Static charged dilaton black cannot be over-charged by gedanken experiments

We consider the new version of the gedanken experiments proposed recently by Sorce and Wald to over-charge a static charged dilaton black hole. First of all, we derive the first-order and second-order perturbation inequalities in Einstein-Maxwell-dilaton gravitational theory based on the Iyer-Wald formalism. As a result, we find that weak cosmic censorship conjecture associated with this black hole can be protected after taking into account the second-order perturbation inequality, although violated by the scene without considering this inequality. Therefore, there is no violation of the weak cosmic censorship conjecture occurs around the charged static dilaton black holes in Einstein-Maxwell-dilaton gravity.


I. INTRODUCTION
When a singularity is not hidden behind a black hole horizon, so as to be seen by a distant observer, then it is called a naked singularity. And the singularity will violate the predictability of general relativity as classical theory. Therefore, Penrose proposed the weak cosmic censorship conjecture (WCC) which asserts that singularities formed by gravitational collapse of matter are hidden behind event horizons [1]. Even though there is still no general proof for this conjecture, many efforts have been taken for decades to test it [2]. Particularly, in a seminal work, Wald proposed a gedanken experiment to test this conjecture by examing whether the black hole horizon could be destroyed by plunging a test particle into a black hole [3]. The result shows that we cannot destroy an extremal Kerr-Newman (KN) black hole in this way. But as initiated by Hubeny in 1999 [4], the nearly extremal KN black hole can be destroyed by inputting the test particle [5][6][7][8][9]. And it also received lots of attention and followed by extensive studies in various theories. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] Motivated by these results, Sorce and Wald [27] have recently suggested a new version of gedanken experiment without proposing the test particle assumption of explicit analyses of trajectories of particle matters. In this version, they apply the Iyer-Wald formalism [28] as well as the null energy condition to the general matter perturbation on the black holes and obtain the first-order and second-order inequalities of the collision matter. After the second-order perturbation inequality of the energy, angular momentum, and charge are taken into account, they showed that the nearly extremal Kerr-Newman black hole cannot be destroyed under the secondorder approximation of the perturbation and no violation of the Hubeny type can ever occur.
Most recently, this new version has also been investigated in the 5-dimensional Myers-Perrt black holes and higherdimensional charged black holes, and they show that WCC is well protected for the nearly extremal black holes when the second-order perturbation inequality is considered [29,30]. These black holes have a lot of remarkable properties, for example, all of them have two horizons. Therefore, it is natural for us to study whether the second-order perturbation inequality can ensure WCC in all kinds of black holes, especially that with different causal structure. As one of the most interesting solutions of the general relativity, the dilaton black hole has many different features from the above cases, where the inner horizon is taken placed by a singular surface after introducing the dilaton field (See Fig.1). Therefore, its subextremal case shares the same causal structure of the Schwarzschild black hole. However, differing from the Schwarzschild case, as shown in [32], the charged static charged dilaton black hole in the Einstein-Maxwell-dilaton theory could be overcharged by the old version of the gedanken experiment since the spacetime causal structure also relies on the electric charges. In this paper, we would like to consider the Hubeny scenario by this new version of gedanken experiment and investigate whether the WCC can be restored when the second-order correction is taken into consideration.
Our paper is organized as follows. In the next section, we review the Iyer-Wald formalism for general diffeomorphism covariant theories and show the corresponding variation quantities. In Sec. III, we focus on dilaton black holes in the Einstein-Maxwell-dilaton theory and derive the relevant quantities in this case. In Sec. IV, we present the setup for the new version of the gedanken experiment, and derive the firstorder and second-order perturbation inequalities for the optimal first-order perturbation of the dilaton black holes. In Sec. V, we examine the Hubeny scenario from the new version of the gedanken experiment when the second-order perturbation inequality is considered, and compare to the result without second-order perturbation. Finally, conclusions are presented in Sec. VI.

II. IYER-WALD FORMALISM AND VARIATIONAL IDENTITIES
In this paper, we would like to use the Noether charge formalism proposed by Iyer and Wald to investigate the gedanken experiments in the charged static black holes of Einstein- Maxwell-dilaton theory. Firstly, we consider a general diffeomorphism-covariant theory on a 4-dimensional spacetime M. The Lagrangian can be given by a 4-form L where the dynamical fields consist of a Lorentz signature metric g ab and other fields ψ. Following the notation in [28], we use boldface letters to denote differential forms and collectively refer to (g ab , ψ) as φ . The variation of L is given by where E φ = 0 gives the equations of motion of this theory, and Θ is called the symplectic potential 3-form which is locally constructed out of φ , δ g ab and their derivatives. The symplectic current 3-form is defined by Let ζ a be the infinitesimal generator of a deffeomorphism. By replacing δ by L ζ in (1), one can define the Noether current 3-form J ζ associated with ζ a , A straightforward calculation yields which indicates that J ζ is closed when the equation of motion are satisfied. On the other hand, it was shown in [33] that the Noether current can be written in the form where Q ζ is called the Noether charge and C ζ = ζ a C a are the constraints of the theory, i.e., C a = 0 when the equations of motion are satisfied. Then, by keeping ζ a fixed and comparing the variations of (3) and (5), one can obtain the first variational identity The second variational identity can further obtained and it can be shown as where we also used the equations of motion and assume ζ a is a symmetry of φ , i.e., L ζ φ = 0. In what follows, we shall consider the globally hyperbolic static solution with a timelike Killing vector ξ a such that L ξ φ = 0. The ADM mass of this black hole is given by Supposing that Σ is a hypersurface with a cross section B of the horizon and the spacial infinity as its boundaries, the integration of the first and second variational identities can be written as where we denote

III. EINSTEIN-MAXWELL-DILATON THEORY AND ITS STATIC SOLUTION
For our purpose, in this section, we consider an Einstein-Maxwell-dilaton theory in 4-dimensional spacetime with the following Lagrangian where F = F ab F ab with the electromagnetic strength F = dA. This model describes a massless dilaton scalar field in coupled to the linear electromagnetic field. The symplectic potential can be given by Here we have defined The Noether charge is given by where If the additional charged matter sources are taken into account, the equations of motion can be written as with Here T ab corresponds to the non-electromagnetic and dilaton part of the stress-energy tensor, j a corresponds to the electromagnetic charge-current, and both of them are non-vanishing after the matter source is introduced. Then, the equations of motion part and constraints in Eq. (1) for the Einstein-Maxwell-dilaton theory are given by If the background spacetime is stationary, the flux of the stress-energy tensors of the electromagnetic field and dilaton field through the horizon must vanish. From (18), it implies that F ab must take the form (20) and ψ satisfies L ξ ψ = 0, where w ab is a purely tangential to the horizon [27]. From (13), the symplectic current for the Einstein-Maxwell-dilaton theory can be written as where in which we denote with P abcde f = g ae g f b g cd − 1 2 g ad g be g f c − 1 2 g ab g cd g e f − 1 2 g bc g ae g f d + 1 2 g bc g ad g e f .
We next restrict on the charged static spherically symmetric solution of the 4-dimensional Einstein-Maxwell-Dilaton theory, which can be described by [31] with the constraint It is also known as the Gibbons-Maeda-Garfinke-Horowitz-Strominger (GMGHS) solution. The Penrose diagrams of this solution are shown in Fig.1. This black hole solution exists as long as censorship condition M > D is satisfied. And the horizon is located at r h = 2M. We shall refer the extremal limit to the case Q 2 = 2M 2 , where the singular radius r = 2D coincides with the horizon. Then, the area of the event horizon with the area is given by One can note that our event horizon is also a Killing horizon which is generated by the Killing field ξ a = (∂ /∂t) a . And the corresponding horizon electric potential and surface gravity can be read off

IV. PERTURBATION INEQUALITIES OF GEDANKEN EXPRIMENTS
As in the new gedanken experiment designed in [27], the situation we plan to investigate is what happens to the above static dilaton black holes when they are perturbed by a oneparameter family of the matter source according to Einstein equation as well as the Maxwell equation around λ = 0 with T ab (0) = 0 and j a [0] = 0. Without loss of generality, we shall assume all the matter goes into the black hole through a finite portion of the future horizon, i.e., the matter source δ T ab and δ j a are non-vanishing only in a compact region of future horizon. In order to obtain the firstorder and second-order perturbations of the black hole, with the similar consideration of [27], we also introduce the following assumption: Additional assumption: The nonextremal, unperturbed static charged dilaton black hole is linearly stable to perturbations, i.e., any source-free solution to the linearized Einstein-Maxwell-dilaton equations approaches a perturbation towards another static charged dilaton black hole at sufficiently late times.
With these in mind, we can always choose a hypersurface Σ = H ∪ Σ 1 such that it starts from the bifurcate surface B of the unperturbed horizon, continues up the horizon through the portion H till the very late cross section B 1 where the matter source vanishes, then becomes spacelike as Σ 1 to approach the spatial infinity. By considering the additional assumption, the dynamical fields satisfy the source-free equation of motion, E [φ (λ )] = 0 on the portion Σ 1 , and the solution is described by Eq. (25).
Then, if we work with the Gaussian null coordinates near the unperturbed horizon, we can further obtain with A B the area of the bifurcate surface [3]. With the above preparation, we now derive the first-order inequality obeyed by the perturbation at λ = 0. Note that for our choice the perturbation vanishes on the bifurcate surface B and , the first equation of Eq. (9) reduces to where we used the fact that T ab = j a = 0 in the background spacetime. Since Φ = −ξ a A a is constant on H , we may pull it out of the integral. The integral δ Q flux = H δ (ε ebcd j e ) is just the total flux of electromagnetic charge through the horizon. Since all of the charge added to the spacetime falls through the horizon, this flux is just equal to the total perturbed charge of the black hole, δ Q flux = δ Q. Combining these observations yields the following formula relating the perturbed parameters of the black hole spacetime: (32) where˜ is the corresponding volume element on the horizon, which is defined by ε ebcd = −4k [eεbcd] with the future-directed normal vector k a ∝ ξ a on the horizon. Then, according to the null energy condition δ T ab k a k b ≥ 0, (32) yields the inequality Obviously, if we want to violate M 2 − 2Q 2 ≥ 0, the optimal choice is to saturate (33) by requiring δ T ab k a k b | H = 0, namely, i.e., the energy flux through the horizon vanished for the first-order non-electromagnetic perturbation. Then, (32) comes The first-order perturbation of Raychaudhuri equation implies that δ ϑ = 0 on the horizon if we choose a gauge in which the first order perturbed horizon coincides with the unperturbed one. Next, we consider the second-order inequality. By performing a similar analysis to the first-order result, the second equation of (9) reduces to Here the integrals in the last two terms only depend on the surface H because δ E and δ 2 C ξ vanishes on Σ 1 by the assumption that there are no source outsider the black hole at late times. Moreover, since ξ a is tangent to the horizon, the first term vanishes. For the second term, together with (19), we have Following the setting of Ref. [27], here we also impose the condition ξ a δ A a | H = 0 by a gauge transformation, we have where δ 2 Q is the second-order change in charge of the black hole. Furthermore, by using the assumption that the first-order perturbation is optimal, we have where we have used energy condition for the second order perturbed non-electromagnetic stress-energy tensor in the last step.
Next, we turn to compute the horizon contribution. It can be decomposed into From the calculation in [27], the gravitational contribution is given by Then, we calculate the contribution for the electromagnetic part. From (22), we have By considering the gauge condition ξ a δ A a = 0 on the horizon as well as (21), the last two term will vanish. Then, Eq. (42) can be written as By considering on the horizon, the integral over H of the first term on the right side will only contribute a boundary term at S = H ∩ Σ 1 .
With the fact the perturbation is stationary at S, i.e., δ G ab has the form (21). Together with the gauge condition ξ a δ A a = 0 on H , the first term of (43) makes no contribution to (42).
Combining above results, we have Finally, we evaluate the dilaton contribution. From (22), we have When pulled back to H , the index d must contribute a k d ∝ ξ d . Then, since the background field is stationary, i.e., L ξ ψ = ξ d ∇ d ψ = 0, the last two terms vanish. Eq. (46) becomes With similar analysis as (43), integral over H of the first term in (47) only contribute a boundary term at S. By considering δ ψ is stationary, i.e., L ξ δ ψ = 0, this term also makes no contribution. Then, we have Together with (45), we have where we have used the null energy condition for the electromagnetic and dilaton stress-energy tensors. Finally, (39) reduces to Now we are left out to evaluate E Σ 1 (φ , δ φ ). To calculate it, we follow the trick introduced in [27], and write where φ DL is introduced by the variation of a family of dilaton black hole solutions (25), where δ M and δ Q chosen to be in agreement with the first order variation of the above optimal perturbation by the matter source. From the variation (51), one can find Thus, from the second expression of (9), we have Since ξ a = 0 on the bifurcation surface B, it can be expressed as Therefore, the second order inequality becomes The right sight of this inequality can be evaluated by taking two variations of the area formula A B = 8π(2M 2 − Q 2 ), and is given by Together with the optimal first-order inequality, the secondorder inequality becomes

V. GEDANKEN EXPERIMENTS TO DESTROY A NEARLY EXTREMAL DILATON BLACK HOLE
In this section, we will explore the gedanken experiments to overcharge a non-extremal black hole by the physical process described above. Therefore, we define a function of λ as Under the second-order approximation of λ , we have Firstly, we would like to analyze the result found in [32] for the old version of gedanken experiments, where they only consider the perturbation of the test particle. Therefore, in their case, there only exist a linear variation of the mass and charge of this black holes, and the second-order variation of black hole mass and charge vanish, i.e., Then, we have By using the optimal first-order inequality, it becomes According this equation, we can see that if we impose that the mass and charge of background black hole have the same order as λ , then we can note that it is possible to make h(λ ) < 0 for the non-extremal black holes, suggesting that the black hole could be overcharged if we neglect the second-order variation of mass and charge. Next, we consider the new version of the gedanken experiments. Using the first-order and second-order inequalities (34) and (56), under the second-order approximation of the perturbation, we can further obtain where we have used the fact that the background spacetime has the black hole geometry. Thus, as we can see, when the second-order correction of the perturbation is taken into account, this static dilaton black hole cannot be over-charged.

VI. CONCLUSION
It is shown in [32] that the old version of the gedanken experiment can destroy the static dilaton black holes in Einstein-Maxwell-dilaton theory if the backreaction or self-energy is ignored. However, in this paper, we following a similar consideration in [27], we showed that after the second-order perturbation inequality are taken into account, the charged static dilaton black hole cannot be overcharged. Therefore, there is no violation of the weak cosmic censorship conjecture occurs around the charged static dilaton black holes in Einstein-Maxwell-dilaton gravity. This result might indicate that once this black hole is formed, it will never be overspun classically. Moreover, the above results indicate that the second-order perturbation inequality might play the role of the backreaction or self-energy for the collision matter.