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Thermodynamics and weak cosmic censorship conjecture in Born-Infeld-anti-de Sitter black holes

We examine the laws of thermodynamics and the weak cosmic censorship conjecture in the normal and extended phase space of Born-Infeld-anti-de Sitter black hole by considering a charged particle absorption. In the normal phase space, the first and second law of thermodynamics as well as the weak cosmic censorship are still valid. However, in the extended phase space, the second law of thermodynamics is violated for double-horizon black holes and part of single-horizon black holes. The first law of thermodynamics and the weak cosmic censorship conjecture are still valid for all black holes. In addition, we find that the shift of the metric function, which determines the locations of the horizons, takes the same form at the minimum point in both the normal and extended phase space, indicating that the weak cosmic censorship conjecture is independent of the thermodynamic phase space.


Introduction
Since the pioneering work of Bekenstein [1,2] and Hawking [3,4], black hole thermodynamics has become a fundamental topic in general relativity. Temperature and entropy in the normal thermodynamical systems are related to the horizon of a black hole. Until now, there have been numerous methods to derive the temperature and entropy of a black hole [5][6][7][8][9][10][11]. It was found that the temperature T of the black hole was related to the surface gravity κ at the horizon, i.e., T = κ/(2π), and the entropy was proportional to the area of the horizon. The horizon is important not only for the thermodynamics, but also for the causality of the spacetime. If there is no horizon, the singularity will be naked for an infinity observer which makes the causality break down. But due to Penrose [12], all singularities arising from gravitational collapse must be hidden by the black hole horizons, which is the so-called "weak cosmic censorship conjecture".
The laws of thermodynamics and weak cosmic censorship conjecture can be tested from a charged particle absorption by a black hole. As a particle drops into the black hole, it has been shown that the first law and the second law of black hole thermodynamics still hold [13,14]. While for the weak cosmic censorship conjecture, there are still some debates. Wald proposed firstly a gedanken experiment to check this conjecture in the extremal Kerr-Newman black hole, which showed that no violations occur as a particle is thrown into a black hole [15]. Nevertheless, the conjecture would be violated in the near-extremal Reissner-Nordstrom [16] black hole and near-extremal Kerr black hole [17]. Later it was found that as the backreaction and self-force effects are taken into account, particles may be escaped from black holes and naked singularities can be avoided [18]. There have been many works discussing about the validity of weak cosmic censorship conjecture in various black hole backgrounds under charged or spinning particles absorption [19][20][21][22][23][24][25][26][27][28][29]. Now we know that including the self-force effect or backreaction, both the extremal black holes and nearextremal black holes cannot be overcharged. This result was confirmed again by considering the second-order corrections to mass recently [30][31][32][33]. It should be stressed that there is also a counter example to the weak cosmic censorship conjecture in four-dimensional anti de Sitter(AdS) space-time [34], in which the curvature grows without bound in the future, leaving regions of spacetime with arbitrarily large curvatures naked to the infinite boundary observers.
Thermodynamics in AdS space now prevails. One possible reason is the application of the AdS/CFT correspondence [35][36][37][38], which relates the gravity theory in D-dimensional AdS spacetime to the conformal field theory in (D − 1)-dimension. Under this duality, the temperature of the black holes is dual to the temperature of the conformal field theory. Currently, there are various applications of AdS/CFT duality, such as holographic superconductors [39,40], holographic Fermi/non-Fermi liquids [41][42][43], and so on [44][45][46][47][48][49][50][51][52][53][54]. Another possible reason that the AdS space is popular is that the phase structures of the AdS sapcetime is more abundant, such as the Hawking-Page phase transition [55], Van der Waals-like phase transition [56,57]. The Van der Waals-like phase transition exists in the extended phase space, where the negative cosmological constant is treated as the pressure while its conjugate acts as the thermodynamical volume in the Einstein gravity. Imposing the cosmological constant as a dynamical variable, the mass of the black hole corresponds to the enthalpy of the black hole system. The Smarr relation and the first law of thermodynamics will hold in this case [58].
In this paper, we will investigate the laws of thermodynamics and the weak cosmic censorship conjecture in the extended phase space in the Born-Infeld-anti-de Sitter spacetime. The first law and phase transition of the Born-Infeld-anti-de Sitter black hole have been investigated extensively. Fernando discussed its thermodynamics and stability in the grand canonical ensemble [59]. Myung investigated its phase transition soon after [60]. The thermodynamics and phase transition were also investigated from the point of view of geometry [61]. Recently, there have been many works to study the thermodynamics of the Born-Infeld-anti-de Sitter black hole in the extended phase space, for in this spacetime there are abundant phases, such as Van der Waals phase transition [62] and reentrant phase transition [63,64]. However, until now, there is still lack of work to discuss the second law of the thermodynamics of the Born-Infeld-anti-de Sitter black hole in the extended phase space. In the Einstein gravity with a negative cosmological constant, a D-dimensional charged AdS black hole with consideration of the pressure and volume has been investigated [65], and it was found that the first law of the thermodynamics holds while the second law is violated for the extremal and near-extremal black holes. They also discussed the weak cosmic censorship conjecture and found that the extremal and near-extremal black holes do not change their configurations, so the weak cosmic censorship conjecture is valid. In this paper, we will investigate the first law and second law of thermodynamics as well as the weak cosmic censorship conjecture of the Born-Infeld-anti-de Sitter black hole in the extended phase space.
Our motivation is two-folded. On one hand, we are going to explore the effect of the other extensive quantities on the laws of thermodynamics and weak cosmic censorship conjecture besides the pressure and volume. In [62], it was found that in order to satisfy the Smarr relation in the Born-Infeld-anti-de Sitter black hole, the Born-Infeld parameter should be treated as a dynamical variable with a conjugate quantity, called Born-Infeld vacuum polarization. In this case, the first law of thermodynamics is modified due to the contribution of the vacuum polarization energy. We want to explore whether the Born-Infeld parameter affects the laws of thermodynamics and weak cosmic censorship conjecture. As a result, the first law and weak cosmic censorship conjecture are found not to be affected. However, the second law of the thermodynamics will be affected for the second law is violated for the double horizon black holes and part of the single horizon black holes. On the other hand, in [65], some approximations are used to investigate the weak cosmic censorship conjecture. The author found that the extremal and near-extremal black holes did not change their configurations under the absorption of a charge particle, which is different from that in the normal phase space where the extremal black holes will deform into non-extremal black holes [13,14]. In this paper, we intend to find an analytical method without any approximation to study the configurations of the black holes under a charged particle absorption. We give a general way to check whether the phase space will affect the configurations of black holes as a charged particle is absorbed. Our results show that configurations of the black holes will be changed in both the normal and extended phase space under a charged particle absorption. In particular, in both cases the extremal black holes will change into non-extremal black holes. This paper is organized as follows. In section 2, the Born-Infeld-anti-de Sitter black hole is introduced, and the motion of a charged particle around the black hole is investigated. In section 4, we establish the first law of thermodynamics under the charged particle absorption in the extended phase space, and further discuss the second law of thermodynamics as well as the weak cosmic censorship conjecture. The second law is violated for the double horizon black holes and part of the single horizon black holes, and the weak cosmic censorship conjecture is valid for all black holes. In section 3, we investigate the first law and second law as well as weak cosmic censorship conjecture in the normal phase space. We find that for all black holes, the first law as well as second law of thermodynamics hold, the weak cosmic censorship conjecture does not violate. Throughout this paper, we will set the gravitational constant G and the light velocity c to be one.
2 Motion of a charged particle in the Born-Infeld-anti-de Sitter black hole

Brief review of the Born-Infeld-anti-de Sitter black hole
The Einstein-Born-Infeld theory in AdS is described by the action [66] in which F = 1 4 F µν F µν , R is scalar curvature, G is the gravitational constant, Λ is the cosmological constant with Λ = −3/l 2 where l is the AdS radius, and b is the Born-Infeld parameter which relates to the string tension α with the relation b = 1/(2πα). The solution of the Born-Infeld AdS black hole can be written as [67][68][69] in which M represents the ADM mass and Q the electric charge, 2 F 1 is the hypergeometric function. From Eq.(2.3), we know that in the limit b → ∞, Q = 0, the solution reduces to the Reissner-Nordström-AdS black hole, and in the limit Q → 0, it reduces to the Schwarzschild AdS black hole. The nonvanishing component of the vector potential is which reminds us that the chemical potential depends on the Born-Infeld parameter.
In [60,70], the author computed the horizon of the extremal black hole as They claimed that bQ ≥ 0.5 should be satisfied in order to have a real root for r 2 e . In particular, they stressed that 0 ≤ bQ < 0.5 is a forbidden region for in this region there is no black hole solution. However, it was later found that there was a single horizon black hole solution in the region 0 ≤ bQ < 0.5 [71]. The action growth of the Wheeler-DeWitt patch for the single horizon black hole has been calculated and it was found that the Lloyd bound was satisfied. In fact, the single horizon black hole exists not only in the region 0 ≤ bQ < 0.5, but also in bQ ≥ 0.5, please refer to Figure ( If the black hole mass is larger than the marginal mass, there are only single horizon black holes.

2.2
Energy and momentum of a particle absorbed by a black hole In this subsection, we will consider the dynamics of a charged particle as it is absorbed by the Born-Infeld-anti-de Sitter black hole. We will concentrate mainly on the relations between the conserved quantities, such as the energy and the momentum. The Hamilton-Jacobi equation for the vector potential A µ is in which u is the rest mass, e is the electric charge, and P µ is the momentum of the particle with the definition where I is the Hamilton-Jacobi action. Considering the symmetry of the black hole, the action can be written as where E and L are the energy and angular momentum of the particle, respectively. They are the conserved quantities with respect to t and φ. From Eq.(2.2), the inverse of the metric is, Therefore, the Hamilton-Jacobi equation can be re-expressed as One can readily separate the angular part from the above Eq.(2.11) and define it as in which K can be solved from the radial part in Eq.(2.11) as Therefore, Eq.(2.9) can be rewritten as with Hence, from Eq.(2.14), the radial momentum P r and angular momentum P θ can be written as We will study how the black hole thermodynamics changes as a charged particle is absorbed by the black hole. Concretely, we focus on the relation between the momentum and energy. In principle, as K is eliminated, we can get their relation at any locations. Our goal is to investigate the thermodynamics on the event horizon, thus we mainly pay attention to the near horizon behavior of the particle. In this case, Eq.(2.16) can be simplified as It should be stressed that a positive sign should be endowed in front of the |P r + | term. This choice is to assure that the signs in front of E and |P r + | are the same and positive in the positive flow of time [72]. From Eq.(2.18), we know that the energy dependents on the electric potential too. However, the potential is independent of the flow of time and only related to the interaction between particle and black hole. Thus, the total value of energy under the sum of the potential is not important, and we simply choose a positive sign in front of |P r + |.
3 Thermodynamics and weak cosmic censorship conjecture in the normal phase space In this section, we will examine whether the first law, the second law as well as the weak cosmic cosponsorship conjecture are valid in the normal phase space of Born-Infeld-anti-de Sitter spacetime. Our discussions are mainly based on the relation between the energy and momentum of the absorbed particle in Eq.(2.18).

Thermodynamics in the normal phase space
In the Born-Infeld-anti-de Sitter black hole, the electrostatic potential difference between the black hole horizon and the infinity is in which r + is the event horizon of the black hole, which is determined from f (r + ) = 0. The Hawking temperature, defined by T = f (r) 4π | r + , can be written as In addition, according to the Bekenstein-Hawking entropy area relation, we can get the black hole entropy At the horizon, the mass M can be expressed as we can obtain the first law of thermodynamics.
In the normal phase space, the cosmological parameter is a constant, and the mass M is the internal energy of the black hole. According to the energy conservation and charge conservation, as a charged particle is absorbed by the black hole, the variation of the internal energy and charge satisfy E = dM, e = dQ, (3.5) so that Eq.(2.18) should be written as In addition, as the charged particle is absorbed by the black hole, the variation of the entropy can be written as where we have used Eq. (3.3). To obtain the last result of the variation of the entropy, we should find dr + firstly. The absorbed particle leads to a variation of the event horizon of the black hole, which further leads to the change of f (r). The variation of f (r), labeled by df + , satisfies Substituting Eq.(3.6) into Eq.(3.8), we can get dr + directly as With Eq.(3.7), the variation of entropy can be expressed as Combining Eqs.(3.2) and (3.10), we find T dS = |P r + |. Therefore the internal energy in Eq.(3.6) can be rewritten as which is the first law of the black hole thermodynamics in the normal phase space.
Next we turn to study the second law of the thermodynamics, which states that the entropy of the black hole never decreases in the arrow of time. As the charged particle is absorbed by the black hole, the entropy of the black hole increases according to the second law of the black hole thermodynamics. We will employ Eq.(3.10) to check whether this is true in the normal phase space.
We first discuss the extremal black hole, for which the inner horizon and outer horizon coincide and the temperature vanishes at the horizon. With Eq.(3.2), we can get the mass of the extremal black hole Substituting Eq.(3.12) into Eq.(3.10), we get The divergency of the variation of the entropy means that it is meaningless to investigate the second law of thermodynamics for the extremal black holes. Therefore we concentrate on mainly the non-extremal black holes thereafter. As the charge Q and the Born-Infeld parameter b are given, we can find the extremal black hole mass M e via asymptotic evaluation method 1 . With the extremal mass, we also can get the extremal radius via Eq.(3.12). Then we can give any mass, which should be larger than the critical mass, to find the corresponding horizon and variation of the entropy of the non-extremal black hole. Throughout this paper, we set l = P r + = 1. For the case Q = 1, b = 0.8, we find that the extremal black hole mass is M e = 1.0302893. For the extremal and non-extremal black holes, the variations of entropy and horizon r h are listed in Table 1. We can see that for all the black holes, the variations of entropy are positive.
That is to say, the second law of black hole thermodynamics always holds as a charged particle is absorbed by the black hole in the normal phase space. (3.14) On the basis of Eq.(3.14), we can plot the relation between the variation of the entropy and the horizon, which is shown in Figure (1). One can see that the variation of entropy is positive for all black holes, which is consistent with the results in Table 1. To confirm that our conclusion is independent of Q and b, we can choose some other values of them. For the case of Q = 0.8, b = 0.8, the extremal mass of the black hole is 0.7671287. For the extremal and non-extremal black holes, the variations of entropy are listed in Table 2, and the relation between the variations of entropy and the horizons are plotted in Figure (2). From them, we can see that the variation of entropy is positive too. In other words, the second law of black hole thermodynamics is satisfied in the normal phase space for the Born-Infeld-anti-de Sitter black hole, which is independent of the charge and Born-Infeld parameter. In addition, form Eq.(2.6), we know that as the mass of the black hole is larger than the critical mass M m , the black hole are single horizon black holes. For the cases Q = 1, b = 0.8 and Q = 0.8, b = 0.8, the critical mass are M m = 1.10556 and M m = 0.791072 respectively. Table 1 and Table 2 show that the second law is satisfied in the normal phase space not only for the double horizon black holes but also for the single horizon black holes.

Weak cosmic censorship conjecture in the normal phase space
The weak cosmic censorship conjecture asserts that there is no singularities visible from future null infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of a black hole. So an event horizon should exist to assure the validity of the weak cosmic censorship conjecture. As a charged particle is absorbed by the black hole, we will check whether there is an event horizon. For the single horizon black holes in the Born-Infeld theory, the horizons will not be broken while for the double horizon black holes, they may be, which can be seen from Figure (3). The weak cosmic censorship conjecture thus is valid for the single horizon black holes always and we only should discuss the double horizon black holes thereafter.
For a double event horizon black hole, there is a minimum value for f (r), the corresponding radial coordinate is labeled as r m . At r m , we find there are always the following relations For the extremal black hole, δ = 0, and for the near extremal black hole, δ is a small quantity. For the extremal and near-extremal black holes, as a charged particle is absorbed by the black hole, f m may move upward or downward, which correspond to that there does not exist or exist horizon respectively. Next, we will find how f m move in the normal phase space.
The absorption of a charged particle will lead to variations of the mass and charge of the black hole. Correspondingly, the locations of the minimum value and event horizon will change as r m → r m + dr m , r + → r + + dr + . The variation of f (r), defined by df m , can be expressed as where we have used f m = 0. In addition, at the new minimum point, there is also a relation which implies With the condition f m = 0, we can obtain M and further dM , which is We are interested in the extremal black holes, for which Eq. . (3.20) In addition, substituting Eq.
which shows that there is a shift of f m in the negative direction as a charged particle is absorbed by the black hole. In other words, the weak cosmic censorship conjecture holds in the normal phase space since there always exist horizons to assure the singularity of the spacetime to be hidden.

Thermodynamics and weak cosmic censorship conjecture in the extended phase space
In this section, we will examine the thermodynamics and weak cosmic censorship conjecture in the extended phase space of Born-Infeld-anti-de Sitter spacetime under a charged particle absorption with Eq.(2.18).

Thermodynamics in the extended phase space
In the extended phase space, the cosmological constant is treated as the pressure Y and the corresponding conjugate quantity is treated as the volume V . In this case, in order to satisfy the Smarr relation, the Born-Infeld parameter b should also be treated as an extensive quantity. The Smarr relation for the Born-Infeld-anti-de Sitter black hole is [62] where B, which is the conjugate quantity of b, is regarded as the Born-Infeld vacuum polarization [62]. It should be stressed that M is not the internal energy but the enthalpy of the thermodynamic system, which relates to the internal energy with the following relation As the charged particle is absorbed by the black hole, the energy and charge are supposed to be conserved. Namely the energy and charge of the particle equal to the varied energy and charge of the black hole. According to Eq.(2.18), we know that the energy of the particle is determined by the charge and radial momentum of the particle near the event horizon.
Our goal is to obtain the first law of thermodynamics, we thus should find some quantities which can be expressed by the charge and radial momentum of the particle. Based on the energy conservation and charge conservation, we can obtain The energy in Eq.(2.18) changes accordingly into dU = The variation of the event horizon of the black hole, denoted by dr + , is determined by the charge, energy, and radial momentum of the absorbed particle, leading to the changes of f (r). However, near event horizon, df (r + ) ≡ df + will not change since f (r + + dr + ) = 0, that is In addition, with the help of Eq.(4.3), Eq.(4.5) can be expressed as From Eq.(4.6), we can obtain dl, and substituting dl into Eq.(4.7), we can delete it directly. Interestingly, dQ, db, and dM are also eliminated at the same time. In this case, there is only a relation between |P r + | and dr + , which is Therefore, the variations of entropy and volume of the black hole can be expressed as Based on the above formulae, we reach The internal energy in Eq.(4.5) thus will change into In the extended phase space, the mass of the black hole have been defined as the enthalpy. From Eq.(4.3), we can get the relation between the enthalpy and internal energy as which is obviously the first law of the black hole thermodynamics in the extended phase space [62]. Namely, the first law of thermodynamics holds in the extended phase space of Born-Infeld-anti-de Sitter spacetime under a charged particle absorbtion. Next we will focus on the second law of the thermodynamics in the extended phase space with Eq.(4.9). It should be stressed that in Eq.(4.2), the variation dB is the function of dr + , dQ, db. However, the existence of dQ, db would affect the definition of Φ and B respectively. Thus, dQ, db may relate to dr + , resulting in that dB is only a function of dr + . Without loss of generality, we will regard dB as ghdr + , in which g ≡ ∂B/∂r + and h are the contributions of ∂B/∂Q and ∂B/∂b. For simplicity, we will set k = gh thereafter, where k is a positive parameter for the energy of the black hole increases as a charged particle is absorbed by the black hole 2 . In this case, Eq.(4.9) would change into Here we can not determine the value of k by the parameters of black holes for we do not know the value h though we know g.
We will use Eq.(4.15) to check the second law of thermodynamics. For the extremal black hole, as Eq.(3.12) is substituted into Eq.(4.15), we obtain dS extreme = − 4πl 2 |P r + |r e 2bkl 2 + 3r 2 e < 0, (4.16) in which r e is the radius of horizon of the extremal black hole. It is obvious that the variation of the entropy is always negative. In other words, the entropy of the black hole decreases and the second law of black hole thermodynamics is violated for the extremal Born-Infeld-anti-de Sitter black hole. Now we turn to the non-extremal black hole. In Table 3, the variation of entropy for different k's are given. For the case of k = 0.01, we find that the variation of the entropy is negative for M ≤ 1.18 while it is positive for M > 1.18. As k increases to 0.05, the dividing point, which determines the negative or positive of dS, jumps to 1.29.  In Figure (3), we plot the relation between f (r) and r. We can see that there exist single horizon black holes and double horizon black holes for different masses M . We are interested in the case M = 1.18, 1.29, which has been marked by red and black in Figure  (3). Obviously, for both cases, the black holes are single horizon black holes. So from Table  3, we can conclude that the variations of entropy for all the double horizon black holes and part of the single horizon black holes are negative. In other words, all the double horizon black holes and part of the single horizon black holes violate the second law of black hole thermodynamics as a charged particle is absorbed by the black hole. As k increases, the mass of the black holes that violate the second law of black hole thermodynamics will be larger. In order to confirm our conclusion, we choose different charges and Born-Infeld parameters in the following. For the case Q = 0.8, b = 0.8, we find that the extremal black hole mass is M e = 0.7671287. The variation of entropy and horizon for different masses are given in Table 4. For k = 0.01, we find that the variation of the entropy is negative for the case M ≤ 0.8 and positive for M > 0.8. As k increases to 0.05, the dividing point jumps to 1.35. From Figure (4), in which the red curve and black curve represent cases for M = 0.8, 0.85, we find that the black holes are single horizon black holes for M = 0.8, 0.85. So we can also conclude that all the double horizon black holes and part of the single horizon black holes violate the second law of black hole thermodynamics.

Weak cosmic censorship conjecture in the extended phase space
In the extended phase space with consideration of pressure, volume and Born-Infeld vacuum polarization, we find that the first law of black hole thermodynamics is valid for both the single horizon black holes and double horizon black holes, but the second law is violated for the double horizon black holes and part of the single horizon black holes. In this section, we are going to check the weak cosmic censorship conjecture in the extended phase space for the Born-Infeld-anti-de Sitter. We want to know whether the horizon of the black hole will shrink to the singularity. As in the normal phase space, we will concentrate on only the double horizon black holes since the weak cosmic censorship conjecture is valid always for the single horizon black holes under charged particle absorbtion. As a charged particle is absorbed by the black hole, the mass M , charge Q, Born-Infeld parameter b, and AdS radius l change into (M +dM, Q+dQ, b+db, l+dl). Correspondingly, the locations of the minimum value and event horizon will change into r m → r m + dr m , r + → r + + dr + . There is also a transformation for f (r), which is labeled as df m . At the new minimum point, according to Eq.(3.17), we can get In addition, at this location, f (r) would change into Eq. (4.22) shows that as a charged particle is absorbed by the black hole, there is a shift of f m in the negative direction. So there always exist horizons to hide the singularity of the space time. In particular, the extremal black holes will change into non-extremal black holes. Our result is different from the result in [65], where they found that the black holes are stable, and the extremal black holes are always extremal black holes in the extended phase space.

Conclusions
In the Born-Infeld-anti-de Sitter black hole, we investigated the motion of a charged particle around the black hole and obtained the relation between the energy and momentum of the particle near the horizon. Furthermore, we investigated the laws of thermodynamics, and weak cosmic censorship conjecture in the normal phase space and extended phase space. As expected, the first law as well as second law of the thermodynamics hold, and the weak cosmic censorship conjecture does not violate in the normal phase space. This is consistent with the previous conclusion that the charged black hole can not be over-spinning as the backreaction is considered, even for the extremal black holes. In the extended phase space, since the cosmological constant and Born-Infeld parameters are not a variable in the action and equations of the motion, the dynamical effect is not easy to predict with the extensive quantities Y, b. As the charged particle is absorbed by the black hole, we obtained the variation of the horizon of the black holes and further the variation of entropy and volume. We found that T dS − Y dV − bdB = |P r + |, therefore, the relation between the energy and momentum can be written as the first law of thermodynamics in the extended phase space. It has been already proven that the satisfaction of the first law of thermodynamics is a necessary condition to ensure the second law of thermodynamics under a particle absorption [13,14] in the normal phase space. But the satisfaction of the first law does not mean the second law is satisfied. So we also discussed the second law in the extended phase space. We found that the second law was violated for the double horizon black holes and part of the single horizon black holes. 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 horizon. The stability can be shown from the change of the minimal value of the function f (r) under the absorption. We found that the variation of the minimal value of f (r) in the extended phase space was consistent with that in the normal phase space, which is −2|P r + |/r m . In particular, the extremal black hole will change into non-extremal black holes. Our results show that the singularity will always be hidden behind the horizons, which implies that the weak cosmic censorship conjecture is valid in the extended phase space. Our result is different from that in [65], where they found that the black holes were stable, and the extremal black holes would always be extremal black holes. But our result is consistent with that in the normal phase space, namely the extremal black holes will change into the non-extremal black holes [13,14].
For the extremal black holes in the normal phase space, we know that the variation of the entropy is infinite, which implies that the horizon is infinite too according to the entropy area relation. So the singularity will be always hidden, which is consistent with the result obtained by weak cosmic censorship conjecture. For the extremal black hole in the extended phase space, we know that the variation of the entropy is negative, the horizons of the black holes thus will shrink. However, we know that the weak cosmic censorship conjecture holds in this case, the horizon thus cannot shrink to the singularity. It is interesting to see that there exists the minimum value of the horizon where the black hole stops to shrink.