Recent research has showed that by accounting for the nonlinear electrodynamic field effects on the motion of test charged particles it is possible to destroy an extremal and nearextremal black hole Phys.Rev.D 101 2020 12 124067. Inspired by this perspective we explore the potential for the destruction of a nonsingular black hole within the framework of general relativity where gravity couples with nonlinear electrodynamic field. By combining the theory of nonlinear electrodynamics in curved spacetime we obtain the conserved charges for test charged particles then investigate the possibility of destroying the event horizon of the AynBeato Garca ABG nonsingular black hole by test charged particle and complex scalar field. We obtain the intervals for the parameter of the particle and scalar field which allow us to destroy the event horizon of an extremal and nearextremal ABG black hole black hole. Due to the nonlinear electrodynamic field effects both an extremal and nearextremal black hole can be destroyed. Interestingly we find that the parameter intervals for the test charged particle and the test complex scalar field that can destroy the event horizon of an extremal black hole are identical. Importantly as the ABG black hole is a nonsingular black hole our findings remain consistent with the weak cosmic censorship conjecture and might provide a viable approach to observe the internal structure of a black hole.
['General Relativity and Quantum Cosmology']
Overcharging a nonsingular black hole in general relativity the nonlinear electrodynamic field effects
{'Overcharging a nonsingular black hole in general relativity: the nonlinear electrodynamic field effects': 'Wei-Jie Miao, a,b,c,d Si-Jiang Yang 1 a,b,c  \n- a Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China\n- b Key Laboratory of Theoretical Physics of Gansu Province, Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China\n- c School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China\n- d Cuiying Honors College, Lanzhou University, Lanzhou 730000, China  \nE-mail: miaowj21@lzu.edu.cn , yangsj@lzu.edu.cn  \nAbstract: Recent research has showed that, by accounting for the nonlinear electrodynamic field effects on the motion of test charged particles, it is possible to destroy an extremal and near-extremal black hole [Phys.Rev.D 101 (2020) 12, 124067]. Inspired by this perspective, we explore the potential for the destruction of a nonsingular black hole within the framework of general relativity, where gravity couples with nonlinear electrodynamic field. By combining the theory of nonlinear electrodynamics in curved spacetime, we obtain the conserved charges for test charged particles, then investigate the possibility of destroying the event horizon of the Ayón-Beato García (ABG) nonsingular black hole by test charged particle and complex scalar field. We obtain the intervals for the parameter of the particle and scalar field which allow us to destroy the event horizon of an extremal and near-extremal ABG black hole black hole. Due to the nonlinear electrodynamic field effects, both an extremal and near-extremal black hole can be destroyed. Interestingly, we find that the parameter intervals for the test charged particle and the test complex scalar field that can destroy the event horizon of an extremal black hole are identical. Importantly, as the ABG black hole is a nonsingular black hole, our findings remain consistent with the weak cosmic censorship conjecture and might provide a viable approach to observe the internal structure of a black hole.  \nKeywords: Classical gravity, nonsingular black hole, weak cosmic censorship conjecture, nonlinear electrodynamic field.', '1 Introduction': "In Einstein's general relativity, Penrose proved that gravitational collapse of matter satisfying reasonable energy condition inevitably leads to spacetime singularity. This is the so-called Hawking-Penrose singularity theorem [1, 2]. To avoid the appearance of 'naked singularities' and rescue general relativity from the loss of predictability, Penrose proposed the weak cosmic censorship conjecture [3]. In the conjecture, it states that singularities generated from gravitational collapse with physically reasonable matter must be covered by event horizon, then singularities cannot be observed by the observers outside the horizon. A general proof of the conjecture is known to be quite challenging and remains beyond reach.  \nIn the pioneer work of testing the weak cosmic censorship conjecture, Wald proposed a gedanken experiment by dropping test particles with large charge or angular momentum into an extremal Kerr-Newman black hole, and demonstrated that an extremal Kerr-Newman black hole cannot be destroyed by test particles [4]. This work is a kind of evidence for the validity of the weak cosmic censorship conjecture. Systematic works by Rocha and Cardoso et al. for Bañados-Teitelboim-Zanelli (BTZ) black holes [5], rotating quantum BTZ black hole [6], higher-dimensional Myers-Perry family of rotating black holes, and a large class of five-dimensional black rings [7] also suggested that extremal black holes cannot be destroyed by test particles. Nevertheless, further investigations of Hod and Hubeny showed that near-extremal black holes might be destroyed by test particles [8, 9]. The work of Jacobson and Sotiriou also indicate that a near-extremal Kerr black hole can be destroyed by test  \nparticles[10, 11]. Similar counterexamples are found in black holes with NUT parameters or magnetized black holes in general relativity [12-15] and other black holes in modified theories of gravity [16-18]. Besides, the investigation of Gao and Zhang suggested that if the second order of the energy, charge and angular momentum of particles are taken into account, even extremal Kerr-Newman black holes can be destroyed [19]. Similar results can be found for the 4 D Einstein-Gauss-Bonnet black hole [20]. Besides destroying the event horizon with test particles, we can also do the gedanken experiment through the scattering of test fields. By shooting a charged scalar field into an extremal magnetized Kerr-Newman black hole, Semiz found that an extremal dyonic Kerr-Newman black hole cannot be destroyed [21]. Systematic works for scalar field scattering suggest the same results [22-28]. Usually, classical fields cannot destroy the event horizon. Recently, Sorce and Wald proposed a new thought experiment that have taken into account the second-order perturbations from the matter fields. The investigation shows that a near-extremal Kerr-Newman black hole cannot be destroyed and the weak cosmic censorship conjecture is valid [29]. The systematical works for other black holes using this new formalism suggest that both extremal and near extremal black holes cannot be destroyed [30-42]. Using Iyer-Wald formalism even to high order perturbations the results indicate that the weak cosmic censorship conjecture still cannot be violated [43-45].  \nSpacetime singularities are widely regarded as the sign of the failure of general relativity. When quantum effects near singularities are taken into account, spacetime singularities might be disappear. But a consistent quantum theory of gravity is still illusive. To solve the problem of black hole singularity, many viewpoints to get nonsingular black holes have been proposed, such as employing 'black-bounce' mechanism, properly modifying the metric or introducing nonlinear electrodynamic field to removing the central singularity [46-51]. The investigation of nonsingular black holes has attracted a lot of attention. By constructing an action for nonlinear electrodynamic field coupled with gravity, Ayón-Beato and García obtained an analytical nonsingular black hole solution in Einstein's general relativity. Then Cai and Miao achieved a kind of generalized ABG related black hole solutions which are dependent on five parameters named the mass, charge and three parameters related to nonlinear electrodynamic field [52].  \nSince there is no spacetime singularity for regular black holes, the destruction of the event horizon of a nonsingular black hole does not violate the weak cosmic censorship conjecture. Thus it is not protected by the weak cosmic censorship conjecture [53]. The destruction of the event horizon of nonsingular black holes are also widespread concerned [54, 55]. Li and Bambi discussed the feasibility to destroy the horizon of rotating nonsingular Bardeen and Hayward black holes [53]. The source of the Bardeen black hole has been interpreted as a magnetic monopole, which can be derived from the nonlinear electrodynamics [56]. Without taking into account the effects of the nonlinearity of the electromagnetic field on the motion of charged test particle, Li and Bambi found that these nonsingular black holes might be destroyed by test particles [53]. When the effects of the nonlinear electrodynamic field are taken into account, the work of Liu and Gao strongly support that nonlinear electrodynamic field corrected black holes can be destroyed by test charged particles [57].  \nMotivated by the work of Liu and Gao, We consider the effects of nonlinear electrodynamics on the motion of the particle and charged scalar field, then explore the possibility of destroying the event horizon of the ABG nonsingular black hole with test particle and test complex scalar field. we consider the extremal and near-extremal ABG black hole and get the parameter region for the energy and charge that can destroy the event horizon. The results demonstrates that both the extremal and near-extremal ABG black hole can be destroyed by charged test particles. We also extend the scattering of a monotonic scalar field to a multifrequency complex scalar field to overcharge the black hole. Interestingly, we find that the parameter interval to destroy an extremal ABG black hole by a charged scalar field is exactly the same with that of a particle.  \nThe outline of the paper is as follows. In Sec. 2, we briefly review the characteristics of the ABG black hole. In Sec. 3, by considering the effects of the nonlinear electrodynamic field effects on the motion of the charged test particle, we investigate the possibility of overcharging the ABG black hole by test charged particle. In Sec. 4, we try to destroy the event horizon of an extremal and near-extremal ABG black holes by a charged scalar field. The last section is devoted to discussion and conclusion.", '2 The nonsingular black hole in general relativity': "Spacetime singularities are windows into physics beyond general relativity. When quantum effects are taken into account, spacetime singular might disappear. There are many viewpoints to avoid spacetime singularity. One of the ways is to assume nonlinear electrodynamic field coupled to gravity. In this section, we presents a brief introduction of the nonsingular black hole in Einstein gravity where nonlinear electrodynamic field coupled to gravity, as proposed by Ayón-Beato and García. The action for the gravity theory is given by [58]  \nS = ∫ d 4 x ( 1 16 π R -1 4 π L ( F ) ) , (2.1)  \nwhere R is the curvature scalar, and L is the Lagrangian for nonlinear electrodynamic field which is a function of the electromagnetic field tensor F = 1 / 4 F µν F µν . Alternatively, the action for the nonlinear electrodynamic field can be described by the Hamiltonian [58]:  \nH = 2 F L F -L , (2.2)  \nwhere L F is the derivative of the Langrangian with respect to F . Defining P µν = L F F µν , then the Hamiltonian H is a function of  \nP = 1 4 PµνP µν = ( L F ) 2 F, (2.3)  \nwhich is determined by the nonlinear electrodynamic source:  \nH ( P ) = P (1 -3 √ -2 q 2 P ) ( 1 + √ -2 q 2 P ) 3 -3 2 q 2 s ( √ -2 q 2 P 1 + √ -2 q 2 P ) 5 / 2 , (2.4)  \nwhere s = | q | / 2 m and the invariant P are non-negative quantities. The corresponding Lagrangian occurs to be  \nL ( P ) = P ( 1 -8 √ -2 q 2 P -6 q 2 P ) ( 1 + √ -2 q 2 P ) 4 -3 4 q 2 s ( -2 q 2 P ) 5 / 4 ( 3 -2 √ -2 q 2 P ) ( 1 + √ -2 q 2 P ) 7 / 2 . (2.5)  \nBy solving the Einstein's field equation and the nonlinear Maxwell equation, Ayón-Beato and García got an exact four-dimensional stationary spherical charged nonsingular black hole solution, which is called the ABG nonsingular spacetime. The metric can be written in the form [59]:  \nds 2 = -f ( r ) dt 2 + f -1 ( r ) dr 2 + r 2 d Ω 2 , (2.6)  \nwhere the metric function  \nf ( r ) = 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 , (2.7)  \nwith the non-vanishing components of electromagnetic tensor:  \nF rt = -F tr = ∂ r A t -∂ t A r = qr 4 ( r 2 -5 q 2 ( r 2 + q 2 ) 4 + 15 2 m ( r 2 + q 2 ) 7 / 2 ) . (2.8)  \nFrom the definition of the electromagnetic field tensor, we can obtain the non-vanishing components of the vector potential:  \nA t = ∫ r ∞ qr 4 ( r 2 -5 q 2 ( r 2 + q 2 ) 4 + 15 2 m ( r 2 + q 2 ) 7 / 2 ) dr = -[ qr 5 ( q 2 + r 2 ) 3 -3 r 5 m 2 q ( q 2 + r 2 ) 5 / 2 + 3 m 2 q ] . (2.9)  \nThe spacetime is globally regular, with m representing the ADM mass and q the electric charge of the black hole. For an appropriate mass-to-charge ratio, the metric describes a black hole with a global structure similar to a Reissner-Nordström black hole but with a regular center instead of a spacetime singularity [60]. The ABG black hole is the nonsingular black hole in general relativity, and the curvature invariants R , R µν R µν , R µναβ R µναβ etc. of the spacetime are regular everywhere [59].  \nThe event horizon is determined by the metric function f ( r ) = 0 , which is equivalent to the following equation  \nf ( r ) = 1 -1 s x 2 (1 + x 2 ) 3 / 2 + x 2 (1 + x 2 ) 2 = 0 , (2.10)  \nwhere s = | q | / 2 m and x = r h / | q | is a continuous function of variable s . The event horizon of the black hole can be solved directly. It is [59]  \nr ± = | q | √ √ √ √ ( 1 4 s + √ h ( s ) 12 s ± √ 6 12 s √ 9 2 -12 s 2 -h ( s ) 6 -9(12 s 2 -1) √ h ( s ) ) 2 -1 , (2.11)  \nwith h ( s ) and g ( s ) given by  \nh ( s ) = 6 ( 3 2 -4 s 2 + sg ( s ) 1 / 3 -4 s ( 11 s 2 -3 ) g ( s ) 1 / 3 ) , g ( s ) = 4 ( 9 s +74 s 3 + √ 27(400 s 6 -112 s 4 +47 s 2 -4) ) , (2.12)  \nwhere s is the parameter that determines the number of horizon of the spaceitme. For the extremal ABG black hole, we have:  \ns = s 0 = t 0 ( t 2 0 -3) 2 t 2 0 -4 ≈ 0 . 317090 , x = x 0 = √ t 2 0 -1 ≈ 1 . 584786 , (2.13)  \nwhere we have defined a constant  \nt 0 = √ √ √ √ 1 3 ( 4 + 3 √ 1 2 ( 83 -3 √ 321 ) + 3 √ 1 2 ( 83 + 3 √ 321 ) ) ≈ 1 . 873912 (2.14)  \nto simplify the results. For s > s 0 , the spacetime has no horizon; when s < s 0 , the spacetime has two horizons. The event horizon of the extremal black hole is r ex = x 0 | q | . For later convenience, we denote the event horizon r + as r h and assume the charge of the black hole q is positive q > 0 .  \nA black hole is not only a strong gravity system, but also a thermodynamic system and it has temperature and entropy. The Hawking temperature for the black IS  \nT = f ' ( r h ) 4 π = r h 8 π [ 2 m ( r 2 h + q 2 ) 3 / 2 -q 2 ( r 2 h + q 2 ) 2 ]( r 2 h -2 q 2 r 2 h + q 2 ) -q 2 r 3 h 8 π ( r 2 h + q 2 ) 3 . (2.15)  \nThe Bekenstein-Hawking entropy of the black hole is  \nS BH = πr 2 h , (2.16)  \nand the electric potential of the event horizon is  \nφ h ( s ) = qr 5 h ( q 2 + r 2 h ) 3 -3 r 5 h m 2 q ( q 2 + r 2 h ) 5 / 2 + 3 m 2 q = x 5 -3 4 s x 5 √ 1 + x 2 (1 + x 2 ) 3 + 3 4 s . (2.17)", '3 Overcharging the nonsingular black hole with test particles': 'As it was shown in the previous section, the ABG black hole is a nonsingular black hole and without spacetime singularity. As proposed by Li and Bambi [53], the destruction of a nonsingular black hole leads no violation of the weak cosmic censorship conjecture and might provided us the possibility to observe the interior of the black hole. In this section, we investigate the possibility of destroying the event horizon of the nonsingular black hole with charged test particles. By employing the method proposed by Liu and Gao [57], which incorporates the effects of nonlinear electrodynamic field on the motion of charged test particles, we derive the conserved quantities of these particles in the ABG nonsingular spacetime. After getting the conserved quantities for the particle in the spacetime, we check whether the nonsingular black hole can be destroyed by the charged test particles.', '3.1 Conserved charges in spacetime with nonlinear electrodynamic field': "We now turn our attention to the motion of a charged test particle in spacetime with nonlinear electrodynamic field, aiming at deriving the conserved quantities for the charged particles along its trajectory.  \nAssuming the Lagrangian for the nonlinear electrodynamic field is a function of F and Y , denoted by L NE ( F, Y ) , where F and Y are defined in terms of the electromagnetic field tensor F µν and its Hodge dual tensor ∗ F µν :  \nF = F µν F µν , (3.1)  \nY = ∗ F µν F µν . (3.2)  \nA general theory of gravity with sourceless nonlinear electrodynamic field can be described by the action [61]:  \nS = ∫ d 4 x √ -gR + S M , S M = -∫ d 4 x √ -g L NE ( F, Y ) , (3.3)  \nwhere R is the curvature scalar. In the absence of magnetic monopoles, the electromagnetic field tensor F µν is a closed form, thus we can introduce the vector potential A µ such that  \nF µν = ∇ µ A ν -∇ µ A ν = ( dA ) µν , (3.4)  \nwhich shows  \n∇ µ ∗ F µν = 0 . (3.5)  \nIn order to facilitate our discussion later, we define the following quantities:  \nh = ∂ L NE ( F, Y ) ∂F , f = ∂ L NE ( F, Y ) ∂Y , X µν = hF µν , H µν = f ∗ F µν , L µν = X µν + H µν . (3.6)  \nIt is evident that L µν is antisymmetric.  \nVarying the action eq. (3.3) with respect to the metric g µν and the vector potential A µ yields the Einstein's field equation and the nonlinear electrodynamic field equation respectively,  \nR µν -1 2 Rg µν = 8 πT µν , (3.7) µν (3.8)  \n∇ µ L = 0 ,  \nwhere the energy momentum tensor of the nonlinear electrodynamic field T µν is given by  \nT µν = -2 √ -g δS M δg µν = 1 4 π ( L α µ F να -1 4 L NE g µν ) . (3.9)  \nA charged test particle in a spacetime with linear electrodynamic field, the particle feels a Lorentz like force. But when considering the nonlinear electodynamic field, the Lorentz like force should be modified as proposed by Liu and Gao [57]. In order to derive the modified electromagnetic force on a charged particle in the spacetime with nonlinear electrodynamic field, it is necessary to calculate the covariant derivative of the energy-momentum tensor:  \n∇ µ T µν = 1 4 π ∇ µ ( L α µ F να -1 4 L NE g µν ) = 1 4 π [ F να ∇ µ L µα +( X µα + H µα ) ∇ µ F να -1 4 ∇ ν L NE ] = 1 4 π [ F να ∇ µ L µα + ( hF µα ∇ µ F να -1 4 h ∇ ν F ) + ( f ∗ F µα ∇ µ F να -1 4 f ∇ ν Y )] . (3.10)  \nCalculations lead us to  \nh ∗ F να ∇ µ ∗ F µα = -hF µα ∇ µ F να + 1 4 h ∇ ν F, (3.11)  \nfF να ∇ µ ∗ F µα = f ∗ F µα ∇ µ F να -1 4 f ∇ ν Y. (3.12)  \nThen eq. (3.10) becomes  \n∇ µ T µν = 1 4 π F να ∇ µ L µα -1 4 π ( h ∗ F να -fF να ) ∇ µ ∗ F µα = 1 4 π F να ∇ µ L µα -1 4 π ∗ L να ∇ µ ∗ F µα . (3.13)  \nIn the sourcefree case, eqs. (3.5) and (3.8) indicate  \n∇ µ T µν = 0 , (3.14)  \nwhich is just the conservation of energy and momentum. However, if the particle carries electric and magnetic charge, we can define the 4-electric and 4-magnetic current densities  \nJ ν = -1 4 π ∇ µ L µν , ˆ J ν = -1 4 π ∇ µ ∗ F µν . (3.15)  \nThen eq. (3.13) becomes  \n∇ µ T µν = -F νµ J µ + ∗ L νµ ˆ J µ . (3.16)  \nAccording to eqs.(3.6) and (3.15), we have  \n∇ µ J µ = 0 , ∇ µ ˆ J µ = 0 , (3.17)  \nwhich are the conservation law of electric and magnetic currents, respectively.  \nFor a point particle with electric charge q and magnetic charge g , integrating eq. (3.16) along the worldline, we can obtain the extended Lorentz force  \nF µ = qF µν U ν -g ∗ L µν U ν . (3.18)  \nwhere U ν is the four-velocity of the test particle, i.e. the tangent vector of the worldline U ν = dx ν /dτ . If the magnetic charges or monopoles are taken into account in Maxwell's theory, the Lorentz force takes exactly the same form as eq. (3.18). The authors of [61] derived the force on a particle with electric charge but without magnetic charge in nonlinear electrodynamics, which is also consistent with eq. (3.18).  \nIn the rest part of this paper, we only consider nonlinear electrodynamic field with Y = 0 , so  \nL µν = X µν . (3.19)  \nIn the sourceless case, eq. (3.8) indicates that ∗ X µν is closed and then we can introduce the potential ˆ A such that  \n∗ X µν = ∇ µ ˆ A ν -∇ ν ˆ A µ = ( d ˆ A ) µν . (3.20)  \nAccording to expression for the extended Lorentz force eq. (3.18), the equation of motion of a particle with electric and magnetic charge in the spacetime with nonlinear electrodynamic field is determined by  \nmU ν ∇ ν U µ = qF µν U ν -g ∗ L µν U ν . (3.21)  \nIf the spacetime possesses a Killing vector field ξ a , then there is a conserved quantity for the particle along the worldline. To achieve this goal, we calculate  \nU ν ∇ ν ( mξ µ U µ ) = mξ µ U ν ∇ ν U µ . (3.22)  \nIn the derivation of the above result, we employed the Killing equation. Combine eqs. (3.20), (3.21) and (3.22), we have  \nU ν ∇ ν ( mξ µ U µ ) = mξ µ U ν ∇ ν U µ = qU ν [ ξ µ ∇ µ A ν -∇ ν ( ξ µ A µ ) + A µ ∇ ν ξ µ ] -gU ν [ ξ µ ∇ µ ˆ A ν -∇ ν ( ξ µ ˆ A µ ) + ˆ A µ ∇ ν ξ µ ] , (3.23)  \nFrom the above equation, we can get  \nD dτ ( mξ µ U µ + qξ µ A µ -gξ µ ˆ A µ ) = qU ν ( ξ µ ∇ µ A ν + A µ ∇ ν ξ µ ) -gU ν ( ξ µ ∇ µ ˆ A ν + ˆ A µ ∇ ν ξ µ ) = qU ν ( ξ µ ∇ µ A ν -A µ ∇ µ ξ ν ) -gU ν ( ξ µ ∇ µ ˆ A ν -ˆ A µ ∇ µ ξ ν ) = qU ν ( L ξ A ν ) -gU ν ( L ξ ˆ A ν ) = 0 . (3.24)  \nTherefore, the quantity associated with the Killing vector field ξ µ  \nP = mξ µ U µ + qξ µ A µ -gξ µ ˆ A µ (3.25)  \nis conserved along the geodesics. The conserved quantity is of significant importance in the following, as the parameter changes of the black hole is closely related to the conserved quantities of the test particle.", '3.2 Destroying the event horizon of the ABG black hole with particle': "After obtaining the conserved quantities of a charged test particle in a spacetime with nonlinear electrodynamic field, we check whether the ABG nonsingular black hole can be destroyed by charged test particles. We shoot a charged test particle with mass δm and charge δq into the black hole from rest at infinity. The particle will move along the radial direction since the angular momentum of the particle is zero. From the metric of the ABG spacetime, the conserved energy of the electrically charged particle along the worldline is  \nδE = -( δm g tt ˙ t + δq A t ) = δm ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) dt dτ + δq [ r 5 ( q 2 -3 2 m √ r 2 + q 2 ) q ( q 2 + r 2 ) 3 + 3 m 2 q ] . (3.26)  \nTo overcharge the ABG black hole, the particle should across the event horizon and captured by the black hole. The worldline of the particle in the spacetime should be timelike and future directed.  \nSince the four-velocity of the particle is a timelike unit vector, we have  \ng µν U µ U ν = ˙ t 2 g tt + ˙ r 2 g rr = -1 , (3.27)  \nfrom the metric of the ABG black hole, we have  \n-˙ t 2 ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) + ˙ r 2 ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) -1 = -1 , (3.28)  \nThen the time component of the four-velocity is  \n˙ t = ± ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) -1 √ √ √ √ ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) + ˙ r 2 , (3.29)  \nwhere the sign choice in the solution eq. (3.29) is dictated by the requirement that the motion of the particle should be future directed, which is dt/dτ > 0 . Therefore the energy of the particle is  \nδE = δm √ √ √ √ ( 1 -2 mr 2 ( r 2 + q 2 ) 3 / 2 + q 2 r 2 ( r 2 + q 2 ) 2 ) + ˙ r 2 + δq [ r 5 ( q 2 -3 2 m √ r 2 + q 2 ) q ( q 2 + r 2 ) 3 + 3 m 2 q ] . (3.30)  \nAt the place where the test particle cross the event horizon, i.e. r = r h , the first term in the right hand side of eq. (3.30) should be non-negative, then we have  \nδE ≥ δE min =   r 5 h ( q 2 -3 2 m √ r 2 h + q 2 ) q ( q 2 + r 2 h ) 3 + 3 m 2 q   δq = ( x 5 -3 4 s x 5 √ 1 + x 2 (1 + x 2 ) 3 + 3 4 s ) δq = φ h δq. (3.31)  \nThe result indicate that the energy of the particle should be large enough to overcome the Columb force to be captured by the black hole, otherwise the charged test particle just 'miss' the black hole due to the Columb reposition force.  \nAfter the test particle being captured by the black hole, the spacetime still described by the metric (2.6) but the mass m and charge q of the black hole change to m ' = m + δE and q ' = q + δq , respectively. To ensure that the event horizon disappears after the particle being captured by the black hole, the energy and charge of the particle should satisfy  \ns ' = q ' 2 m ' = q + δq 2( m + δE ) > s 0 . (3.32)  \nThe equation can be simplified as  \nδE < δE max = δq 2 s 0 + m ( s s 0 -1 ) . (3.33)  \nThe result shows that to destroy the event horizon of the ABG black hole the charge of the test particle should be large enough.  \nHence, only the the energy δE and charge δq of the particle satisfy eqs. (3.31) and (3.33) simultaneously, can the event horizon of the nonsingular ABG black hole be destroyed by the charged test particle.  \nFor an extremal ABG nonsingular black hole, we have q = 2 ms 0 and x = x 0 . Then  \nδE min = ( x 5 0 -3 4 s 0 x 5 0 √ 1 + x 2 0 1 + x 2 0 3 + 3 4 s 0 ) δq = φ h ( s 0 ) δq, (3.34)  \n( ) δE max = δq 2 s 0 . (3.35)  \nEvaluating the difference between the upper and lower bound of energy, we have  \nδE max -δE min = ( 1 2 s 0 -φ h ( s 0 ) ) δq ≈ 0 . 003975 δq > 0 . (3.36)  \nThe result shows that there are charged test particles with energy δE and charge δq satisfy the two conditions (3.31) and (3.33) simultaneously. It demonstrates that the event horizon of an extremal ABG black hole can be destroyed by a charged test particle. The associated interval for charge and mass is relatively narrow, with an order of magnitude of approximately 10 -3 .  \nFor a near-extremal ABG nonsingular black hole, we define a positive infinitesimal parameter /epsilon1 to characterize the deviation from an extremal black hole  \n/epsilon1 = s 0 -s. (3.37)  \nIt describes an extremal ABG black hole when /epsilon1 = 0 and a near-extremal ABG black hole when 0 < /epsilon1 /lessmuch 1 .  \nTo destroy the event horizon of a near-extremal ABG nonsingular black hole, the charge of the test particle should be large enough. To ensure that the horizon can be destroyed after capturing the charged test particle, the energy and charge of the particle should satisfy eq.(3.30) and eq.(3.33) simultaneously. For a near-extremal ABG nonsingular black hole, we have  \n1 2 s 0 -φ h = 0 . 003975 + O ( /epsilon1 ) (3.38)  \nIt is evident that there are charged test particles with energy δE and charge δq satisfied eq.(3.30) and eq.(3.33) simultaneously. So as with the case of an extremal black hole, a nearextremal ABG nonsingular black hole can be destroyed by a charged test particle. This is a reasonable result given that the parameter in question is subject to continuous change. The above investigation shows that when the nonlinear electrodynamic field effects on the motion of the charged test particles are taken into account, both an extremal and near-extremal ABG nonsingular black holes can be overcharged by charged test particles.", '4 Overcharging the nonsingular black hole with complex scalar field': 'Besides testing the possibility of overcharging the charged ABG nonsingular black hole with test charged particles, we can also check the possibility of destroying the event horizon with charged scalar field. In this section, we examines the possibility of destroy the event horizon of the nonsingular black hole with charged scalar field. After investigating the possibility of destroying the nonsingular black hole with monotonic complex scalar field, we extend the investigation to multi-frequency complex scalar field.', '4.1 Massive complex scalar field in the ABG spacetime': 'We now consider the scattering of a massive complex scalar field in the ABG spacetime background. The action for the massive complex scalar field Ψ with mass µ s and charge Q minimally coupled to the gravity is  \nS ψ = ∫ L √ -gd 4 x (4.1)  \nwith the Lagragian density L  \nL = -1 2 D µ Ψ( D µ Ψ) ∗ -1 2 µ S ΨΨ ∗ , (4.2)  \nwhere D µ is the covariant derivative D µ = ∂ µ -iQA µ . The equation of motion for the complex scalar field can be derived from the action. It is  \n( ∇ µ -iQA µ ) ( ∇ µ -iQA µ ) Ψ -µ 2 s Ψ = 0 . (4.3)  \nExpanding the above equation, we obtain  \n1 √ -g ( ∂ µ -iQA µ ) [ √ -gg µν ( ∂ ν -iQA ν )Ψ ] -µ 2 s Ψ = 0 . (4.4)  \nSince the spacetime is static and spherically symmetric, the complex scalar field can be decomposed into the following form  \nΨ( t, r, θ, φ ) = e -iωt R lm ( r ) Y lm ( θ, φ ) , (4.5)  \nwhere Y lm ( θ, φ ) is the spherical harmonic function. Substituting eq. (4.5) into eq. (4.4), then the equation of motion for the complex scalar field can be divided into the radial part and the angular part. The radial part of the equation is  \n1 r 2 d dr [ r 2 f ( r ) dR lm dr ] + [ ( ω + QA t ) 2 f ( r ) -l ( l +1) r 2 -µ 2 s ] R lm = 0 , (4.6)  \nand the angular part is  \n[ 1 sin θ ∂ ∂θ ( sin θ ∂ ∂θ ) + 1 sin 2 θ ∂ 2 ∂φ 2 ] Y lm = -l ( l +1) Y lm , (4.7)  \nwhere l ( l + 1) is the eigenvalue of the spherical harmonic function and l is positive integral value. The solution to the angular part of the equation is the spherical harmonic function. Given that the angular solution is well established and can be normalised to unity in the calculation of the energy and charge fluxes, our focus is on the radial part of the equation.  \nTo solve the radial equation, we define the tortoise coordinate  \ndr dr ∗ = f ( r ) . (4.8)  \nWhen r varies from the event horizon r h to infinity ∞ , the tortoise coordinate r ∗ ranges from -∞ to ∞ , and hence covers the whole spacetime outside the event horizon. After the transformation, the radial equation can be simplified as  \nd 2 R lm dr 2 ∗ + 2 f ( r ) r dR lm dr ∗ + [ ( ω + QA t ) 2 -f ( r ) ( l ( l +1) r 2 -µ 2 s )] R lm = 0 . (4.9)  \nNear the event horizon, we have r ≈ r h and f ( r h ) = 0 . Equation (4.9) can be approximated as  \nd 2 R lm dr 2 ∗ +( ω -Qφ h ) 2 R lm = 0 . (4.10)  \nThe above equation represents a harmonic oscillatory system and can be solved directly. The solution to the eq. (4.10) is  \nR lm ( r ) = exp [ ± i ( ω -Qφ h ) r ∗ ] . (4.11)  \nThe positive sign on the exponent corresponds to outgoing wave modes, while the negative sign on the exponent corresponds to the ingoing wave modes. It is appropriate to select the negative sign, since the ingoing wave mode is the physically acceptable solution. Consequently, the charged complex scalar field near the event horizon has the following simple form  \nΨ = e -i ( ω -Qφ h ) r ∗ Y lm ( θ, φ ) e iωt . (4.12)  \nOnce the field function has been obtained, it is possible to calculate the change of the parameter for the black hole by analyzing the energy-momentum and charge flux of the complex field.', '4.2 Destroying the ABG black hole with a monotonic complex scalar field': "Since the ABG black hole is non-rotating, we shoot a monotonic complex scalar field with frequency ω and azimuthal angular momentum m = 0 into the black hole. The variation of parameters of the black hole can be calculated from the fluxes of the charged scalar field during the scattering process.  \nThe energy-momentum tensor of the charged complex scalar field is given by  \nT µ ν = -∂ L ∂ ( ∂ µ Ψ) ∂ ν Ψ -∂ L ∂ ( ∂ µ Ψ ∗ ) ∂ ν Ψ ∗ + δ µ ν L = 1 2 D µ Ψ ∂ ν Ψ ∗ + 1 2 ( D µ Ψ) ∗ ∂ ν Ψ -δ µ ν ( 1 2 D α Ψ( D α Ψ) ∗ -1 2 µ s ΨΨ ∗ ) , (4.13)  \nFrom the action of the complex scalar field (4.2), we can get the energy flux through the event horizon  \ndE dt = ∫ H T r t √ -gdθdφ = ∫ H ω ( ω -Qφ h ) Y 2 00 ( θ, φ ) r 2 sin θdθdφ = ω ( ω -Qφ h ) r 2 h . (4.14)  \nThe Lagrangian for the complex scalar field is invariant under a U (1) transformation. The electric current density is the conserved current, which is  \nj µ = -1 2 iQ (Ψ ∗ D µ Ψ -Ψ( D µ Ψ) ∗ ) . (4.15)  \nConsequently, the electric current through the event horizon is  \ndq dt = -∫ H j r √ -gdθdφ = ∫ H Q ( ω -Qφ h ) Y 2 00 ( θ, φ ) r 2 sin θdθdφ = Q ( ω -Qφ h ) r 2 h , (4.16)  \nwhere we have used the normalization condition for the spherical harmonic functions Y lm ( θ, φ ) in the calculation.  \nWithout loss of generality, we focus on a small time interval during the scattering process. For a long time period of scattering process, it can be decomposed into a series of such small time intervals, with each interval analyzed independently by changing the black hole parameters.  \nDuring this small time interval, the changes in the mass and charge of the black hole are  \ndm = dE = ω ( ω -Qφ h ) r 2 h dt, (4.17)  \ndq = Q ( ω -Qφ h ) r 2 h dt. (4.18)  \nThe ratio of the energy dE and charge dq absorbed by the black hole is  \ndE dq = ω Q . (4.19)  \nDuring the scattering process, the black hole absorbs the complex scalar field and its parameters changes to  \nm → m ' = m + dm, (4.20)  \nq → q ' = q + dq, (4.21)  \nand the parameter s changes to s '  \ns ' = q + dq 2( m + dm ) = s -q 2 m 2 dE + 1 2 m dq. (4.22)  \nSubstituting eqs. (4.17) and (4.18) into eq. (4.22), we have  \ns ' -s = -q 2 m 2 ω ( ω -Qφ h ) r 2 h dt + 1 2 m Q ( ω -Qφ h ) r 2 h dt = -Q 2 m s ( ω Q -1 2 s )( ω Q -φ h ) r 2 h dt. (4.23)  \nConsidering an initial extremal black hole, which is characterized by s = s 0 . becomes  \nEquation (4.23)  \ns ' -s 0 = -Q 2 m s 0 ( ω Q -1 2 s 0 )( ω Q -φ h ) r 2 h dt. (4.24)  \nThe electrostatic potential of the extremal black hole is  \nφ h = x 5 0 -3 4 s 0 √ 1 + x 2 0 x 5 0 (1 + x 2 0 ) 3 + 3 4 s 0 ≈ 1 . 572935 < 1 2 s 0 ≈ 1 . 576839 . (4.25)  \nIf we shoot the complex scalar field with frequency  \nφ h < ω Q < 1 2 s 0 , (4.26)  \nThen we have s ' -s 0 > 0 . Hence, an extremal black hole can be overcharged.  \nAs can be observed, the two sides of the interval are analogous to the particle situation  \n1 2 s 0 -φ h ≈ 0 . 003975 , (4.27)  \nthe allowed window for the ratio of energy and charge for the complex scalar field ω/Q to destroy an extremal ABG black hole is quite small. As with the case of charged particle to destroy the extremal ABG black hole, the interval for the ratio of the energy to charge (4.26) for the complex scalar field to destroy the black hole are the same as that of charge particle case (3.36).  \nSuppose the electromagnetic field is linear, then the ABG black hole reduce to the Reissner-Nordstrom black hole. The parameters of the black hole become  \ns 0 = q 2 m = 1 2 , x 0 = r h q = r h m = 1 , (4.28)  \nand the electrostatic potential of the extremal Reissner-Nordstrom black hole is  \nφ h = q m = 1 . (4.29)  \nThe result shows that the event horizon of an extremal Reissner-Nordstrom black hole cannot be destroyed by charged complex scalar field. Which is consistent with previous research that an extremal Reissner-Nordstrom black hole cannot be overcharged by a complex scalar field [23]. Besides, the results show that due to the nonlinear electrodynamic field correction, the extremal spherical charged nonsingular black hole can be overcharged.", '4.3 Destroying the ABG black hole with a non-monotonic complex scalar field': "The energy momentum tensor of the charged scalar field is a second order of the function of the scalar field and it is not linear dependent. For multi-frequency mode complex scalar field scattering, we can not use the simple rule of superposition. In this subsection, we consider the situation where the complex scalar field is not monotonic. For a linear combination of complex scalar field with various frequencies, the complex scalar field Ψ can be written as  \nΨ = n ∑ k =1 A k exp[ -i ( ω k -Qφ h ) r ∗ ] Y lm ( θ, φ ) e iω k t . (4.30)  \nSimilar to the case of monotonic complex scalar field scattering, we can get the energy flux and charge flux crossing the event horizon, which are  \ndE dt = { n | A k | 2 ω k ( ω k -Qφ h )  \n∑ k =1 (4.31) + ∑ 1 ≤ k<l ≤ n [ ω k ( ω l -Qφ h ) + ω l ( ω k -Qφ h )] /Rfractur [ A k A ∗ l exp ( i ( ω l -ω k )( t + r ∗ ))]    r 2 h , dq dt = Q { n ∑ k =1 | A k | 2 ( ω k -Qφ h ) (4.32) + ∑ 1 ≤ k<l ≤ n ( ω k + ω l -2 Qφ h ) /Rfractur [ A k A ∗ l exp[ i ( ω l -ω k )( t + r ∗ )]]    r 2 h . For the extremal black hole, we have  \ns ' -s 0 = -q 2 m 2 dm + 1 2 m dq = -Q 2 s 0 m n ∑ k,l =1 ( ω k Q -1 2 s 0 )( ω l Q -φ h ) /Rfractur { A k A ∗ l exp[ i ( ω l -ω k )( t + r ∗ )] } r 2 h dt. (4.33)  \nEvidently, there exists complex scalar field  \nφ h < ω k Q < 1 2 s 0 , (4.34)  \nsuch that there are situations  \ns ' -s 0 > 0 . (4.35)  \nIt is clear that after the extremal black hole absorbs the complex scalar field, the event horizon disappears. It means that the extremal nonsingular black hole can be overcharged by a multifrequency complex scalar field.", '5 Conclusion and Discussion': 'The weak cosmic censorship conjecture is one of the cornerstones for black hole physics. The destruction of a nonsingular black hole does not contravene the weak cosmic censorship conjecture. Consequently, it is possible to destroy a nonsingular black hole in principle.  \nIn this paper, we investigated the possibility of destroying a nonsingular black hole in Einstein gravity with gravity coupled to the nonlinear electrodynamic field. Since equation of motion for the electromagnetic field is no long the linear Maxwell equation, the Lorentz force for a charged particle in the spacetime with nonlinear electrodynamic field might modified. We derived the conserved charge for a charged test particle in the spacetime by combining nonlinear electrodynamics. Having obtained the conserved energy of the test particle in the spacetime, we proceeded to examine the possibility of destroying an extremal and nearextremal ABG black holes with test particles and complex scalar field, thereby identifying the parameter region. By analysing the scalar field in ABG spacetime, we obtain the field equation of the scalar field. For the scattering of the complex scalar field, we found that test field could destroy an extremal ABG black hole. It is inspiring that the parameter region for charged scalar field destroying the event horizon of an extremal ABG black hole is the same with that of a test charged particle. The investigation of destroying the event horizon of a nonsingular black hole in general relativity by taking into account the nonlinear electrodynamic field effects into account might give us new insight into the weak cosmic censorship conjecture and the posibility of destroying nonsingular black holes.', 'Acknowledgments': "This work was supported by National Key Research and Development Program of China (Grant No. 2021YFC2203003), the National Natural Science Foundation of China (Grants No. 12305065, No. 12247178 and No. 12247101), the China Postdoctoral Science Foundation (Grant No. 2023M731468), the 111 Project (Grant No. B20063), and Lanzhou City's scientific research funding subsidy to Lanzhou University.", 'References': "- [1] R. Penrose, Gravitational collapse and space-time singularities , Phys. Rev. Lett. 14 (1965) 57.\n- [2] S.W. Hawking and R. Penrose, The Singularities of gravitational collapse and cosmology , Proc. Roy. Soc. Lond. A 314 (1970) 529.\n- [3] R. 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