New Gedanken experiment on RN-AdS black holes surrounded by quintessence

In this paper, we use the new version of Gedanken experiment to investigate the weak cosmic censorship conjecture(WCCC) for RN-AdS black holes surrounded by quintessence. The process of matter fields falling into the black hole can be regarded as a dynamic process. Since the perturbation of matter fields doesn’t affect the spacetime geometry, we propose the stability condition and assume the process of matter fields falling into the black hole satisfies the null energy condition. Based on the stability condition and the null energy condition, the first-order and second-order perturbation inequalities are derived. As a result, we show that the WCCC for RN-AdS black holes surrounded by quintessence cannot be violated under the second-order approximation of matter fields perturbation.


Introduction
The most fundamental prediction of general relativity is the existence of black holes. Black holes have event horizon and a gravitational singularity inside the event horizon. If the event horizon vanishes, the naked singularity will destroy the well-define spacetime and the law of causality. To solve this problem, Penrose [1] proposed the weak cosmic censorship conjecture (WCCC), which indicates the singularity should be hidden in the event horizon and the observer at infinity cannot receive any information from the singularity.
Even though there is no general method to prove this conjecture, many efforts have been made to test it. The Gedanken experiment was firstly proposed by Wald [2] In 1974 to verify the applicability of WCCC in extremal Kerr-Newman (KN) black holes. The experiment shows that the conjecture will not be destroyed under the first-order perturbation when considering a particle with enough charges (or angular a e-mail: quyang@stu.scu.edu.cn b e-mail: taojun@scu.edu.cn (corresponding author) c e-mail: wujiayi777@stu.scu.edu.cn momentum) falling towards the extremal KN black holes. Over the last few years, many similar works based on this experiment have been considered in different works [3][4][5][6][7][8][9][10][11][12][13]. But there are some inherent flaws in this approach. It only considers the process of particles falling into black holes but does not consider the background spacetime's effect on the test particle and the analysis only focus on the first-level perturbation of the process. Later, Hubery [14] considered the second-order perturbation of test particles to show that the KN black holes can be destroyed in the Gedanken experiment. After that, the WCCC for other kinds of black holes is examined, and it's found that the conjecture can be violated during the process [15][16][17][18][19][20][21][22][23].
Observations over the last century have shown that the universe is dominated by an energetic component with the negative pressure [61,62]. One of the candidates for this component is the quintessence, which is a slow-changing and spatially non-uniform component of the negative pressure [63][64][65][66][67][68]. The quintessence is described by an ordinary scalar field minimally coupled to gravity, whose potential leads to the expansion. Quintessence must be coupled to ordinary matter, which results in long-range interaction and time dependent natural constants, even if it's compressed to Planck's scale. Kiselev [69] proposed the typical dark energy with black holes to explain this phenomenon. Since then, relevant studies have been conducted successively [70][71][72]. We consider the contribution of the space-time gravitational field of extra energy, the momentum tensor, which consists of dark energy. Then we can study the thermodynamic effects of black holes surrounded by the dark energy [73][74][75][76][77][78][79][80][81][82][83][84]. Next, we will consider the thermodynamics of the black hole surrounded by quintessence and use the Gedanken experiment to test the WCCC.
The organization of the paper is as follows. In Sect. 2, we study the spacetime geometry of RN-AdS black holes surrounded by quintessence under the matter fields perturbation. In Sect. 3, we discuss the Iyer-Wald formalism and derive the first-order and second-order variational identities. In Sect. 4, based on the variational identities and null energy condition, we derive the first-order and the second-order perturbation inequalities. In Sect. 5 we use the first-order optimal option and the second-order perturbation inequality to examine the WCCC for nearly extremal RN-AdS black holes surrounded by quintessence. In Sect. 6, some discussions and conclusions are given.

Spacetime gemogery of RN-AdS black hole surrounded by quintessence
In this paper, we consider the RN-AdS black holes surrounded by quintessence in four-dimensional spacetime. The Lagrangian of this case could be described as follows [69,83] where F = d A is the strength of the electromagnetic field and A is the potential of the electromagnetic field. is the cosmological constant with a negative value, ε is the volume element, L q is the quintessence dark energy such as a barotropic perfect fluid and defined by [85] where ρ 0 is an integral constant, ρ q is the energy density, and ω is the quintessence dark energy barotropic index. The value range of ω is −1 < ω < − 1 3 for the quintessence dark energy.
We use the Eddingdon-Finkelstein coordinate in this work [35]. Compared with Schwarzschild gauge, there is no coordinate singularity at the event horizon. When we consider the matter fields following into the black hole, it is convenient to use this coordinate to calculate. The solution of RN-AdS black holes surrounded by quintessence is where here M and Q can be interpreted as the mass and charge of the black hole respectively, l is the radius of AdS space and is related to the cosmological constant by = − 3 l 2 . The parameter a is the normalization factor associated with the quintessence density of dark energy and it's always positive. The relationship between a and ρ q can be expressed as [75,77] By imposing the condition f (r ) = 0, we can derive the radius of event horizon r + . Based on the first law of thermodynamics in the extended space [75,86], we can obtain that where P is the thermodynamic pressure, V is the thermodynamic volume, and A is conjugate to the parameter a.
The process of matter fields falling into the black hole can be considered as a completely dynamic process. We further assume that cosmological constant and normalization factor a can be regarded as effective parameters determined by the matter source coupled to the Einstein-Maxwell gravity. Then, we can represent the Lagrangian as L mt is Lagrangian of matter field, and the spacetime geometry can be simply derived from the Lagrangian if the matter field satisfies the stable solution. Then we can obtain the energy-momentum tensor of matter fields [83], Based on the Iyer-Wald formalism [24], n-form Lagrangian can be regarded as the function of metric g ab and other tensor fields ψ in spacetime. We use ϕ to represent the collection of the dynamical field, i.e. ϕ = (g ab , ψ). In this case, we set ϕ to be the collection of dynamical field ϕ = (g ab , A) and use λ to express the perturbation of the matter field to φ. The spacetime field equations are where T E M ab is the energy-momentum tensor of electromagnetic field.
Considering the perturbation process of matter fields, the spacetime geometry can be described as [35] We assume that the spacetime satisfies the stability condition [24], which means that after enough long time, the spacetime geometry should be consistent with the RN-AdS black hole surrounded by quintessence. We utilize M, Q, to describe the properties of black holes and represent these quantities by the parameter λ. Thus, the dynamical fields can be expressed as [35] where The energy-momentum tensor of matter fields is When the parameter λ is zero, the spacetime should still be the solution of RN-AdS black holes surrounded by quintessence, i.e. when f (r, 0) = f (r ), metric can be restored to the original spacetime form.

The linear variational identities
In this paper, we would like to use the Iyer-Wald formalism [24,25] to investigate the Gedanken experiments in RN-AdS black holes surrounded by quintessence. Consider the Einstein-Maxwell Theory, the Lagrangian is expressed as We use ϕ = (g ab , A) to represent the collection of the dynamical field. The variation of Lagrangian is where E ϕ = 0 gives the equations of motion. is the threeform of the symplectic potential and is locally composed of ϕ and its derivatives. In the Einstein-Maxwell theory, can be linearly expressed with gravitational field part and electromagnetic field part as Then we define the three-form of symplectic current as It also can be linearly expressed by two parts, where with For an arbitrary vector field, the Noether current associated with ζ a is defined by The variation of Noether current [25] is As shown in Ref. [87], we can write the Noether current as where Q ζ is the Noether charge and C ζ = ζ a C a is the constraint to the theory. We can obtain C a = 0 and d J ζ = 0 from the equation of motion. The Noether charge Q ζ is linearly expressed by where Considering the Einstein-Maxwell Theory, the equations of motion and constraints are given by where T ab = 1 8π G ab − T E M ab and j b = 1 4π ∇ a F ba are the energy-momentum tensor and electric current respectively.
By differentiating Eq. (23) and substituting Eq. (22), we can obtain the first-order variational identity, In the same way, by differentiating Eq. (27), the second-order variational identity can be shown as

First-order and second-order perturbation inequalities
In this section, we calculate the integral of first-order and second-order variational identities to obtain the perturbation inequalities. Due to the stability condition, we can choose a hypersurface = H ∪ 1 . H is a portion of the horizon r = r + in the background spacetime, starting from the unperturbed horizon's bifurcate surface B and ending up on the cross-section B 1 . 1 approaches infinity along the timeslice (ν=constant) as a space-like hypersurface. And the event horizon is a Killing horizon generated by the Killing field ζ a . For the first-order variational equation, we utilize the condition L ζ ϕ = 0 and integrate it on the hypersurface Using the Stokes theorem and the condition that B is unperturbed horizon's bifurcate surface, it can be rewritten as Since the integral diverges as the integral region approaches infinity, we apply a cut-off method at sphere S o with radius r o and let the limit of S o approach asymptotic infinity. Then, we calculate each integration term separately. Firstly, we evaluate the first term of Eq. (30). Considering Eqs. (11), (16) and (25), we have As the S e approaching asymptotic infinity, the second term vanishes, and then the first term of Eq. (30) is given by Secondly, we integrate the second part of Eq. (30) by considering ϕ(0) as a globally hyperbolic, asymptotically flat solution of the equations of motion, which means that the integral of the second term equal to zero [24].
Then, we utilize the condition j a = 0 on 1 and Eq. (26) to show that where Finally, with A a ζ a | H = − + and H ε ebcd δ j = δ Q, we can further obtain the expression as Since both the normal vectors n a and the time-like Killing vectors ζ a on the horizon become null and ζ a ∝ n a , we can use the null energy condition δT ab n a n b 0. On the horizon, we have ε ebcd = −4n [e ε bcd] , where n a is the normal vector and ε bcd is the volume element, so the first term of Eq. (35) can be written as H ε bcd n e ζ a δT e a (λ) 0. Therefore, from Eqs. (32,33,35) and the null energy condition, we can obtain the first-order perturbation inequality In our work, we use the second-order perturbation inequality to examine whether the WCCC can be violated under the second-order perturbation approximation of matter fields. When the first-order inequality is satisfied, it can be proved that WCCC is valid under the first-order approximation. But when the first-order perturbation takes the optimal option the WCCC can not be examined sufficiently by only considering the first-order approximation. Then we need to derive the second-order perturbation inequality. Integrating Eq. (28) on the hypersurface , it yields where W H (ϕ, δϕ) = H ω(ϕ, δϕ, L ζ δϕ), Following the previous calculation steps, the second term of Eq. (38) equals to 0, the third and fourth term of Eq. (38) can be expressed respectively Similarly, the first term can be written as where the second term on the last line vanishes if S e approaches asymptotic infinity, and then the first term of Eq. (38) is given by Finally, we calculate the fifth term of Eq. (38). It can be linearly expressed by two parts From [24], we can obtain the similar result, and then we have Therefore, from Eqs. (40), (41), and (44), we can get Following the same method in Ref. [24], we can build an auxiliary spacetime to calculate. Because of the stability condition, the spacetime geometry on 1 is still the RN-AdS spacetime surrounded by quintessence, and the configuration of dynamical fields under the perturbation of matter field can be described by one parameter λ. Hence, the metric function and electromagnetic strength for the auxiliary spacetime are where Next we only consider the first-order variation of matter field on auxiliary spacetime, and then the M R A (λ), Q R A (λ), R A (λ) and a Q R (λ) are given as follows, Considering only the first-order variation of the matter field, we obtain Then we obtain δϕ Q R = δϕ on hypersurface 1 , which implies that W 1 (ϕ, δϕ) = W 1 (ϕ, δϕ Q R ). Thus we can calculate them straightly in auxiliary spacetime. Integrating the second-order variation identity on 1 , we have (49) Following the previous calculation steps and the gauge condition of the electromagnetic field such that ζ a δ A a = 0 at H, the Eq. (49) can be expressed by Then, the Eq. (45) can be rewritten as We consider the null energy condition δ 2 (T E M ab +T ab )n a n b 0. under the second-order approximation. Then the second-order perturbation inequality can be reduced as

Test the WCCC of RN-AdS black hole surrounded by quintessence
In this section, we will apply the new version of the Gedanken experiment to discuss the WCCC of nearly extremal RN-AdS black holes surrounded by quintessence. We assume that the spacetime satisfies the stability condition. The condition of the existing event horizon r + is metric factor satisfy f (r + ) = 0. We suppose that there exists one minimum point at r = r 0 for f (r ), and the existence of the event horizon is consistent with condition f (r 0 ) 0. We can take f (r 0 (λ), λ) and use the discriminant function to represent the change of extremum of f (r ) under the matter field perturbation. r 0 (λ) is the minimum point of f (r 0 (λ), λ), which satisfies the condition ∂ r f (r 0 (λ), λ) = 0. We can expand the function to second-order at λ = 0, With ∂ r f (r 0 (λ), λ) = 0 and the zero-order approximation of λ, one can obtain Considering the matter fields and taking the first-order variation of ∂ r f (r 0 (λ), λ) = 0, we have Therefore, by applying Eqs. (54) and (55), we can yield the detailed expression of Eq. (53) as The event horizon r + and r 0 satisfy the relation r + (1 − ε) = r 0 with ε 1 [24] for the case of the nearly extremal black hole. With the relation of ∂ r f (r 0 ) = 0, f (r + ) = εr + f (r + ), and we can obtain the relation of f (r 0 ) = − 1 2 ε 2 r 2 + f (r + ) under the second-order approximation of ε. We can conclude these relations, Therefore, we can rewrite the expression of Eq. (56) as Utilizing the Eqs. (37) and (52) together with above results, we can further simplify the expression as With the condition that f ((1 + ε)r 0 ) = 0 and f (r 0 ) = 0, and considering the zero-order approximation of ε, we can derive and M, Together with the relation r + = (1 + ε)r 0 , we can expressed Eq. (59) as where Because r 0 is the minimum point that satisfies the condition f (r 0 ) > 0. The above expression gives f (r 0 (λ), λ) 0, which implies that the event horizon of near extremal RN-AdS black holes surrounded by quintessence still exists when the second-order perturbation is taken into account, therefore the WCCC cannot be violated under the second-order approximation of matter fields perturbation.

Conclusion
In this paper, we discuss the WCCC of nearly extremal RN-AdS black holes surrounded by quintessence with the new version of Gedanken experiment. Based on the stability condition and the null energy condition, we utilize Iyer-Wald formalism to derive the first-order and the second-order perturbation inequalities. With the first-order optimal option and the second-order inequality, we prove that the event horizon of nearly extremal RN-AdS black holes surrounded by quintessence still exists under the second-order approximation of matter fields perturbation, which implies that the WCCC can not be violated for this case. This result is the same as that of previous papers [79][80][81]. Furthermore, we can prove the event horizon still exists under the higher perturbation or another black hole, this gives us a broader perspective and methods to examine the WCCC.

Data Availability Statement
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