm d}S_h+\Phi_h {\rm d}Q+V_h {\rm d}P. \end{align} $

(31)

Here, we demonstrate that the first law of thermodynamics is still satisfied for the black hole surrounded by quintessence dark energy under the charged particle absorption.
As the absorption is an irreversible process, the entropy of the final state should be greater than the initial state of the black hole. Hence, under the charged particle absorption, the variation of the entropy is $ {\rm d}S > 0 $. Subsequently, we will test the validity of the second law of thermodynamics with Equation (27).
We first study the case of the extremal black holes, for which the temperature is zero. On the basis of this fact and Equation (27), we can obtain

$ \begin{align} {\rm d}S_h = -\frac{4\pi p^r l^2}{3r_h}. \end{align} $

(32)

There is a minus sign in Equation (32). Therefore, the entropy is decreased for the extremal black hole. Hence, the second law of thermodynamics is violated for the extremal black hole in the extended phase space. Furthermore, it is worth noting that the parameters $ a $ and $ \omega $ are not present in the above equation. In other words, the violations about the second law do not depend on the parameters $ a $ and $ \omega $.
Here, we focus on the near-extremal black hole. We will verify whether the second law is valid in the extended phase space by numerically studying the variation of entropy. We set $ M = 0.5 $ and $ l = p^r = 1 $. For the case $ \omega = -2/3 $ and $ a = 1/3 $, we find the extremal charge is $ Q_e = 0.48725900857 $. In the case that the charge is less than the extremal charge, we take different charge values to produce the variation of entropy. In Table 1, we provide the numeric results of $ r_h $ and $ {\rm d}S $ for different charges. In Table 1, when the charge $ Q $ of the black hole decreases, the event horizon of the black hole increases along with the variation of entropy. Interestingly, there are two regions where the entropy increases are $ {\rm d}S > 0 $ and $ {\rm d}S < 0 $. This indicates that there exists a phase transition point that divides the variation of entropy into positive and negative values.

Q	rh	dSh
0.48725900857	0.432041	?9.69538
0.487259	0.432111	?9.70328
0.48	0.494855	?22.9663
0.46	0.567768	?1006.44
0.445	0.5757	386.566
0.4	0.628743	44.8667
0.3	0.695449	24.9494
0.2	0.731872	21.0262
0.1	0.751055	19.5985

Table1.Relation between dS_h, Q and r_h .

We can also obtain the relation between $ {\rm d}S $ and $ r_h $, which is shown in Fig. 1. From Fig. 1, we can clearly see that a phase transition point exists making $ {\rm d}S_h $ positive and negative. When the electric charge is close to the extremal charge, the variation of entropy is negative, whereas when the electric charge is far away from the extremal charge, the entropy increases. Therefore, the second law of thermodynamics is violated for near-extremal black holes and valid for the far-extremal black holes in the extended phase space.
Figure1. Relation between $ {\rm d}S_h $ and $ r_h $, whose parameter values are $ M = 0.5, l = p^r = 1 $ and $ \omega = -2/3, a = 1/3 .$

In Fig. 2 and Table 2, we set $ \omega = -1/2 $ and $ a = 1/2 $. We want to explore whether the values of state parameter of dark energy affect the laws of thermodynamics. For $ \omega = -1/2 $ and $ a = 1/2 $, the extremal charge is $ Q_e = $$ 0.525694072 $. Form Fig. 2 and Table 2, we also find that the second law of thermodynamics fails for near-extremal black holes when a particle is absorbed by the black hole. Furthermore, by comparing Fig. 1 and Fig. 2, we find that the magnitudes of the violations are related to values of the parameters $ \omega $ and $ a $, however parameters $ \omega $ and $ a $ do not determine whether the second law of thermodynamics will eventually be violated.
Figure2. Relation between $ {\rm d}S_h $ and $ r_h $, whose parameter values are $ M = 0.5, l = p^r = 1 $ and $ \omega = -1/2 $, $ a = 1/2 .$

Q	rh	dSh
0.525694072	0.483844	?8.65919
0.525	0.504022	?10,2582
0.5	0.599642	?27.2971
0.43	0.69007	?766.564
0.425	0.694382	20923.7
0.4	0.713732	171.512
0.3	0.767156	50.6936
0.2	0.798062	37.5343
0.1	0.814681	33.2405

Table2.Relation between dS_h, Q and r_h.

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3.3.Weak cosmic censorship conjecture in the extended phase space

In the extended phase space with consideration of the thermodynamic volume, although the particle absorption is an irreversible process, the second law of thermodynamics for the extremal and near-extremal black holes are violated. The definition of entropy is $ S_h = \dfrac{A_h}{4} $, and the event horizon of black holes is closely relevant to entropy. Because of this violation of the second law of thermodynamics, the case considering thermodynamic volume is considered. This implies that, the violation of the second law of thermodynamics can be related to the weak cosmic censorship conjecture, which is related to the stability of the event horizon. Therefore, it is necessary to verify the validity of the weak cosmic censorship conjecture in these cases.
If the event horizon cannot wrap the singularity after the charged particle is absorbed, the weak cosmic censorship conjecture will be invalid. Hence, the event horizon should exist to assure the validity of the weak cosmic censorship conjecture. We will verify whether there is an event horizon after the charged particle is absorbed by the black hole. Further, we pay attention to how $ f(r) $ changes. The function $ f(r) $ has a minimum point at $ r_{\min } $. There are three possibilities. For the case $ f(r)_{\min }<0 $, there are two roots of $ f(r) $. Then we have an usual black hole with $ {r_+} $ and $ {r_-} $, as the inner and outer horizon. For the case $ f(r)_{\min } = 0 $, the two roots coincide, and the black hole becomes an extremal black hole. For the case $ f(r)_{\min }>0 $, the function $ f(r) $ has no real root, and there is no event horizon. At $ r_{\min } $, we have

$ \begin{split} &f(M,Q,l,r)|_{r = r_{\min }}\equiv f_{\min } = \delta\leqslant 0,\\ & \partial_{r} f(M,Q,l,r)|_{r = r_{\min }}\equiv f'_{\min } = 0, \end{split} $

(33)

and

$ \begin{align} (\partial_{r})^2 f(M,Q,l,r)|_{r = r_{\min }}>0. \end{align} $

(34)

For the extremal black hole, $ \delta = 0 $, $ r_h $ and $ r_{\min } $ are coincident. For the near-extremal black hole, $ \delta $ is a small quantity. When the black hole absorbs a charged particle, there is an infinitesimal change in the mass, charge, and AdS radius, which are $ (M+{\rm d}M, ~ Q+{\rm d}Q, ~ l+{\rm d}l) $. Because of these changes, there are also movements for the minimum value and event horizon of the black hole, namely $ r_{\min }\rightarrow r_{\min }+{\rm d}r_{\min } $ and $ r_h\rightarrow r_h+{\rm d}r_h $. At the new minimum point, the function $ f(r) $ satisfies

$ \begin{align} \partial_{r} f(M+{\rm d}M,Q+{\rm d}Q,l+{\rm d}l,r)|_{r = r_{\min }+{\rm d}r_{\min }} = f'_{\min }+{\rm d}f'_{\min } = 0. \end{align} $

(35)

Using the known condition $ f'_{\min } = 0 $, we can obtain $ {\rm d}f'_{\min } = 0 $. Expanding it further, we obtain

$ \begin{align} {\rm d}f'_{\min } = \frac{\partial f'_{\min }}{\partial M}{\rm d}M+\frac{\partial f'_{\min }}{\partial Q}{\rm d}Q+\frac{\partial f'_{\min }}{\partial l}{\rm d}l+\frac{\partial f'_{\min }}{\partial r_{\min }}{\rm d}r_{\min } = 0. \end{align} $

(36)

At $ r_{\min }+{\rm d}r_{\min } $, the function $ f(r) $ takes the form as

$ \begin{split} &f(M+{\rm d}M,Q+{\rm d}Q,l+{\rm d}l,r)|_{r = r_{\min }+{\rm d}r_{\min }} = f_{\min }+{\rm d}f_{\min }\\ & = \delta+\left(\frac{\partial f_{\min }}{\partial M}{\rm d}M+\frac{\partial f_{\min }}{\partial Q}{\rm d}Q+\frac{\partial f_{\min }}{\partial l}{\rm d}l\right). \end{split} $

(37)

For the extremal black hole, we know $ f'_{\min } = 0 $ and $ f_{\min } = \delta = 0 $. Substituting Equation (25) into Equation (37), we obtain

$ \begin{align} {\rm d}f_{\min } = -\frac{2 p^r}{r_{\min }}-\frac{3 r_{\min } {\rm d}r_{\min } }{l^2}. \end{align} $

(38)

Form Equations (29) and (38), we find

$ \begin{align} {\rm d}f_{\min } = 0. \end{align} $

(39)

For the near-extremal black hole, Equation (25) is not applicable any longer. However, we can expand it near the minimum point, as there is a relation $ r_h = r_{\min }+\epsilon $. To the first order, we find

$ \begin{split} {\rm d}M = &\frac{r_{\min }^{-2-3 \omega } {{\rm d}r}_{\min }\left({al}^3 r_{\min }+3 a l^3 \omega r_{\min }-2 l^3 Q^2 r_{\min }^{3 \omega }+2 l^3 M r_{\min }^{1+3 \omega }+2 l r_{\min }^{4+3 \omega }\right)}{2 l^3}+\frac{r_{\min }^{-2-3 \omega }\left(2 l^3 Q r_{\min }^{1+3 \omega }{{\rm d}Q}-2r_{\min }^{5+3 \omega }{{\rm d}l}\right)}{2 l^3} \end{split} $

$ \begin{split} & -\frac{r_{\min }^{-3-3 \omega }{{\rm d}r}_{\min } \left(a l^3 r_{\min }+6 a l^3 \omega r_{\min }+9 a l^3 \omega ^2 r_{\min }\right) \epsilon }{2 l^3} -\frac{r_{\min }^{-3-3 \omega }{{\rm d}r}_{\min } \left(-4 l^3 Q^2 r_{\min }^{3 \omega }+2 l^3 M r_{\min }^{1+3 \omega }-4 l r_{\min }^{4+3 \omega }\right) \epsilon }{2 l^3} \\ & -\frac{r_{\min }^{-3-3 \omega }\left(2 l^3 Q r_{\min }^{1+3 \omega }{{\rm d}Q}+6 r_{\min }^{5+3 \omega }{{\rm d}l}\right) \epsilon }{2 l^3}+\mathcal O(\epsilon )^2. \end{split} $

(40)

Substituting Equation (40) into (37), we have

$ \begin{align} {{\rm d}f}_{\min } \!\!= &\frac{2 Q {{\rm d}Q}}{r_{\min }^2}\!-\!\frac{2 r_{\min }^2{{\rm d}l}}{l^3}\!-\!\frac{r_{\min }^{-3-3\omega } \left(2 l^3 Q r_{\min }^{1+3 \omega }{{\rm d}Q}-2 r_{\min }^{5+3\omega }{{\rm d}l}\right)}{l^3} \!-\!\frac{r_{\min }^{-3-3 \omega }{{\rm d}r}_{\min }\left({al}^3r_{\min }+3a l^3\omega r_{\min }\!-\!2 l^3 Q^2 r_{\min }^{3 \omega }\!+\!2 l^3 M r_{\min }^{1+3 \omega }\!+\!2 l r_{\min }^{4+3 \omega }\right)}{l^3} \\ & +\frac{r_{\min }^{-4-3 \omega } {{\rm d}r}_{\min }\left(a l^3 r_{\min }+6 a l^3 \omega r_{\min }+9 a l^3 \omega ^2 r_{\min }\right) \epsilon }{l^3}+\frac{r_{\min }^{-4-3 \omega } {{\rm d}r}_{\min }\left(-4 l^3 Q^2 r_{\min }^{3 \omega }+2 l^3 M r_{\min }^{1+3 \omega }-4 l r_{\min }^{4+3 \omega }\right) \epsilon }{l^3} \\ & +\frac{r_{\min }^{-4-3 \omega } \left(2 l^3 Q r_{\min }^{1+3 \omega }{{\rm d}Q} +6 r_{\min }^{5+3 \omega } {{\rm d}l}\right) \epsilon }{l^3}+\mathcal O(\epsilon )^2. \end{align} $

(41)

For the equation $ f(r_h) = 0 $, which we can also expand and solve, then we find

$ \begin{align} l = \frac{\sqrt{3} r_{\min }^{\frac{1}{2} (4+3 \omega )}}{\sqrt{-3 a \omega r_{\min }+Q^2 r_{\min }^{3 \omega }-r_{\min }^{2+3 \omega }}}, \end{align} $

(42)

Differentiating both sides of this equation, we can further obtain

$ \begin{align} {{\rm d}l} =& -\frac{\sqrt{3} Q r_{\min }^{3 \omega +\frac{1}{2} (4+3 \omega )}{{\rm d}Q}}{\left(-3 a \omega r_{\min }+Q^2 r_{\min }^{3 \omega }-r_{\min }^{2+3 \omega }\right){}^{3/2}} \\ & +\frac{\sqrt{3} (4+3 \omega ) r_{\min }^{-1+\frac{1}{2} (4+3 \omega )}{{\rm d}r}_{\min }}{2 \sqrt{-3 a \omega r_{\min }+Q^2 r_{\min }^{3 \omega }-r_{\min }^{2+3 \omega }}} \\ &\!\! -\!\frac{\sqrt{3} r_{\min }^{\frac{1}{2} (4+3 \omega )} \left(-3 a \omega \!+\!3 Q^2 \omega r_{\min }^{-1+3 \omega }\!-\!(2+3 \omega ) r_{\min }^{1+3 \omega }\right){{\rm d}r}_{\min }}{2 \left(-3 a \omega r_{\min }+Q^2 r_{\min }^{3 \omega }\!-\!r_{\min }^{2+3 \omega }\right){}^{3/2}}. \end{align} $

(43)

Substituting Equations (43) and (42) into Equation (41), we find

$ \begin{align} {\rm d}f_{\min } = \mathcal O(\epsilon^2 ), \end{align} $

(44)

then, we can obtain

$ \begin{align} f_{\min }+{\rm d}f_{\min } = \delta +\mathcal O(\epsilon^2 ). \end{align} $

(45)

In Ref. [48], $ \delta $ is a very small negative value and $ \epsilon\ll 1 $. When we chose $ \delta = \epsilon = 0 $, the Equation (45) returns to the form of the extremal black hole. This result also proves the correctness of the Equation (39) from the side. Hence, the authors thought that the extremal and near-extremal black holes remain at their minimum. For the near-extremal black hole, they ignore the $ O(\epsilon^2 ) $ due to it is high order small value, such that the weak cosmic censorship is also valid for the case of the near-extremal black hole. However, because we do not know the magnitude of $ |\delta | $ and $ O(\epsilon^2 ) $, we can not easily ignore $ O(\epsilon^2 ) $. Therefore, it is important to determine whether there is a relationship $ |\delta |>\mathcal O(\epsilon^2 ) $. For the near-extremal black hole, the minimum point is located slightly to the left of the outer horizon. The value of the minimum is obtained at $ r_{\min }+\epsilon $ as

$ \begin{align} &f{\left(r_{\min }+\epsilon \right)} = \left(1-\frac{Q^2}{r_{\min }^2}+\frac{3 r_{\min }^2}{l^2}+3 a \omega r_{\min }^{-1-3 \omega }\right) \\ &+\left(\frac{3}{l^2}+\frac{1}{2} r_{\min }^{-4-3 \omega } \left(-3 a \omega (1+3 \omega ) r_{\min }+2 Q^2 r_{\min }^{3 \omega }\right)\right) \epsilon ^2+\mathcal O(\epsilon^3), \end{align} $

(46)

where we skip to write the constant term. Namely

$ \begin{align} \delta = -\left(\frac{3}{l^2}+\frac{1}{2} r_{\min }^{-4-3 \omega } \left(-3 a \omega (1+3 \omega ) r_{\min }\!+\!2 Q^2 r_{\min }^{3 \omega }\right)\right) \epsilon ^2\!-\!\mathcal O(\epsilon^3). \end{align} $

(47)

We can also obtain

$ {\rm d}f_{\min} = \left( \begin{gathered} \frac{{\left( {8{Q^2} - 2{r^2_{{{\min }}}}} \right){\rm d}{r_{\min }} - 4Q{r_{\min }}{\rm d}Q}}{{{r^5_{{{\min }}}}}} - \frac{{27}}{2}a\omega {(1 + \omega )^2}{r^{ - 4 - 3\omega }_{{{\min }}}}{\rm d}{r_{\min }} \end{gathered} \right){\epsilon ^2} + \mathcal{O}({\epsilon ^3}).$

(48)

Equation (45) can be represented as

$ \begin{split} f{\left(r_{\min }+{\rm d}r_{\min } \right)} = & \left(\frac{\left(8 Q^2 -2 r_{\min }^2\right){{\rm d}r}_{\min }-4 Q r_{\min }{{\rm d}Q}}{r_{\min }^5}-\frac{27}{2} a \omega (1+\omega )^2{ }r_{\min }^{-4-3 \omega }{{\rm d}r}_{\min }\right)\epsilon ^2 \\ & -\left(\frac{3}{l^2}+\frac{1}{2} r_{\min }^{-4-3 \omega } \left(-3 a \omega (1+3 \omega ) r_{\min }+2 Q^2 r_{\min }^{3 \omega }\right)\right) \epsilon ^2. \end{split} $

(49)

Here, we can have $ \mathcal F = \frac{f{\left(r_{\min }+{{\rm d}r}_{\min }\right)}}{\epsilon ^2} $ and $ e = {\rm d}Q $. Then, the above equation becomes

$ \begin{split} \mathcal F \!\!=&\frac{\left(8 Q^2 -2{ }r_{\min }^2\right){{\rm d}r}_{\min }-4 Q r_{\min }e}{r_{\min }^5} \!-\!\frac{27}{2} a \omega (1+\omega )^2{ }r_{\min }^{-4-3 \omega }{{\rm d}r}_{\min }\\ & -\frac{1}{2} r_{\min }^{-4-3 \omega } \left(-3 a \omega (1+3 \omega ) r_{\min }+2 Q^2 r_{\min }^{3 \omega }\right)-\frac{3}{l^2}, \end{split} $

(50)

Obviously, from above equation we find that the values of $ \mathcal F $ are linked with the values of the parameters $ e, a, l, r_{\min}, \omega, Q, {\rm d}r_{\min} $. Mostly, we find that $ e $ has a more direct affect on $ \mathcal F $. To gain an intuitive understanding, we provide the plots in Fig. 3.
Figure3. (color online) Value of $ \mathcal F $ for $ l = 1, Q = 0.5, e = 0.2 .$

When we assume different values of these parameters, the region where $ \mathcal F $ is greater than zero always appears in the graph. $ \mathcal F>0 $, hence, $ f{\left(r_{\min }+{\rm d}r_{\min } \right)} > 0 $. Thus, the positive region shows that there exists a range of the particle charges, which allows us to overcharge black holes into naked singularities. Similarly, for different values of the parameters $ a, l, r_{\min}, \omega, Q, {\rm d}r_{\min} $, the configurations of $ \mathcal F $ are different, hence, the violation about the weak cosmic censorship depends on the parameters, and the magnitude of the violation is related to the parameters. In particular, when we take $ e < <1 $, as shown in Fig. 4, we find the same violation. However, the degree of violation differs with the change of the charge $ e .$
Figure4. (color online) Value of $ \mathcal F $ for $ l = 1, Q = 0.5, e = 0.00005 .$

In the extended phase space, we find that the change of values of $ f(r_{\min }) $ vanishes always after the charged particle is absorbed for the extremal black hole. Hence, for the extremal black holes, the function $ f(r) $ always has a root. Therefore, the black hole has an event horizon covering its singularity. The weak cosmic censorship conjecture thus is valid for the configuration of the black hole that does not change. However, we derive that the near-extremal black holes can be overcharged by absorbing charged particles. Hence, the cosmic censorship conjecture would be violated for near-extremal black holes in the extended phase space.
When $ a $ is zero, the black hole becomes RN-AdS black holes. Hence, we set $ a = 0 $, and the result is shown in Fig. 5.
Figure5. (color online) Value of $ \mathcal F $ for $ l = 1, Q = 0.5, e = 0.2 .$

Interestingly, the results show that not only $ \mathcal F<0 $ but also $ \mathcal F>0 $, which means that the weak cosmic censorship may be invalid for the RN-AdS black holes, and the results are somewhat different from those obtained in previous studies [48] on the weak cosmic censorship. The reason is that they did not consider high-order corrections to the energy.

In this section, we mainly discuss the laws of thermodynamics and examine the weak cosmic censorship conjecture of the black hole in normal phase space. We want to explore whether the phase space affects thermodynamics and the weak cosmic censorship conjecture.
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4.1.Thermodynamics in normal phase space

In normal phase space, the mass $ M $ of the black hole is defined as energy. Meanwhile, we assume that there is no energy loss in the process of particle absorption. As the charged particle is absorbed by the black hole, the change of the internal energy and charge of the black hole satisfy

$ \begin{align} E = {\rm d}M, \quad e = {\rm d}Q. \end{align} $

(51)

Then, Equation (19) can be rewritten as

$ \begin{align} {\rm d}M = \frac{Q}{r_h} {\rm d}Q+p^r. \end{align} $

(52)

Because of the absorption of a charged particle, the event horizon and function $ f(r_h) $ will change, and there is always a relation

$ \begin{align} {\rm d}f_h = \frac{\partial f_h}{\partial M}{{\rm d}M}+\frac{\partial f_h}{\partial Q}{{\rm d}Q}+\frac{\partial f_h}{\partial r_h}{{\rm d}r}_h = 0. \end{align} $

(53)

To eliminate the $ {\rm d}M $ term, we can combine Equations (52) and (53). We find that the $ {\rm d}Q $ term is also eliminated in this process, and we lastly obtain

$ \begin{align} {{\rm d}r}_h = \frac{2 l^2 p^r r_h{}^2}{2l^2 r_h{}^2-2Ml^2 r_h+4r_h{}^4+a l^2r_h{}^{-(3\omega -1)} (3\omega -1)}. \end{align} $

(54)

After substituting Equation (54) into Equation (22), we have

$ \begin{align} {{\rm d}S}_h = \frac{4 \pi l^2 p^r r_h^3}{2l^2 r_h{}^2-2Ml^2 r_h+4r_h{}^4+a l^2 r_h{}^{-(3\omega -1)} (3\omega -1)}. \end{align} $

(55)

Form Equations (4) and (55), we obtain

$ \begin{align} T_h {{\rm d}S}_h = p^r. \end{align} $

(56)

Combining Equations (4), (6) and (55), we obtain

$ \begin{align} {{\rm d}M} = T_h {{\rm d}S}_h+\Phi _h{{\rm d}Q}. \end{align} $

(57)

In normal phase space, the first law of thermodynamics of the black hole surrounded by quintessence dark energy is valid when a particle is absorbed by the black hole.
With Equation (55), we also can investigate the second law of thermodynamics. For the extremal black holes, we find that the variation of the entropy is divergent, which is meaningless. Thus, we are interested in the near-extremal black hole thereafter. We also set $ M = 0.5 $ and $ l = p^r = 1 $. For case of $ \omega = -1/2 $ and $ a = 1/2 $, we obtain the extremal charge $ Q_e = 0.525694072 $. In the case that the charge is less than the extreme charge, we obtain values of $ r_h $ and $ {\rm d}S_h $ for different charges in Table 3. From this table, it can be clearly seen that when the charge is smaller than the extremal charge, while the variation of the entropy is always positive. In Fig. 6, we present the relation between $ {\rm d}S_h $ and $ r_h $, and the entropy increases too. Therefore, the second law of thermodynamics is valid in normal phase space.

Q	rh	dSh
0.525694072	0.483844	39998.1
0.525	0.504022	43.7772
0.5	0.599642	9.3879
0.4	0.713732	5.67468
0.3	0.767156	4.92923
0.2	0.798062	4.60478
0.1	0.814681	4.45287

Table3.Relation between dS_h, Q and r_h.

Figure6. Relation between $ {\rm d}S_h $ and $ r_h $, whose parameter values are $ a = 1, M = 0.5, l = p^r = 1 .$

For the case $ \omega = -2/3, a = 1/3 $, we find the extremal charge is $ Q_e = 0.48725900875 $. the values of $ r_h $ and $ {\rm d}S_h $ for a different charge are given in Fig. 7 and Table 4 From these, we also find that the entropy increases, implying that the second law is valid.
Figure7. Relation between $ {\rm d}S_h $ and $ r_h $, whose parameter values are $ a = 1/3, M = 0.5, l = p^r = 1 .$

Q	rh	dSh
0.48725900875	0.432041	3.47995×106
0.487259	0.434211	322.509
0.48725	0.434319	307.346
0.4	0.628743	5.80081
0.3	0.695450	4.85183
0.2	0.731837	4.49911
0.1	0.751055	4.34168

Table4.Relation between dS_h, Q and r_h.

Thus far, both the first and second laws of thermodynamics hold in normal phase space for the black hole surrounded by quintessence dark energy under charged particle absorption.
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4.2.Weak cosmic censorship conjecture in normal phase space

In normal phase, the examination of the validity of the weak cosmic censorship conjecture should also return to the value of the function $ f(r_{\min }) $. Similarly, we study how $ f(r_{\min }) $ changes as charged particles are absorbed. At $ r_{\min }+{{\rm d}r}_{\min } $, there is also a relation $ \partial _r f(r_{\min }+{{\rm d}r}_{\min }) = 0 $, implying

$ \begin{align} {{\rm d}f}'_{\min } = \frac{\partial f'_{\min }}{\partial M} {\rm d}M+\frac{\partial f'_{\min }}{\partial Q} {\rm d}Q+\frac{\partial f'_{\min }}{\partial r_{\min }} {\rm d}r_{\min } = 0. \end{align} $

(58)

In addition, at the new minimum, $ f\left(r_{\min }+{\rm d}r_{\min }\right) $ can be expressed as

$ \begin{align} f\left(r_{\min }+{\rm d}r_{\min }\right) = f_{\min }+{\rm d}f_{\min }, \end{align} $

(59)

where

$ \begin{align} {\rm d}f_{\min } = \frac{\partial f_{\min }}{\partial M} {\rm d}M+\frac{\partial f_{\min }}{\partial Q} {\rm d}Q. \end{align} $

(60)

For the extremal black holes, Equation (52) can be applied. In this case, we have $ f_{\min } = \delta = 0 $, inserting Equation (52) into Equation (60), we can finally obtain

$ \begin{align} {\rm d}f_{\min } = -\frac{2 p^r}{r_{\min }}. \end{align} $

(61)

When it is an extreme black hole, $ r_h $ and $ r_{\min } $ are tightly coincident. In addition, we have $ T_h = 0 $. Incorporating Equation (61) and Equation (56), the minimum value of $ f\left(r_{\min }+{\rm d}r_{\min }\right) $ becomes

$ \begin{align} f\left(r_{\min }+{\rm d}r_{\min }\right) = 0, \end{align} $

(62)

which shows that $ f_{\min }+{\rm d}f_{\min } = 0 $, such that the charged particle does not change the minimum value. Therefore, the weak cosmic censorship conjecture is valid for the extremal black hole surrounded by quintessence dark energy. This result is the same as that in the extended phase space, the configuration of the black hole has not changed after the absorption, and the extremal black hole is still an extremal black hole.
For the near-extremal black hole, Equation (52) can not be used. With the condition $ r_h = r_{\min }+\epsilon $, we can expand Equation (52) at $ r_{\min } $, which leads to

$ \begin{split} {\rm d}M =& \frac{Q}{{{r_{\min }}}}{\rm d}Q + \left( {{Q^2}{\rm d}{r_{\min }} + \frac{3}{2}{r_{\min }}\left( {\frac{{2{r_{\min }}^3}}{{{l^2}}} - a{r_{\min }}^{ - 3\omega }\omega (1 + 3\omega )} \right){\rm d}{r_{\min }} - Q{r_{\min }}{\rm d}Q} \right){r_{\min }}^{ - 3} \hfill \\ & + \frac{1}{4}{r_{\min }}^{ - 4 - 3\omega } \left( {4Q{r_{\min }}^{1 + 3\omega }{\rm d}Q + {\rm d}{r_{\min }}\left( { - 8{Q^2}{r_{\min }}^{3\omega } + 9a{r_{\min }}\omega \left( {1 + 4\omega + 3{\omega ^2}} \right)} \right)} \right){\epsilon ^2} + \mathcal{O}({\epsilon ^3}). \hfill \\ \end{split}$

(63)

By combining Equations (60) and (63) we have

$ \begin{split} {\rm d}f_{\min } =& \frac{1}{2}{ }r_{\min }{}^{-4-3 \omega } \left(4 r_{\min }{}^{1+3 \omega }-9 a \omega \left(2 \omega +3 \omega ^2-1\right)\right)\\ & \times \epsilon ^2{{\rm d}r}_{\min } +\mathcal O(\epsilon^3). \end{split} $

(64)

We performed the same calculation as in the extended phase space. For the near-extremal black hole, we also can define $ \mathcal F_N = \frac{f{\left(r_{\min }+{{\rm d}r}_{\min }\right)}}{\epsilon ^2} $. Hence, we can obtain the expression of $ \mathcal F_N $ in the normal phase as

$ \begin{align} \mathcal F_N = \frac{4 r_{\min } {{\rm d}r}_{\min }-2 Q^2+3 a r_{\min }{}^{-3 w} \omega \left(r_{\min }+3 r_{\min } \omega -3 {{\rm d}r}_{\min } (1+\omega ) (3 \omega -1)\right)}{2 r_{\min }{}^4}-\frac{3}{l^2}, \end{align} $

(65)

From Equation (65), it is not easily determine the value of $ \mathcal F_N $ as positive or negative. Similarly with the extended phase space, we still have Fig. 8 and Fig. 9.
Figure8. (color online) Value of $ \mathcal F_N $ for $ Q = 0.5, l = a = 1 $ and $ \omega = -0.5 .$

Figure9. (color online) Value of $ \mathcal F_N $ for $ Q = 0.5, l = a = 1 $ and $ \omega = -2/3 .$

In Fig. 8 and Fig. 9, there is no region where the value of $ \mathcal F_N $ is positive. Concerning this result, it is important to note that the conclusion is very different from the extended phase space. In other words, we always have $ f\left(r_{\min }+{\rm d}r_{\min }\right) < 0 $ for the near-extremal black hole in normal phase space. Therefore, the weak cosmic censorship conjecture of the near-extremal black hole surrounded by quintessence dark energy is valid under charged particle absorption in normal phase space.

In the extended phase space, the cosmological parameter has been set to be a dynamic variable and interpreted as the pressure. When the charged particle is absorbed by the black hole surrounded with quintessence dark energy, the change of the energy and the charge of the black hole were supposed to be equal to the conserved quantity of the charged particle. In this case, we found that the first law of thermodynamics was completely valid, however the second law of thermodynamics was violated for extremal and near-extremal black holes. We also studied the validity of the weak cosmic censorship conjecture. We mainly calculated the change of the minimum value of the function $ f(M,Q,l,r) $ under the charged particle absorption by studying the minimum value of the function $ f(M,Q,l,r) $. First, we calculated the case of the extremal black holes. We found that the minimum value of the function was not changed under the charged particle absorption. Thus, after the charged particle absorption, the extremal black hole stays extremal. Therefore, the weak cosmic censorship conjecture is valid for the extremal black hole in the extended phase space. However, our results show that the minimum value of the function under the absorption would be greater than zero. The near-extremal black hole could be overcharged, which was different from the extremal black hole. Hence, the cosmic censorship conjecture would be violated for the near-extremal black hole surrounded by quintessence dark energy after absorbing the charged particles in the extended phase space.
In this study, the results for the violation of the weak cosmic censorship conjecture in the extended phase space are dissimilar from previous conclusions. In previous research [48], the second order $ \mathcal O(\epsilon^2) $ of $ {\rm d}f_{\min} $ was regarded as a very small quantity, and the contribution of $ \mathcal O(\epsilon^2) $ to $ {\rm d}f_{\min} $was neglected. Here, we made precise comparisons and calculations at the minimum value of the function, and we obtained the magnitudes relationship between $ \delta $ and $ \mathcal O(\epsilon^2) $. We found there was always a case of $ f(r_{\min}+{\rm d}r_{\min}) > 0 $, which means that the weak cosmic censorship conjecture may be violated. We still observe such violations while assuming $ e<<1 $. Interestingly, when we take $ a = 0 $, the black hole returns to the RN-AdS black hole, and the cosmic censorship conjecture was also invalid, which is different from the result obtained in Ref. [48].
In normal phase space, the cosmological constant is definite. We found that the first and second laws of thermodynamics are both satisfied under the particle absorption. Moreover, the weak cosmic censorship conjecture was also confirmed. For the case of the extremal black hole, we found that the configuration of the black hole surrounded by quintessence dark energy did not change after absorbing charged particles, which is the same as in the extended phase space. The result implies that the extremal black hole cannot be overcharged in normal phase space. Interestingly, for the case of near-extremal black hole, the minimum value of the function is still negative when the particle is absorbs into the black hole, which is a different form than that in the extended phase space. Thus, the weak cosmic censorship conjecture is valid for the near-extremal black holes in normal phase space.