New version of the gedanken experiments to test the weak cosmic censorship in charged dilaton-Lifshitz black holes
NNew version of the gedanken experiments to test the weak cosmic censorship in chargeddilaton-Lifshitz black holes
Jie Jiang
1, * and Ming Zhang
2, † Department of Physics, Beijing Normal University, Beijing, 100875, China Department of Physics, Jiangxi Normal University, Nanchang 330022, China (Dated: August 31, 2020)In this paper, based on the new version of the gedanken experiments proposed by Sorce and Wald, we examinethe weak cosmic censorship in the perturbation process of accreting matter fields for the charged dilaton-Lifshitzblack holes. In the investigation, we assume that the black hole is perturbed by some extra matter sourcesatisfied the null energy condition and ultimately settle down to a static charged dilaton-Lifshitz black hole inthe asymptotic future. Then, after applying the Noether charge method, we derive the first-order and second-order perturbation inequalities of the perturbation matter fields. As a result, we find that the nearly extremalcharged dilaton-Lifshitz black hole cannot be destroyed under the second-order approximation of perturbation.This result implies that the weak cosmic censorship conjecture might be a general feature of the Einstein gravity,and it is independent of the asymptotic behaviors of the black holes.
I. INTRODUCTION
The general relativity predicts the existence of the blackhole. There is a central singularity for most of the black holes.However, the singularity will make the spacetime ill-definedand destroy the law of causality. Therefore, Penrose proposedthe weak cosmic censorship conjecture (WCCC) to ensure thepredictability in the physical spacetime region [1]. This con-jecture states that all singularities caused by the gravitationalcollapsing body must be hidden inside an event horizon suchthat it does not affect the causality outside the black hole. Italso means that the black holes cannot be destroyed by anyphysical process once it is formed if there is a singularity in-side the event horizon. To test this conjecture, Wald proposeda gedanken experiment to check whether the Kerr-Newman(KN) black hole can be destroyed by absorbing a test particle[2]. As a result, they found that the extremal KN black holescannot be overspun or overcharged in this process under thefirst-order approximation. However, there are two drawbacksto this discussion, i.e., the initial black hole is extreme and it isonly at the level of the first-order perturbation. For this storyto be truly consistent, Hubeny extended the discussion to thesecond-order case in the nearly extremal KN black holes andshowed that the nearly black hole can be destroyed in this case[3] when the second-order effects are neglected. Their resultattracted lots of researchers to extend it into various theories[4–24].Recently, Sorce and Wald pointed out that if we considerthe second-order correction, the spacetime cannot be easilytreated as a background and we need to consider the fulldynamical process of the spacetime and perturbation matter.Therefore, they proposed a new version of the gedanken ex-periments to overspin or overcharge the nearly KN black holes[25]. Based on the Noether charge method [26], they consid-ered the second-order corrections of the energy, angular mo-menta, and charge of the RN black hole, and derived a pertur- * [email protected] † [email protected] (corresponding author) bation inequality at second-order approximation. Then, theyconcluded that these nearly extremal KN black holes cannotbe destroyed after the second-order inequality are taken intoconsideration.Most recently, the discussion of the new version has alsobeen extended into some other stationary black holes [27–37].Although all of them showed the validity of the WCCC fornearly extremal black holes under the second-order approxi-mation of perturbation, there is still a lack of the general proofof the WCCC. However, most of the researches only focus onthe asymptotic flat spacetimes. We want to ask whether theWCCC is a general property for the Einstein gravity and itis independent of the asymptotic behaviors of the spacetime.Therefore, it is necessary for us to test the WCCC in the sit-uation with different asymptotic behaviors. In this paper, wewould like to consider the asymptotic Lifshitz black hole so-lution in Einstein-dilaton gravity and check whether the blackhole can be destroyed by the new version of the gedanken ex-periments.The remainder of this paper is organized as follows: insection II, we review the spacetime geometry of the chargeddilaton-Lifshitz black holes and discuss the perturbation fromthe physical process of accreting matter in this spacetime. Insection III, based on the Noether charge method as well as thenull energy condition for all of the matter fields, we derivethe first two order perturbation inequalities of the perturba-tion matter fields. In section IV, we discuss the possibility todestroy the nearly extremal black holes in the above physicalprocess under the second-order approximation of the pertur-bation. Section V is devoted to our conclusions. II. LINEARLY CHARGED DILATON-LIFSHITZ BLACKHOLES WITH THE SPHERICAL PERTURBATION
In this paper, we would like to test the WCCC of theasymptotic Lifshitz black holes by using the new version ofthe gedanken experiments proposed by Sorce and Wald [25].In this section, we first review the four-dimensional chargeddilaton-Lifshitz black hole solution in Einstein-dilaton gravity a r X i v : . [ g r- q c ] A ug coupled to a linear Maxwell electrodynamics and two Lifshitzsupporting gauge fields [38]. The Lagrangian four-form ofthis theory can be expressed as L = (cid:15) π (cid:32) R − Λ − ∇ a Φ∇ a Φ − ∑ i = e − α i Φ H i (cid:33) + L mt , (1)in which α i is some coupling constant fixed as α = −√ z − , α = √ z − , α = √ z − z > Φ is the dilaton field, R is the Ricci scalar related tothe metric g ab , and L mt is the Lagrangian of the extra mattersource, H = F ab F ab , and H i = ( H i ) ab ( H i ) ab with i = , F = d A is the strength of the electromagnetic field A ,and H i = d B i with i = , B i . The equations ofmotion derived from the variation of above Lagrangian aregiven by R ab − Rg ab − Λ g ab = π (cid:0) T EM ab + T B ab + T DIL ab + T mt ab (cid:1) , ∇ a G bai = π j bi , ∇ Φ + ∑ i = α i e − α i Φ H i = ψ (3)with G ab = e − α Φ F ab , G ab , = e − α , Φ H ab , , (4)in which we have denoted the stress-energy tensors of theelectromagnetic field, dilaton field, and supporting gaugefields as T EM ab = e − α Φ π (cid:20) F ac F bc − g ab H (cid:21) , T DIL ab = π (cid:18) ∇ a Φ∇ b Φ − g ab ∇ c Φ∇ c Φ (cid:19) , T B ab = ∑ i = T iab , (5)with T iab = e − α i Φ π (cid:20) ( H i ) ac ( H i ) bc − g ab H i (cid:21) , (6)with i = ,
3. Moreover, here T mt ab is the stress-energy tensor ofthe accreting matter fields, j a and j a , correspond to the cur-rent of the electromagnetic field and supporting gauge fieldsseparately, and ψ is the source of the dilaton field.In the following, we consider the four-dimensional chargeddilaton-Lifshitz solutions which are expressed as ds = − r z L z f ( r ) dv + r z − L z − dvdr + r (cid:0) d θ + sin θ d ϕ (cid:1) , Φ ( r ) = √ z − (cid:16) rb (cid:17) , A = − qb ( z − ) zr z dv , B = q r z + ( z + ) b dv , B = q r z zb dv (7) with the blackening factor f ( r ) = + L z r − Mr z + + q L z b ( z − ) zr ( z + ) , (8)and the constants q = b ( z − )( z + ) L z , q = b ( z − ) L ( z − ) z , Λ = − ( z + )( z + ) L . (9)Here b is some positive integral constant, q and M correspondto the electric charge and mass of the spacetime, separately. Ifthere exists at least one root of the blackening factor f ( r ) ,the solution describes a black hole. Otherwise, it describesa naked singularity. For the black hole case, the radius ofthe event horizon is the largest root of f ( r ) . If we also have f (cid:48) ( r h ) =
0, the black hole becomes extreme and the conservedquantities have the constraints, M = b L r zh + ( z + ) b r zh [ L + z ( z + ) r h ]( z + ) z b r zh , q = r zh [ L + z ( z + ) r h ] zL z b ( z − ) . (10)In this paper, we would like to consider the situation whenthe static charged dilaton-Lifshitz black hole is perturbed bythe spherically accreting matter fields which satisfy the nullenergy condition and it settles down to the dilaton-Lifshitzblack hole with different parameters in the asymptotic future.A concrete example of this process is that a static black holeslowly accreting matter for a finite time and finally becomesanother static black hole. In the following, we consider a fam-ily of above physical processes labeled by λ . Then, the dy-namical fields φ ( λ ) satisfies the equations of motion in (3).Here we denote φ to the collection of g ab , Φ , A , B , as wellas some extra matter fields. Generally, the spacetime in thisphysical process can be described by ds = − r z L z f ( r , v , λ ) dv + µ ( r , v , λ ) drdv + r ( d θ + sin θ d ϕ ) , (11)which satisfies f ( r , v , ) = f ( r ) , µ ( r , v , ) = r z − L z − (12)for the background geometry. As mentioned above, at suffi-ciently late times, we assume that the black hole can also bedescribed by the charged dilaton-Lifshitz solution with differ-ent parameters which can be labeled by λ , i.e., the line ele-ment can be expressed as f ( r , v , λ ) = f ( r , λ )= + L z r − M ( λ ) r z + + q ( λ ) L z b ( λ ) ( z − ) zr ( z + ) , µ ( r , v , λ ) = r z − L z − (13)at sufficiently late times. Then, the vector field ξ a = ( ∂ / ∂ v ) a becomes an Killing vector at late times. In this physical pro-cess, we also assume that the extra matter contains the sourcesof the dilaton field as well as the supporting gauge fields.Therefore, in the asymptotic future, the spacetime can be de-scribed by some different parameter b ( λ ) , mass M ( λ ) andcharge q ( λ ) . By virtue of this assumption, testing the weakcosmic censorship in this process is equivalent to checkingwhether the line element at late times describes a black holegeometry. III. PERTURBATION INEQUALITIES
In this section, we would like to derive some inequalities ofthe physical quantities for the perturbation at sufficiently latetimes under the second-order approximation. Different fromthe case of the asymptotic flat black holes, the mass of theblack holes cannot be easily expressed like that in asymptoticspacetime. For simplification, we only consider the off-shellvariation of the Einstein part. The Lagrangian four-form con-sidered is given by L = (cid:15) π R . (14)Following the notations in [25], we will denote η = η ( ) , δ η = d η d λ (cid:12)(cid:12)(cid:12)(cid:12) λ = , δ η = d η d λ (cid:12)(cid:12)(cid:12)(cid:12) λ = (15)for the physical quantity η ( λ ) in the family labeled by λ . Thevariation of above action gives δ L = E abg δ g ab + d Θ ( g , δ g ) , (16)in which E abg = − (cid:15) G ab , Θ abc ( g , δ g ) = π (cid:15) dabc g de g f g (cid:0) ∇ g δ g e f − ∇ e δ g f g (cid:1) . (17)Here G ab = R ab − / Rg ab is the Einstein tensor. Using aboveexpressions, the symplectic current three-form ω ( g , δ g , δ g ) = δ Θ ( g , δ g ) − δ Θ ( g , δ g ) , (18)can be expressed as ω abc = π (cid:15) dabc w d (19)with w a = P abcde f (cid:0) δ g bc ∇ d δ g e f − δ g bc ∇ d δ g e f (cid:1) (20)and P abcde f = g ae g f b g cd − g ad g be g f c − g ab g cd g e f − g bc g ae g f d + g bc g ad g e f . (21) B B h rr vv fieldsmatter accreting )( FIG. 1. Plot showing a dynamical configuration φ ( λ ) where thestatic nonextremal black hole is perturbed by the spherically accret-ing matter fields. Σ is a hypersurface determined by r = r h , where r h is the horizon radius of the background geometry φ ( ) . Differentfrom the hypersurface H in [25], Σ is not a null hypersurface in theconfiguration φ ( λ ) with the line element ds ( λ ) because r h is onlythe horizon radius of the background geometry. Using the Killing vector field ξ a = ( ∂ / ∂ v ) a of the back-ground spactime, the Noether current three-form is defined as J ξ = Θ ( g , L ξ g ) − ξ · L . (22)According to the calculation in [2], it can be also written as J ξ = C ξ + d Q ξ , (23)in which C ξ = ξ · C with C dabc = (cid:15) eabc G de , (cid:16) Q ξ (cid:17) ab = − π (cid:15) abcd ∇ c ξ d . (24)Based on above results, the first two order variational identi-ties can be calculated and they are expressed as d [ δ Q ξ − ξ · Θ ( g , δ g )] + ξ · E abg δ g ab + δ C ξ = , d [ δ Q ξ − ξ · δ Θ ( g , δ g )] = ω (cid:0) g , δ g , L ξ δ g (cid:1) − δ [ ξ · E abg δ g ab ] − δ C ξ , (25)in which we used the fact that ξ a is the Killing vector of thebackground geometry.Since we assume that the process of accreting matter fieldsis in a finite time. We can choose a hypersurface which ismade up of two portions Σ and Σ (as shown in Fig.1), where Σ starts at the cross-section B where the perturbation vanishesand goes through the event horizon of the background space-time (i.e., the hypersurface determined by r = r h = r h ( λ = ) )to another cross-section B at asymptotic future, and Σ istime-slice ( v = constant) connected B and spatial infinity S ∞ at sufficiently late times.Then, integration of the variational identities in (25) gives (cid:90) S ∞ (cid:2) δ Q ξ − ξ · Θ ( g , δ g ) (cid:3) + (cid:90) Σ ξ · E abg δ g ab + (cid:90) Σ δ C ξ + (cid:90) Σ δ C ξ = (cid:90) S ∞ (cid:2) δ Q ξ − ξ · δ Θ ( g , δ g ) (cid:3) + (cid:90) Σ δ [ ξ · E abg δ g ab ]+ (cid:90) Σ δ C ξ + (cid:90) Σ δ C ξ − E Σ − E Σ = , (27)where we denote E Σ i = (cid:90) Σ i ω ( g , δ g , L ξ δ g ) (28)with i = , (cid:90) S ∞ (cid:104) δ Q ξ − ξ · Θ ( g , δ g ) (cid:105) = δ ML z + . (29)Also, we have T ab ( λ ) dg ab ( λ ) d λ = − ( z − ) M (cid:48) ( λ ) L z + r + ( z − ) q ( λ ) b ( λ ) z − zL − z r ( z + ) (cid:2) b ( λ ) q (cid:48) ( λ ) + ( z − ) q ( λ ) b (cid:48) ( λ ) (cid:3) (30)on Σ . Then, the second term can be further obtained and it isexpressed as (cid:90) Σ ξ · E abg δ g ab = ( z − ) ln r h δ Mr z + h + ( z − ) qb z − L z − z r zh [ b δ q + ( z − ) q δ b ] . (31)Using Eq. (24), we have (cid:90) Σ C ξ ( λ ) = − ( z − ) q ( λ ) b ( λ ) ( z − ) L z − z r zh − r zh z L z + (cid:18) z r h z + + L ( z − ) (cid:19) − ( z − ) M ( λ ) ln r h L z + (32)Using the above result, the third term can be obtained as (cid:90) Σ δ C ξ = − ( z − ) ln r h δ Mr z + h − ( z − ) qb z − L z − z r zh [ b δ q + ( z − ) q δ b ] . (33)Summing these results, the first-order variational identity be-comes δ ML z + − qb z − L z − zr zh [ b δ q + ( z − ) q δ b ] = − (cid:90) Σ δ C ξ = δ (cid:20) (cid:90) Σ ˜ (cid:15) G ab ( dr ) a ξ b (cid:21) = δ (cid:20) (cid:90) Σ T ab k a k b dv ˆ (cid:15) (cid:21) , (34) where T ab is the stress energy tensor for all of the matter fields(including the dilaton field, electromagnetic field, supportinggauge fields, and extra matter fields), k a is a null vector fieldon Σ which is defined by k ( λ ) = (cid:18) ∂∂ v (cid:19) a + r zh f ( r h , v , λ ) L z µ ( r h , v , λ ) (cid:18) ∂∂ r (cid:19) a , (35)and the volume elements ˆ (cid:15) and ˜ (cid:15) are defined byˆ (cid:15) = r sin θ d θ ∧ d ϕ , ˜ (cid:15) = dv ∧ ˆ (cid:15) . (36)As mentioned in the last section, we assume that allof the matter fields satisfy the null energy condition, i.e., T ab ( λ ) k a ( λ ) k b ( λ ) ≥
0. Under the first-order approximationof perturbation, it gives δ [ T ab k a k b ] ≥
0. Then, the first-ordervariational identity (34) reduces to δ M − qb z − L z zr zh [ b δ q + ( z − ) q δ b ] ≥ . (37)The main purpose of this paper is to test whether the aboveperturbation process can destroy a nearly extremal black hole.When the first-order perturbation inequality is satisfied, in thenext section, we will show that the WCCC cannot be violatedunder the first-order approximation. However, if the pertur-bation satisfies the optimal condition which saturates the first-order perturbation inequality (37), i.e., δ M − qb z − L z zr zh [ b δ q + ( z − ) q δ b ] = , (38)the WCCC cannot be examined only considering the first-order approximation and the second-order approximationshould be taken into account. Therefore, in the follow-ing, we would like to derive the second-order perturbationinequality under the first-order optimal condition. Fromthe above discussions, the optimal condition also implies δ (cid:2) √− gT ab ( dr ) a ξ b (cid:3) = Σ by virtue of the null en-ergy condition. With a straightforward calculation, this gives ∂ v δ f ( r h , v ) =
0, in which we have defined the notation δ η ( r , v ) = ∂ η ( r , v , λ ) ∂ λ (cid:12)(cid:12)(cid:12)(cid:12) λ = . (39)for the scalar function η ( r , v , λ ) .Next, we turn to evaluate the second-order variational iden-tity in (27). According to our assumption that the spacetime isstatic at sufficiently late times, the fifth term of (27) vanishes.For the first term, the straight calculation gives (cid:90) S ∞ (cid:104) δ Q ξ − ξ · δ Θ ( g , δ g ) (cid:105) = δ ML z + . (40)According to the results in Eqs. (30) and (32), the second andthird terms reduce to (cid:90) Σ δ [ ξ · E abg δ g ab ] + (cid:90) Σ δ C ξ = − qb z − L z − zr zh [ b δ q + ( z − ) q δ b ] − b ( z − ) L z − zr zh × [( z − )( z − ) q δ b + ( z − ) bq δ q δ b + b δ q ] , (41)Considering the optimal condition of the first-order pertur-bation inequality, using the explicit expression of the metric(11), it is easy to verify that E Σ =
0. Summing the aboveresults, the second-order variational identity can be shown as δ ML z + − qb z − L z − zr zh [ b δ q + ( z − ) q δ b ] − b ( z − ) L z − zr zh × [( z − )( z − ) q δ b + ( z − ) bq δ q δ b + b δ q ]= − (cid:90) Σ δ C ξ = δ (cid:20) (cid:90) Σ ˜ (cid:15) T ab ( dr ) a ξ b (cid:21) = δ (cid:20) (cid:90) Σ T ab k a k b dv ˆ (cid:15) (cid:21) . (42)Because of the optimal condition of the first-order per-turbation inequality, the null energy condition becomes δ [ T ab k a k b ] ≥ δ M − qb z − L z zr zh [ b δ q + ( z − ) q δ b ] − b ( z − ) L z zr zh × [( z − )( z − ) q δ b + ( z − ) bq δ q δ b + b δ q ] ≥ . (43) IV. GEDANKEN EXPERIMENTS TO DESTROY THENEARLY EXTREMAL BLACK HOLES
Now we shall discuss the possibility to destroy the nearlyextremal charged dialton-Lifshitz black holes in the physi-cal process introduced in the previous sections. Because weassume that the spacetime settles down to a static state inthe asymptotic future, checking the validity of the WCCC isequivalent to see whether the line element at sufficient latetimes also describes a black hole, i.e., there exists at least oneroot of the blacking factor f ( r , λ ) . To make it computable, wedefine a function h ( λ ) = f ( r m ( λ ) , λ ) (44)to describe the minimal value of the blackening factor in theasymptotic future. Here r m ( λ ) is the minimal radius of theblackeing factor, and it can be obtained by ∂ r f ( r m ( λ ) , λ ) = . (45)Using the explicit expression of blackening factor in Eq. (13),the above identity becomes M ( λ ) = z ( z + ) q ( λ ) b z ( λ ) L z + b ( λ ) L r zm ( λ ) z ( z + ) b ( λ ) r zm ( λ ) . (46)Under the zero-order approximation of λ , we have M = z ( z + ) q b z L z + b L r zm z ( z + ) b r zm . (47)Taking the first-order variation of Eq. (46), we can furtherobtain δ M = b z − L z ( z + ) q [ b δ q + ( z − ) q δ b ] z ( z + ) r zm + ( b L r zm − q z ( z + ) b z + L z ) δ r m z ( z + ) r z + m b , (48) which implies δ r m = z ( z + ) r z + m b δ Mb L r zm − q z ( z + ) b z + L z − q ( z + ) L z r m ( b δ q + ( z − ) q δ b )] b − z L r zm − q z ( z + ) bL z . (49)Under the second-order approximation of perturbation, theminimal value of the blackening factor at late times can beexpressed as h ( λ ) (cid:39) + L z ( z + ) r m − L q ( z + ) b ( z − ) r ( z + ) m − λ r z + m (cid:18) δ M − qb z − L z zr zm [ b δ q + ( z − ) q δ b ] (cid:19) − λ r z + m (cid:18) δ M − qb z − L z zr zm [ b δ q + ( z − ) q δ b ] (cid:19) + λ L z b ( z − ) zr ( z + ) m [ b δ q + ( z − )( z − ) q δ b ]+ λ ( z − ) b z − L z δ q δ bzr ( z + ) m , (50)where we have used Eq.(47) to replace M by r m , q and b . Be-cause the physical process is only a perturbation of the back-ground spacetime, the physical quantities at late times are onlythe small correction. Therefore, in order to destroy the blackhole, the initial state must be a nearly extremal black hole. Inthe following, we consider the situation when the backgroundspacetime is a nearly extremal black hole. Then, the positionof the minimal value can be expressed as r m = ( − ε ) r h . Witha similar setup as [25], we assume that the parameter ε is agreewith the first-order approximation of perturbation. Then, wehave f ( r m ) = f (( − ε ) r h ) (cid:39) − ε r h f (cid:48) ( r h ) + ε r h f (cid:48)(cid:48) ( r h ) (cid:39) − ε r h f (cid:48) ( r m ) − ε r h f (cid:48)(cid:48) ( r m ) + ε r h f (cid:48)(cid:48) ( r h )= − ε r h f (cid:48)(cid:48) ( r h ) (cid:39) − ε r h f (cid:48)(cid:48) ( r m ) (51)under the second-order approximation of ε , i.e., we have ne-glected the higher-order term O ( ε ) of ε . Using the explicitexpression of the blackening factor in Eq. (8), the left-handside of the above equation gives f ( r m ) = + L z r m − Mr z + m + q L z b ( z − ) zr ( z + ) m = + L z ( z + ) r m − L q ( z + ) b ( z − ) r ( z + ) m , (52)where we have used Eq.(47) to replace M by r m , q and b . Forthe right-hand side, we have f (cid:48)(cid:48) ( r m ) = − ( z + )( z + ) Mr z + + L z r m + ( + z )( + z ) z − b z − L z q r − ( z + ) m = − L zr m + ( z + ) L z q b ( − z ) r ( z + ) m . (53)Summing the above results, Eq. (51) becomes1 + L z ( z + ) r m − L q ( z + ) b ( z − ) r ( z + ) m = ε r h (cid:32) L zr m − ( z + ) L z q b ( − z ) r ( z + ) m (cid:33) (cid:39) L ε zr h − ( z + ) ε L z q b ( − z ) r ( z + ) h , (54)under the second-order approximation of ε . In the last step,we have replaced r m = ( − ε ) r h by r h and neglected thehigher-order term O ( ε ) . Using the above results, under thefirst-order approximation of perturbation, we have h ( λ ) (cid:39) − λ r z + m (cid:18) δ M − qb z − L z zr zm [ b δ q + ( z − ) q δ b ] (cid:19) , (55)where we have neglected the higher-order terms O ( ε ) , O ( λ ) and O ( λ ε ) . Considering the first-order perturbation inequal-ity (37), we can see that h ( λ ) ≤ h ( λ ) is totally determined by its leading order. If h ( λ ) < h ( λ ) = h ( λ ) cannotbe determined by the first-order approximation and thereforewe need to consider the second-order approximation of h ( λ ) . Combining the second-order perturbation inequality (43), inthe first-order optimal condition, we can easily obtain h ( λ ) ≤ − [ b r zh z ε [ L + ( z + )( z + ) r h ] − λ z b r zh δ M ] b z r z + h [ L + ( z + )( z + ) r h ] ≤ r m = ( − ε ) r h by r h be-cause their difference only contributes some higher-order cor-rections. We can see that h ( λ ) ≤ V. CONCLUSION
There are a lot of investigations to test the weak cosmic cen-sorship in various spacetimes background based on the newversion of the gedanken experiments. All of them showedthe validity of the WCCC. However, there is still a lack ofgeneral proof of the WCCC even in general relativity. It isnatural for us to ask whether its validity is independent of theasymptotic behaviors of spacetime. Therefore, in this paper,we examined the WCCC in the situation when the chargeddilaton-Lifshitz black holes are perturbed by the sphericallyaccreting matter which satisfies the null energy condition andfinally settles to down to a static state at asymptotic future.Based on the Noether charge method, we first derived the first-order and second-order perturbation inequalities. As a result,we found that the charged dilaton-Lifshitz black holes cannotbe destroyed by the above physical process under the second-order approximation of perturbation. Our result implies thatthe WCCC might be a general feature of the general relativity,and its validity does not depend on the asymptotic behavior ofthe black hole.
ACKNOWLEDGEMENT
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