Weak cosmic censorship conjecture for the novel $4D$ charged Einstein-Gauss-Bonnet black hole with test scalar field and particle

Motivated by the recent research of black holes in $4D$ Einstein-Gauss-Bonnet (EGB) gravity, we investigate the possibility of destroying the event horizon of a charged $4D$ EGB black hole with a test massive complex scalar field and a test charged particle, respectively. For the scalar field scattering gedanken experiment, we consider an infinitesimal time interval process. The result shows that both extremal and near-extremal $4D$ charged EGB black holes cannot be overcharged. For the test particle thought experiment, the study suggests that extremal $4D$ charged EGB black hole cannot be overcharged to first order and can be overcharged to second order due to the existence of the Gauss-Bonnet coupling constant. While, near-extremal $4D$ EGB black hole can be overcharged to both first and second order.


I. INTRODUCTION
It is well known, in four dimensional Einstein-Gauss-Bonnet gravity, the Gauss-Bonnet term is a topological invariant and has no contribution to the gravitational field equations, which results in trivial black hole solutions in four dimensions. Recently, Glavan and Lin proposed a general covariant Einstein-Gauss-Bonnet modified theory of gravity in four-dimensions and presented a novel vacuum black hole solution [1]. Their theory circumvents the Loverlock's theorem and avoids Ostrogradsky instability by rescaling the Gauss-Bonnet coupling parameter α → α/(d − 4) and then taking the limit d → 4. Thus, the Gauss-Bonnet term has nontrivial contribution to the gravitational field equations. Following the spirit of Glavan and Lin, Fernandes generalized their black hole solution to include electric charge in an anti-de Sitter space and get a 4D charged spherical solution [2]. Later, applying the modified Newman-Janis algorithm, Kumar and Ghosh obtained a 4D rotating Einstein-Gauss-Bonnet black hole solution [3].
In this paper, we study the possibility of destroying the horizon of a charged black hole in the novel 4D Einstein-Gauss-Bonnet gravity. The destruction of the horizon of a black hole could provide us the possibility of checking the validity of Penrose' weak cosmic censorship conjecture [37]. The milestone to consider the destruction of a black hole horizon is the work of Wald [38], in which he proposed a plausible way to destroy the horizon of extremal Kerr-Newman black hole by throwing a particle with large charge or angular momentum into the black hole. The result shows that particles would cause the destruction of event horizon will not be captured by the black hole. Later, Hubeny's pioneer work shows that the horizon of a near-extremal charged black hole can be destroyed by droping a charged test particle [39], the result is the same for near-extremal Kerr black holes [40]. For other works to destroy the horizon of black holes, see Ref. . The aim of this paper is to investigate the possibility of destroying the horizon of a 4D novel charged Einstein-Gauss-Bonnet black hole by the scattering of a massive charged scalar field and a charged test particle, respectively. For scattering of charged scalar field, our result suggests that both extremal and near-extremal charged Einstein-Gauss-Bonnet black hole cannot be overcharged. For test particle injection, the study suggests that extremal 4D charged Einstein-Gauss-Bonnet black hole cannot be overcharged to first order and can be overcharged to second order. While, near-extremal 4D Einstein-Gauss-Bonnet black hole can be overcharged to both first and second order.
The outline of the paper is as follows. In Sec. II, we briefly review the 4D charged Einstein-Gauss-Bonnet black hole and its thermodynamics. In Sec. III, we explore the scattering of a massive complex scalar field in 4D charged Einstein-Gauss-Bonnet black hole background and obtain the energy and charge fluxes of the complex scalar field. In Sec. IV, we check the validity of the weak cosmic censorship conjecture for extremal and near-extremal black holes by scattering of the charged scalar field. In Sec. V, we check the possibility of overcharge the black hole by injection of a test charged particle. The last section is devoted to discussions and conclusions.

II. THE NOVEL 4D CHARGED EINSTEIN-GAUSS-BONNET BLACK HOLE AND ITS THERMODYNAMICS
The action of the Einstein-Gauss-Bonnet gravity with electromagnetic field in a d-dimensional spacetime is with the Gauss-Bonnet term where R is the curvature scalar, F µν is the field strength tensor F µν = ∂ µ A ν −∂ ν A µ , with A µ being the electromagnetic 4-vector potential of the black hole. The black hole solution was obtained by solving the field equations and adopting the limit d → 4 in Ref. [2]: with the electromagnetic 4-vector potential A = −Q/rdt. The parameters M and Q are the black hole mass and charge respectively, and α is the Gauss-Bonnet coupling constant. Taking the limit α → 0, the Reissner-Nordström will be recovered. The black hole solution has the same form with the one obtained in a comformal anomaly gravity [71,72]. This spacetime is singular at r = 0 due to the divergence of curvature scalar. The metric function can be written as f (r) = 2(r 2 − 2M r + Q 2 + α) where we have defined The event horizon is determined by the equation f (r) = 0, which is equivalent to the following equation For a non-extremal charged EGB black hole, the above equation gives the inner and outer horizons The two horizons coincide for an extremal black hole, and the degenerate horizon locates at r ex = M . The horizon disappears for M 2 < Q 2 + α . In this case, there is no black hole and the metric describes a charged naked singularity. In the following we denote the event horizon r + as r h . The temperature of the black hole can be calculated as The area of the event horizon of the black hole is and the electric potential of the event horizon is The first law of thermodynamics for the black hole is [13] with the entropy and the conjugate quantity A to the coupling parameter α It is worth noting that the entropy of the 4D charged EGB black hole is modified from the Bekenstein-Hawking entropy-area law by the Gauss-Bonnet coupling constant α, and this entropy is consistent as that obtained from the Iyer-Wald formula [27].

III. MASSIVE SCALAR FIELD IN CHARGED EINSTEIN-GAUSS-BONNET SPACE-TIME
A. The scattering for massive scalar field The dynamics of a massive scalar field Ψ with mass µ and charge q in the charged EGB spacetime minimally coupled to the gravity is governed by the equation of motion which can be written as To make the problem more easier to solve, it is convenient to make the following decomposition for the complex scalar field [73,74] where Y lm (θ, φ) are spherical harmonical functions and R lm (r) is the radial functions. Inserting the above equation into the equation of motion Eq. (16, we get the angular part of the equation and the radial part where l(l + 1) is the separation constant and l takes positive integra values. The solutions to the angular part of the equation are the spherical harmonical functions. Since the angular solution is well known and it can be normalized to unity, we are more interested in the radial part. The radial equation can be simplified by introducing the tortoise coordinate by the definition The tortoise coordinate ranges from −∞ to +∞ when r varies from the horizon r h to infinity, and thus covers the whole space outside the horizon. Then, we can write the radial equation as the following form It is convenient to investigate the radial equation near the horizon since we are more concerned with waves incident into the black hole. Near the horizon, Eq. (21) can be written as which is It has the following solution with positive and negative signs corresponding to the outgoing and ingoing wave modes, respectively. Since ingoing wave mode is the physically acceptable solution , we choose the negative sign. Thus, the field near the horizon is After obtaining the wave function, we can calculate the changes of the energy and charge of the black hole through the flux of the scalar field.

B. Thermodynamics during scattering of the charged scalar field
We shoot a single wave mode (l, m = 0) into the charged Einstein-Gauss-Bonnet black hole to investigate the changes of the parameters of the black hole. The energy and charge carried by the complex scalar field can be estimated from their fluxes at the event horizon. The energy-momentum tensor of the charged scalar field is given by with The energy flux through the event horizon is [75] and the charge flux through the event horizon is [76] where we have used the normalization condition of the spherical harmonical functions Y lm (θ, φ) in the integration. From the above two equation, we can see that waves with ω > qφ h , the energy and charge flow into the event horizon; while, for waves with ω < qφ h , the energy and charge fluxes are negative, which implies that this wave modes extract energy and charge from the black hole. This is called black hole superradiance [77]. During an infinitesimal time interval dt, the changes in the mass and charge of the black hole are For the black holes far from extremal, after the absorption of the infinitesimal energy and charge of the complex scalar field, the final state is still a black hole. The change in the black hole configuration can be represented in terms of the frequency ω and charge q of the complex scalar field. The change in the location of the horizon dr h can be obtained from the condition Then, we obtain the change of the horizon for the scattering process The change of the black hole area is which is always positive. This indicates that the area of the event horizon increases during the scattering of the complex scalar field, and it is consistent with Hawking's area increasing theorem, which states that during any classical process, the area of the black hole event horizon never decrease [78,79].

IV. OVERCHARGE THE BLACK HOLE WITH THE MASSIVE COMPLEX SCALAR FIELD
In this section, we examine the validity of the weak cosmic censorship conjecture by shooting a monotonic classical test complex scalar field with frequency ω and azimuthal harmonic index m = 0 into the extremal and near-extremal charged Einstein-Gauss-Bonnet black holes, and argue whether we can push the resulting composite object over the extremal limit, thus destroy the event horizon.
The event horizon of the black hole is determined by the metric function where we have defined with the minimum of ∆ at the point r min = M. For a black hole, the minimum of the metric function ∆ is negative or zero; while, there is no black hole for positive minimal value of ∆. In the process of absorbing the complex scalar field with energy dE and charge dQ, the change of the parameters of the black hole are After the absorption of the test complex scalar field, the minimum of the metric function ∆ min changes to ∆ ′ min , ∆ ′ min = ∆ ′ min (M + dM, Q + dQ, α) To check the validity of the weak cosmic censorship conjecture, we consider the extremal and near-extremal black holes. Now, the question is whether ∆ = 0 has a positive solution after the black hole absorbs the test field, or equivalently, whether the minimum ∆ min of the metric function is positive.
Since the event horizon radius r h of a near-extremal black hole is extremely close to the minimal radius r min = M , we can define an infinitesimal distance ǫ between r h and r min : It is clear that ǫ > 0 and ǫ = 0 correspond to the near-extremal and extremal black holes, respectively. Before the absorption of the scalar field, the minimum of the metric function ∆ can be written as It is convenient for us to consider an infinitesimal time interval dt. For a long period of time, it can be divided into a lot of small time intervals dt, and we can consider the scattering process for each time interval separately by only changing the black hole parameters.
After the absorption of the scalar field, the minimum of the metric function ∆ min becomes ∆ ′ min : To first order in dt, we have where φ h is the electric potential of the horizon defined in Eq. (11). Now, we can check whether the horizon of the charged Einstein-Gauss-Bonnet black hole can be destroyed and become naked singularity. This can be done by judging whether ∆ ′ min in (43) is positive. The metric describes a naked singularity for ∆ ′ min > 0, while for a black hole, ∆ ′ min ≤ 0. For the extremal charged Einstein-Gauss-Bonnet black hole, the relation between the electric potential of the black hole φ h = Q/M and the minimum of ∆ ′ is which can never be positive. For ω = qφ h , there will be two horizons after the absorption of the field and so the extremal charged Einstein-Gauss-Bonnet black hole will become a non-extremal one after the scattering. While for ω = qφ h , it will still be extremal. This indicates the horizon of the extremal charged Einstein-Gauss-Bonnet black hole cannot be destroyed, and the weak cosmic censorship conjecture is preserved. For a near-extremal 4D charged Einstein-Gauss-Bonnet black hole, it is obvious that, for wave modes with the value of ∆ ′ min is the largest. Thus, if these wave modes cannot destroy the horizon of the near-extremal 4D charged Einstein-Gauss-Bonnet black hole, all the wave modes cannot destroy the near-extremal black hole either. We shoot one of these wave modes into the near-extremal black hole. Then, we have By choosing the infinitesimal time interval dt ∼ ǫ, we can see that which shows that it is impossible to overcharge the black hole and the event horizon cannot be destroyed. Thus, both the extremal and near-extremal 4D charged Einstein-Gauss-Bonnet black holes cannot be overcharged by test complex scalar fields, and the weak cosmic censorship conjecture is preserved.

V. OVERCHARGE THE BLACK HOLE WITH TEST PARTICLE
Another way to check the validity of the weak cosmic censorship conjecture is throwing a test particle with large charge or angular momentum into an extremal or near-extremal black hole. This gedanken experiment was first envisaged by Wald [38], it was shown that the event horizon of an extremal Kerr-Newman black hole cannot be destroyed. This method was further developed by Hubeny, Jacobson and Sotiriou [39,40]. Their research suggested that the event horizon of a near-extremal Reissner-Nordström black hole or a Kerr black hole can be destroyed. In this section, we use this method to check whether we can destroy the event horizon of the 4D charged Einstein-Gauss-Bonnet black hole. We drop a test particle with rest mass m and charge δQ from rest at infinity. The particle moving in the 4D charged Einstein-Gauss-Bonnet space-time can be described by the equation of motion which can be derived from the Lagrangian Since we drop the particle at rest from infinity, the angular momentum of the test particle δJ = 0. The energy δE of the particle is and the angular momentum of the particle We first find the condition for a particle with energy δE and charge δQ to enter the black hole, then check whether such particle can overcharge the black hole and destroy the event horizon.
The four velocity of a massive particle is a time-like and unit vector, Substituting the energy δE (50) and angular momentum δJ = 0 into the above equation, we get g 00 δE 2 + 2g 00 A t δQδE + g 00 A 2 t δQ 2 + g 11 P 2 r + g 22 P 2 θ = −m 2 .
Then the energy of the particle is For a massive particle outside the event horizon, the motion should be future directed and time-like. In the above equation we have chosen the future directed solution dt/dτ > 0, which is equivalent to the following condition If the particle enters the black hole, it must cross the event horizon. On the event horizon, the condition becomes Thus, for the particle to be absorbed by the black hole, the energy of the particle must satisfy On the other hand, to destroy the horizon of the black hole, the minimum of the metric function ∆ should be positive after the absorption of the particle.
To first order, the condition is Then, we have When the two conditions (57) and (59) are satisfied simultaneously, the horizon will be destroyed and the weak cosmic censorship conjecture will be violated. For the extremal 4D charged Einstein-Gauss-Bonnet black hole, we have M 2 − Q 2 − α = 0 and φ h = Q/M . Therefore, which means that the two conditions (57) and (59) can not be satisfied simultaneously. Thus, to first order, the horizon of the extremal 4D charged Einstein-Gauss-Bonnet black hole cannot be destroyed as previous research suggested for other extremal black holes [38,40]. For the near-extremal charged Einstein-Gauss-Bonnet black hole, since the event horizon r h = M + M 2 − Q 2 − α > M , then, we have φ h = Q/r h < Q/M . To first order, it is clear that there exist particles with charge δQ such that δE max > δE min , which shows that the two conditions (57) and (59) can be satisfied simultaneously.
For positive coupling constant α, the condition to destroy the horizon of the black hole is The above inequality describes a region below the upper branch of a hyperbola (for positive coupling constant). It is clear that the two conditions (57) and (59) can be satisfied simultaneously both for extremal and near-extremal charged Einstein-Gauss-Bonnet black holes.We emphasize here, that due to the existence of the positive coupling constant, extremal charged Einstein-Gauss-Bonnet black hole can be overcharged, which is contrary to previous results of research that other extremal black holes cannot be overcharged or overspun by test particle [38,40]. Hence, to first order, extremal 4D charged Einstein-Gauss-Bonnet black hole cannot be overcharged; while, nearextremal 4D charged Einstein-Gauss-Bonnet black hole can be overcharged. However, to second order, both extremal and near-extremal 4D charged Einstein-Gauss-Bonnet black hole can be destroyed, and the weak cosmic censorship conjecture is violated.

VI. DISCUSSION AND CONCLUSIONS
In this paper, we have investigated the possibility of destroy the event horizon of a 4D charged Einstein-Gauss-Bonnet hole by test charged scalar fields and charged particles. For the test charged scalar field scattering gedanken experiment, we considered an infinitesimal time interval. The result suggests that both extremal and near-extremal 4D charged Einstein-Gauss-Bonnet black holes cannot be overcharged. For the test particle thought experiment, the study suggests that extremal 4D charged Einstein-Gauss-Bonnet black hole cannot be overcharged to first order and can be overcharged for second order. While, near-extremal 4D Einstein-Gauss-Bonnet black hole can be overcharged both to first and second order.