Static charged Gauss-Bonnet black holes cannot be overcharged by the new version of gedanken experiments

Based on the new version of the gedanken experiments proposed by Sorce and Wald, we examine the weak cosmic censorship conjecture (WCCC) under the spherically charged infalling matter collision process in the static charged Gauss-Bonnet black holes. After considering the null energy condition and assuming the stability condition, we derive the perturbation inequality of the matter source. As a result, we find that the static charged Gauss-Bonnet black holes cannot be overcharged under the second-order approximation of the perturbation when the null energy condition is taken into account, although they can be destroyed in the old version of gedanken experiments. Our result shows that the WCCC holds for the above collision process in the Einstein-Maxwell-Gauss-Bonnet gravity and indicates that WCCC may also be valid in the higher curvature gravitational theories.


I. INTRODUCTION
The weak cosmic censorship conjecture (WCCC) is one of the most important open questions in classical gravitational theory. It states that the singularities must be surrounded by the event horizon and cannot be seen by distant observes [1]. One way of examining this conjecture is to see whether the event horizon can be destroyed by some physical processes. In 1974, Wald suggested a gedanken experiment and proved that the extremal Kerr-Newman black hole cannot be destroyed via dropping a test particle [2]. However, there are two crucial assumptions in his setup: the background black hole is extremal and the analysis is only at the level of the first-order perturbation. Therefore, Hubeny [3] illustrated that the near extremal Kerr-Newman black holes can be destroyed under the secondorder approximation of the test particle. This attracted lots of attention and was then studies in various theories [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
However, the method which performs a test particle has some inherent defects. In the process of the particle dropping into the black hole, the spacetime is just treated as a background. When considering the second-order approximation, the backreaction, self-force and finite size effects should be taken into consideration. To solve the above defects, Sorce and Wald [21] has proposed a new version of the gedanken experiments to examine the WCCC, in which they consider a full dynamical process of the spacetime and collision matter fields based on the Iyer-Wald formalism [22]. After the null energy condition of the collision matter fields is taken into account, they derived a second-order perturbation inequality on the second-order correction of black hole mass δ 2 M. As a result, they showed that the nearly extremal Kerr-Newman black hole cannot be destroyed under the second-order approximation after considering this new inequality. Moreover, because the process considered is full dynamical, all self-force, finite size effects, and backreaction are automatically taken into account [22].
Most recently, following the setup of this new version, WCCC has also been investigated in the five-dimensional * jiejiang@mail.bnu.edu.cn Myers-Perry black holes [23], higher-dimensional charged black holes [24], charged dilaton black holes [25], RN-AdS black holes [26], static Einstein-Born-Infeld black hole, Kerr-Sen black holes [27] as well as the scalar-hairy RN black holes [28]. Although all of them showed the validity of the WCCC in the new version of gedanken experiments, there is still a lack of the general proof of the WCCC. Therefore, it is necessary for us to test it in various theories. We can see that all of the analyses above are performed in the context of the Einstein gravity, relatively little is in higher curvature gravitational theories. After the quantum effect or string modification are taken into account, higher curvature term should be added to the Einstein-Hilbert action. In order for this story to be truly consistent, it is necessary to check whether the WCCC is also valid in higher curvature gravity. As one of the most interest higher curvature gravitational theories, the Einstein-Maxwell-Gauss-Bonnet (EMGB) gravity is an important generalization of Einstein gravity, where the Gauss-Bonnet term can be regarded as a correction from the heterotic string theory [29,30]. In [31], the authors investigated the old version of the gedanken experiments in the charged static Gauss-Bonnet black holes and found that the near extremal case can also be overcharged. Actually, they also neglected the self-force, finite size, and backreaction effects in their discussion. Therefore, in this paper, we would like to check whether the WCCC can be restored in the EMGB gravity after the second-order perturbation inequality is considered.
Our paper is organized as follows. In the next section, we discuss the spacetime geometry of static charged Gauss-Bonnet black holes perturbed by the spherically infalling matter source. In Sec. III, based on the Iyer-Wald formalism as well as the null energy condition, we derived the perturbation inequality of the matter fields under the second-order approximation. In Sec. IV, we utilize the new version of the gedanken experiment to destroy the near extremal static charged Gauss-Bonnet black holes under the second-order approximation of perturbation. Finally, the conclusions are presented in Sec. V.

II. PERTURBED CHARGED STATIC GAUSS-BONNET BLACK HOLE GEOMETRY
In this paper, we would like to investigate WCCC for ndimensional charged Gauss-Bonnet black holes in Einstein-Maxwell-Gauss-Bonnet (EMGB) gravity. The action of the EMGB gravity can be expressed as [32][33][34] with in which F = dA is the strength of the electromagnetic field A, α GB is the Gauss-Bonnet coupling constant, ǫ is the volume element, and L mt is the Lagrangian of the extra matter fields. The equation of motion (EOM) of the EMGB gravity is given by with Here G ab is the Einstein tensor, T EM ab and T ab are the stressenergy tensor of the electromagnetic field and perturbation matter fields, separately, and j a is electric current of the perturbation matter fields. The static solution of above theory can be written as [32][33][34] with the blackening factor and the line element of the unit (n − 2)-dimensional sphere Here the constant α is related the Gauss-Bonnet coupling con- denotes the volume of the unit (n − 2)-dimensional sphere, and Φ 0 is a constant related the gauge freedom of the electrodynamics. The parameters µ and q are the parameters related to the mass and charge of the black hole as The horizon of the black hole is determined by the equation f (r) = 0 and the radius of the event horizon is the largest root of blackening factor f (r). The surface gravity, area, and electric potential of the static charged Gauss-Bonnet black hole are given by If the blackening factor satisfies the condition f ′ (r h ) > 0, there exists two Killing horizons of the black hole solutions. However, if f ′ (r h ) = 0, these two horizons overlap and the black hole becomes extreme. For the extremal case, the conserved quantities of the black hole satisfy the constraints f (r e ) = f ′ (r e ) = 0 with the horizon radius r h = r e , and the constraints can be explicitly express as In the following, we consider a one-parameter family spherically charged infalling matter perturbation in the Gauss-Bonnet black hole, where the perturbation matter fields are only non-zero in a compact region of the future horizon and vanishing on the bifurcation surface B. Under this perturbation, the dynamical fields φ (λ ) are labeled by the variation parameter λ . Here we denote φ to g ab , A and the other fields related to the matter source. Then, the EOM in this setup can be expressed as where ∇ (λ ) a denotes the derivative operator related to the metric g ab (λ ), T ab (λ ) and j b (λ ) are only nonvanishing in a compact region. Since the case we considered here only contains an spherically symmetric infalling matter source, the spacetime can be generally described by the Vaidya solution with the line element Because the background spacetime is Gauss-Bonnet black hole, we have f (0) (r, v) = f (r) and T ab (0) = j a (0) = 0. For simplification, in the following, we will denote χ = χ(0) to the quantity χ in the background, and its first two order variation is expressed by With similar consideration of Ref. [21], here we also assume that the static charged Gauss-Bonnet black hole is linearly stable to above perturbation, which means that after a sufficient time of the perturbation, the spacetime can also be described by the static charged Gauss-Bonnet solution. Then, we choose a hypersurface Σ = Σ 0 ∪ Σ 1 . Here Σ 0 is a portion of the future horizon in the background spacetime (i.e., the hypersurface with radius r = r h for the background spacetime horizon r h = r h (0)), and it is bounded by the bifurcation surface B as well as the very late cross section B 1 . Σ 1 is a spacelike hypersurface connected B 1 and spatial infinity where the dynamical fields are described by static charged Gauss-Bonnet solution. Note that Σ 0 (i.e., r = r h ) is not the horizon of the spacetime with the metric g ab (λ ). For later convenience, by considering the gauge freedom of the electromagnetic field, we impose the gauge condition such that with the Killing vector ξ a = (∂ /∂ v) a of the background spacetime. Then, under this gauge condition, the dynamical fields can be expressed as with the blackening factor on the hypersurface Σ 1 . The strength of the electromagnetic field on Σ 1 is given by

III. PERTURBATION INEQUALITY
In this section, we would like to derive a perturbation inequality of above collision matter sources. Following the calculation as [21], we also consider the off-shell variation of the EMGB gravity neglected the matter fields part, i.e., The variation of above action gives where with the symplectic potential of the gravity part and electromagnetic part Θ GB a 1 ···a n−1 = 1 8π ǫ ba 1 ···a n−1 P a cbd δ Γ a cd + δ g bd ∇ a P acbd , and P abcd = 1 2 g ac g bd − g ad g bc + α GB The symplectic current (n − 1)-form is defined by Utilizing (21), one can further obtain with ω GB a 1 ···a n−1 (φ , δ 1 φ , δ 2 φ ) = 1 8π δ 1 ǫ ba 1 ···a n−1 P a cbd δ 2 Γ a cd −δ 2 ǫ ba 1 ···a n−1 P a cbd δ 1 Γ a cd + δ 1 (ǫ ba 1 ···a n−1 ∇ a P acbd )δ 2 g bd Replacing the variation by a infinitesimal diffeomorphism generated by the vector field ζ a , according to Eqs. (18) and (19), we can define a Noether current (n − 1)-form as It has been shown in [2] that it can be expressed as is the constraint of the Gauss-Bonnet gravity, and with is the (n − 2)-form Noether charge. Replacing ζ a by the Killing vector ξ a = (∂ /∂ v) a of the background geometry in Eqs. (26) and (27), the first two order variational identities can be further obtained, Here we have used the fact that T ab = j b = 0 for the background fields and ξ a is a Killing vector so that L ξ φ = 0. Integrating the first variational identity on the hypersurface Σ as introduced in the last section, we have where we used the assumption that the perturbation vanishes on the bifurcation surface B. For the first term, the integration is on the infinity boundary of Σ 1 , therefore, here the fields can be described by (15). Then, the gravity part can be straightly calculated and it gives For the EM part, using (15), we have on S ∞ , whereǫ is the volume element of the (n − 2)dimensional sphere, which can be expressed aŝ The above expression then gives Integrating it, we can further obtain Using (21), we can also obtain Then, we have Using the stability assumption that T ab (λ ) = j a (λ ) = 0 on Σ 1 , the third term of (33) vanishes. Combining above results, we have The last step we used the gauge condition ξ a A a (λ )| r=r h = 0.
Hereǫ is the volume element on Σ 0 , which is defined byǫ = dv ∧ǫ . Next, we turn to calculate the second-order variational identity in (19). With similar consideration and integrating it on Σ, we have where we denote Here we used that L ξ φ (λ ) = T ab (λ ) = j a (λ ) = 0 on Σ 1 from (15), and ξ a is a tangent vector on Σ 0 . With Straight calculation based on the explicit expressions on Σ 1 , it is not difficult to verify E 0 = 0 in our case. For the gravity part in left side in (41), similar calculation gives For the EM part, according to (34), we can further obtain With similar calculation, we also have Then, the second-order variational identity reduces to Finally, we consider the null energy condition. Note that in the above setup, ξ a is not a null vector on Σ 0 for the geometry with the metric g ab (λ ). Therefore, in order to obtain a relevant null energy condition, here we choose the null vector field as l a (λ ) = ξ a + β (λ )r a . with We can see that β = β (0) = 0 on Σ 0 since f (0) (r h , v) = f (r h ) = 0. Then, we have T ab (λ )l a (λ )l b (λ ) = T ab (λ )ξ a (dr) b + β 2 (λ )T ab (λ )r a r b .
when the matter fields satisfy the null energy condition and the collision satisfies the stability condition. Different with the result in [21], here we did not utilize the optimal condition of the first-order identity. As a result, we found that the static charged Gauss-Bonnet black holes cannot be overcharged by above collision process under the second-order approximation of perturbation, although they can be destroyed by the old version of the gedanken experiments as shown in [31]. Our result at some level implies that the WCCC can also be restored in the EMGB gravitational theory.