Examining the Weak Cosmic Censorship Conjecture by Gedanken Experiments for an EMDA Black Hole 3

Home , Hua (surname)

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2 H-F Ding, X-H Zhai

a gedanken experiment to examine the WCCC for an extremal Kerr-Newman black hole.19 It was shown that no violations of WCCC can occur by throwing particle matter into an extremal Kerr-Newman black hole. But in 1999, Hubeny proposed that violations of WCCC might still occur by over-charging a near-extremal charged black hole.20 Recently, Sorce and Wald21 suggested that the analysis of Hubeny’s experiment is insufficient at the linear order, so that the second order correction of the perturbation must be taken into account to check the WCCC, and a new version of the gedanken experiments has been proposed. In the new version of the gedanken experiments,21 the detailed physical process of falling matter in Kerr-Newman spacetime has been analyzed based on the Iyer- Wald formalism.22 After the null energy condition of the falling matter was taken into account, they derived the first order and second order inequalities relating energy, angular momentum and electric charge of the black hole. Importantly, the second order inequality of mass automatically takes all effects on energy into account, including self-force and finite-size effects, and it is valid not only for particle- like matter but also for general matter entering a Kerr-Newman black hole. They showed that the near-extremal Kerr-Newman black hole cannot be over-charged or over-spun when the second order correction of the perturbation was taken into account. So far, there are no general procedures to prove the validity of WCCC for arbi- trary black holes in any gravitational theories. Hence, its validity has to be demon- strated for black holes case by case. Most recently, by using the new version of the gedanken experiments, the WCCC has been tested for several stationary black hole solutions in Refs. 23–35, and the results show that the WCCC is valid under the second order approximation. Usually, in the new version of the gedanken experiments, the background conserved charges include the mass, angular momentum and electric charge. It is worth further study whether the increase of the background solution parameters will affect the validity of WCCC. The Einstein-Maxwell-Dilaton-Axion (EMDA) black hole36 with stationary and axial symmetries is equipped with six con- tinuous free parameters and two discrete constants. In particular, it contains the generalized Sen black hole with mass, Newman-Unti-Tamburino parameter, charge, angular momentum, dilaton and axion parameters. So, to examine the WCCC for such a rich balck hole will be an important and interesting work. In this paper we will use the new version of the gedanken experiments to investigate the WCCC for this EMDA black hole. The paper is organized as follows. In Sect. 2, we review the Iyer-wald formalism to derive the variational identities. In Sect. 3, we focus on the EMDA theory of gravity and calculate the relevant quantities for our further analysis. In the end of this section the EMDA black hole solution are introduced. In Sect. 4, we present the set-up for the new version of the gedanken experiments, and derive the first order and second order perturbation inequalities for EMDA black holes. Whereafter, we investigate the new version of the gedanken experiments to destroy a near-extremal EMDA black hole. We prove the WCCC is valid under the second order correction December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

Examining the weak cosmic censorship conjecture by gedanken experiments for an EMDA black hole 3

of the perturbation. The conclusions are presented in Sect. 5.

2. Iyer-Wald Formalism and Varational Identities Firstly, we review the Iyer-Wald formalism to derive the variational identities.21 We consider a general diffeomorphism covariant theory of gravity on a 4-dimensional spacetime , where the Lagrangian 4-form L = Lǫ is constructed locally by the M metric field gab and matter fields ψ’s, with ǫ the volume element compatible with the metric gab. We denote the dynamic fields jointly as Φ = (gab, ψ). The variation of L leads to

δL(Φ) = EΦδΦ + dΘ(Φ,δΦ), (1)

where EΦ = 0 gives the equations of motion (EOM), and Θ is called the symplectic potential 3-form. The symplectic current 3-form is defined by ω(Φ,δ Φ,δ Φ) = δ Θ(Φ,δ Φ) δ Θ(Φ,δ Φ). (2) 1 2 1 2 − 2 1 The Noether current 3-form Jζ associated with a vector field ζ is defined as

Jζ = Θ(Φ, Lζ Φ) ζ L, (3) − · where Lζ is the Lie derivative along the vector ﬁeld ζ. A straightforward calculation gives

dJζ = E LζΦ, (4) − Φ which implies dJζ = 0 when the EOM are satisﬁed. From Ref. 37, the Noether current can also be expressed as

Jζ = Cζ + dQζ , (5)

where Qζ is called the Noether charge and Cζ are the constraints of the theory, i.e., Cζ = 0 when the EOM are satisfied. Comparing the variations of Eqs.(3) and (5) with ζ fixed, we obtain the first variational identity

d [δQζ ζ Θ(Φ,δΦ)] = ω(Φ,δΦ, Lζ Φ) ζ E δΦ δCζ . (6) − · − · − The variation of (6) further gives the second variational identity 2 2 d δ Qζ ζ δΘ(Φ,δΦ) = ω(Φ,δΦ, LζδΦ) ζ δE δΦ δ Cζ . (7) − · − · − In what follows, we are interested in stationary axial-symmetric EMDA black hole solutions with horizon killing field a a a ξ = t +ΩHϕ , (8) a a where t , ϕ and ΩH are the timelike killing field, axial killing field and the angular velocity of the horizon, respectively. For asymptotically flat black hole solutions, the perturbation of mass and angular momentum are given by

δM = [δQt t Θ(Φ,δΦ)] , Z∞ − ·

δJ = δQϕ. (9) Z∞ December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

4 H-F Ding, X-H Zhai

a We restrict consideration to the case where (a) Φ satisfies the EOM EΦ = 0, (b) ξ is also a symmetry of the matter fields ψ, so that LξΦ = 0. Let Σ be a hypersurface with a cross section B on the horizon and with the spatial infinity as its boundaries. Using the Stokes theorem, we can obtain

δM ΩHδJ = [δQξ ξ Θ(Φ,δΦ)] δCξ, (10) − ZB − · − ZΣ 2 2 2 δ M ΩHδ J = δ Qξ ξ δΘ(Φ,δΦ) − Z − · B 2 ξ δE δΦ δ Cξ + EΣ(Φ,δΦ), (11) − ZΣ · − ZΣ

where the canonical energy is deﬁned by

EΣ(Φ,δΦ) = ω(Φ,δΦ, LξδΦ). (12) ZΣ

3. Einstein-Maxwell-Dilaton-Axion Theory and Black Hole Solutions The EMDA black hole solution is given by the Lagrangian 4-form36

1 2 1 4φ 2 −2φ ab ab L = R 2(∂φ) e (∂K) e FabF K Fab ⋆ F ǫ, (13) 16π − − 2 − −

which is obtained from a low-energy action of heterotic string theory, where ⋆Fab = 1 cd 2 ǫabcdF is the dual of the electromagnetic field Fab, φ is the massless dilaton field, and K is the axion field dual to the three-index antisymmetric tensor H = exp(4φ) ⋆ dK/4. In this paper we use the notation in Ref. 38 and the ǫ-tensor is − defined as √ gǫ0123 = 1. − − The variation of the Lagrangian 4-form gives

1 ab a E δΦ= ǫ T δgab + j δAa + Eφδφ + E δK , (14) Φ − 2 K

where

8πT ab =Gab 8πT ab 8πT ab 8πT ab , − EM − DIL − AX a 1 ab j = bF˜ , 4π ∇ 1 4φ 2 a −2φ ab Eφ = 2e (∂K) 4 a φ 2e FabF , 16π − ∇ ∇ − 1 ab 4φ a E = Fab ⋆ F a(e K) , (15) K 16π − ∇ ∇ December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

Examining the weak cosmic censorship conjecture by gedanken experiments for an EMDA black hole 5

with 1 Gab =Rab Rgab, − 2 ab 1 ac b 1 ab cd T = F F˜ c g FcdF˜ , EM 4π − 4 1 T ab = 2 aφ bφ (∂φ)2gab , DIL 8π ∇ ∇ − e4φ T ab = 2 aK bK (∂K)2gab , (16) AX 32π ∇ ∇ − in which

−2φ F˜ab e Fab +K ⋆ Fab. (17) ≡ Here T ab corresponds to the non-electromagnetic, -dilaton, and -axion (non-EDA) part of stress-energy tensor, and ja corresponds to the electromagnetic charge current. The symplectic potential 3-form is given by Θ(Φ,δΦ) = ΘGR(Φ,δΦ) + ΘEM (Φ,δΦ) + ΘDIL(Φ,δΦ) + ΘAX (Φ,δΦ), (18) where

GR 1 de fg Θ (Φ,δΦ) = ǫdabcg g ( gδgef eδgfg), abc 16π ∇ − ∇ EM 1 de Θ (Φ,δΦ) = ǫdabcF˜ δAe, abc − 4π DIL 1 d Θ (Φ,δΦ) = ǫdabc( φ)δφ, abc − 4π ∇ AX 1 4φ d Θ (Φ,δΦ) = ǫdabc(e K)δK. (19) abc − 16π ∇ From Eq.(2) we can obtain the symplectic current ω ωGR ωEM ωDIL ωAX abc(Φ,δ1Φ,δ2Φ) = abc + abc + abc + abc , (20) where 1 ωGR = ǫ wd, abc 16π dabc ωEM 1 ˜de ˜de abc = δ2(ǫdabcF )δ1Ae δ1(ǫdabcF )δ2Ae , 4π h − i DIL 1 d d ω = δ (ǫdabc φ)δ φ δ (ǫdabc φ)δ φ , abc 4π 2 ∇ 1 − 1 ∇ 2 AX 1 4φ d 4φ d ω = δ (ǫdabce K)δ K δ (ǫdabce K)δ K , (21) abc 16π 2 ∇ 1 − 1 ∇ 2 with a abcdef w = P [δ gbc dδ gef δ gbc dδ gef ] , 2 ∇ 1 − 1 ∇ 2 1 1 1 1 P abcdef = gaegfbgcd gadgbegfc gabgcdgef gbcgaegfd + gbcgadgef . (22) − 2 − 2 − 2 2 December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

6 H-F Ding, X-H Zhai

b c By using Lξgab =2 ξ , LξAa = ξ Fba + a(ξ Ac) and doing a straightfor- ∇(a b) ∇ ward calculation in Eq.(3), we get the Noether current 3-form

1 [e d] ed f d d f Jabc = ǫdabc e ξ +2F˜ Af ξ + ǫdabc T f + Af j ξ . (23) 8π ∇ ∇ h i Comparing it with Eq.(5), we obtain the Noether charge GR EM (Qξ)ab = (Qξ )ab + (Qξ )ab, (24) with

GR 1 [c d] (Q )ab = ǫabcd ξ , ξ − 16π ∇ EM 1 cd e (Q )ab = ǫabcdF˜ Aeξ , (25) ξ − 8π and the constraint term d d (Cf )abc = ǫdabc T f + Af j . (26) Now, we focus on the EMDA black hole solution given in Refs. 36, 39. The line element can be read oﬀ Ξ a2 sin2θ 2a sin2θ ds2 = − dt2 (r2 2Dr + a2) Ξ dt dϕ − ∆ − ∆ − − ∆ sin2θ + dr2 + ∆dθ2 + (r2 2Dr + a2)2 Ξa2 sin2θ dϕ2, (27) Ξ ∆ − − where ∆=r2 2Dr + a2 cos2θ, Ξ= r2 2mr + a2, − − W ω e2φ = = (r2 + a2 cos2θ), ω = e2φ0 , ∆ ∆ 2aD cosθ 1 K =K + , At = (Qr ga cosθ), 0 W ∆ − 1 2 2 2 2 Ar =Aθ =0, Aϕ = ( Qra sin θ + g(r + a )a cosθ). a∆ − The ADM mass M, angular momentum J, and dilaton charge D are given by Q2 M = m D, J = a(m D),D = , (28) − − −2ωM respectively. The two horizons are

r± = M + D (M + D)2 a2, (29) ± p − in which r+ is the event horizon. The surface gravity, area, angular velocity and electric potential of the horizon are given by r M D a κ = + − − , Ω = , r2 2Dr + a2 H r2 2Dr + a2 + − + + − + 2DM Φ = − , A =4π(r2 2Dr + a2). (30) H Q(r2 2Dr + a2) H + − + + − + December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

Examining the weak cosmic censorship conjecture by gedanken experiments for an EMDA black hole 7

The EMDA black hole becomes extremal when (2ωM 2 Q2)2 4ω2J 2 = 0. If − − (2ωM 2 Q2)2 4ω2J 2 0, (31) − − ≥ the metric describes a black hole solution, whereas the metric describes a naked singularity for the violation of the inequality.

4. Gedanken Experiments to Destroy a Near-Extremal EMDA Black Hole 4.1. Perturbation Inequalities of Gedanken Experiments In this section, we use the new version of the gedanken experiment to obtain the ﬁrst order and second order perturbation inequalities for the above EMDA black holes. We consider that the EMDA black holes are perturbed by a one-parameter family of the matter sources. The corresponding EOM can be expressed as

ab ab ab ab ab G (λ) =8π TEM (λ)+ TDIL(λ)+ TAX (λ)+ T (λ) ,

a 1 ab j (λ)= bF˜ (λ), 4π ∇ Eφ(λ) =0, EK(λ)=0, (32)

with T ab(0) = 0 and ja(0) = 0 for background spacetime. In this paper we consider the perturbation matter contains only the electromagnetic matter source, i.e., the sources of dilaton field φ and axion field K vanish, implied by Eφ(λ) = 0 and EK(λ) = 0. As in Ref. 21, we also assume the gedanken experiment admits the following assumptions. (a) All the perturbation matter goes into the black hole through a finite portion of the future horizon, i.e., the matter source δT ab and δja are non-vanishing only in a compact region of future horizon. (b) Linear stability assumption. The non-extremal, unperturbed EMDA black hole is linearly stable to perturbation, i.e., any source free solution to the linearized EOM approaches a perturbation towards another EMDA black hole at sufficiently late times. (c) We choose a hypersurface Σ = H Σ to perform our analysis. It starts ∪ 1 from the bifurcate surface B of the unperturbed horizon H, continues up the future horizon through the matter source region of H till the very late cross section B1 where the matter source vanishes, then becomes the spacelike hypersurface Σ1 and continues out towards infinity. The boundaries of Σ are located at the bifurcate surface and the spatial infinity. The linear stability assumption implies that the dynamic fields satisfy the source- free EOM, E[Φ(λ)]=0onΣ1, and the solutions are described by Eq.(27). With the above set-up, we now derive the first order perturbation inequality at λ = 0. Similar to the analysis in Ref. 21, for a non-extremal black hole the horizon December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

8 H-F Ding, X-H Zhai

will be of bifurcate type, the second term of the ﬁrst integral in Eq.(10) vanishes since ξa = 0 on the bifurcate surface B. Therefore

[δQξ ξ Θ(Φ,δΦ)] = δQξ. (33) ZB − · ZB For the gravitational part, from the ﬁrst expression of (25) we have

GR κ δQξ = δAB , (34) ZB 8π

Where AB is the area of B and κ is the surface gravity of the event horizon. For the electromagnetic part, from the second expresson of (25) we obtain

EM 1 e ˜cd e ˜cd δQξ = ξ Aeδ(ǫabcdF )+ ξ (δAe)ǫabcdF , (35) ZB −8π ZB h i a e where the second term vanishes at B by ξ B = 0, but ξ Ae does not vanish since e | Φ ξ Ae(λ) must be constant on the horizon at λ = 0. So, H ≡− EM 1 ˜cd δQξ = ΦH δ(ǫabcdF )=ΦHδQB, (36) ZB 8π ZB

Where QB is the electric charge ﬂux integral over B. The assumption that the perturbation vanishes on the bifurcate surface B leads to δAB = δQB = 0. This also holds for extremal black holes. Therefore, the ﬁrst integral vanishes in Eq.(10). By using the fact that T ab = ja = 0 in the background spacetime (since E[Φ(0)] = 0), then the Eq.(10) can be written as

δM ΩHδJ = δCξ − − ZΣ d e e d = ǫdabcδT eξ Aeξ ǫdabcδj − ZH − ZH d e = ǫãbcδTdek ξ +ΦHδQ ZH Φ δQ, (37) ≥ H whereǫ ãbc is the volume element on H, which is defined by ǫdabc = 4k[dǫãbc] with a a d − the future-directed normal vector k ξ , and ǫdabcδj = δQflux = δQ is the ∝ H total flux of charge through the horizon. In theR last line we used the null energy a b condition δTabk k 0 for the non-EDA stress-energy tensor. Thus, we obtain |H ≥ the first order perturbation inequality δM Ω δJ Φ δQ 0. (38) − H − H ≥ Under the first order perturbation, if we want to violate (2ωM 2 Q2)2 4ω2J 2 0, a b − − ≥ the optimal choice is to saturate (38) by requiring δTabk k = 0, i.e., the energy |H flux through the horizon vanishes for the first order non-EDA perturbation. Then, (37) reduces to δM Ω δJ Φ δQ =0. (39) − H − H December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

Examining the weak cosmic censorship conjecture by gedanken experiments for an EMDA black hole 9

Next, we derive the second order perturbation inequality. In exact parallel to the derivation of the ﬁrst order inequality (37), Eq.(11) becomes

2 2 2 δ M ΩHδ J = ξ δE δΦ δ Cξ + EΣ(Φ,δΦ), (40) − − ZH · − ZH with

d 1 ef e (ξ δE δΦ)abc = ξ ǫdabc δT δgef + δj δAe , · − 2 2 2 d e 2 e d (δ Cξ)abc =δ (ǫdabcT eξ )+ δ (ǫdabcAeξ j ). (41)

In Eq.(40), the integrals in the ﬁrst two terms only depend on the surface H since 2 δE = δ Cξ =0onΣ1 by the assumption that there are no sources outside the black hole at late times. In addition, since ξa is tangent to the horizon, the ﬁrst term of a the right side of (40) vanishes. By using the condition ξ δAa = 0 on H from a gauge transformation, Eq.(40) becomes

2 2 e d 2 2 δ M ΩHδ J =EΣ(Φ,δΦ) + ǫ˜abcξ k δ Tde +ΦHδ Q − ZH E (Φ,δΦ) + E (Φ,δΦ)+Φ δ2Q, (42) ≥ Σ1 H H a b where we have used the optimal choice δTabk k H = 0 and the null energy condition 2 a b | δ Tabk k 0 for the second order perturbed non-EDA stress-energy tensor. To |H ≥ obtain EH(Φ,δΦ), we split it into

GR EM DIL AX EH(Φ,δΦ) = ω + ω + ω + ω . (43) ZH ZH ZH ZH The contribution of the gravitational part has already been calculated in Ref. 21

GR 1 a bc ω = ǫ˜(ξ au)δσbcδσ 0. (44) ZH 4π ZH ∇ ≥ For the electromagnetic, dilaton and axion parts, according to (21), the symplectic currents ω(Φ,δΦ, LξδΦ) are given as

ωEM 1 L ˜de ˜deL abc = ǫdabc ξδF δAe δF ξδAe 4π h − i 1 de de + (Lξδǫdabc)F˜ δAe δǫdabcF˜ LξδAe , 4π h − i DIL 1 d d ω = ǫdabc Lξδ( φ)δφ δ( φ)Lξδφ abc 4π ∇ − ∇ 1 d d + Lξδǫdabc( φ)δφ δǫdabc( φ)Lξδφ , 4π ∇ − ∇ AX 1 4φ d 4φ d ω = ǫdabc Lξδ(e K)δK δ(e K)LξδK abc 16π ∇ − ∇ 1 4φ d 4φ d + Lξδǫdabc(e K)δK δǫdabc(e K)LξδK . (45) 16π ∇ − ∇ December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

10 H-F Ding, X-H Zhai

Through the similar calculations to those in Refs. 21 and 26, the corresponding contributions to the canonical energy give the following inequalities

ωDIL ǫ a b 2 DIL = ˜ξ k δ Tab 0, ZH ZH ≥ ωAX ǫ a b 2 AX = ˜ξ k δ Tab 0, ZH ZH ≥ ωEM 1 ǫ f ˜de ǫ a b 2 EM = ˜kdξ δF δFef = ˜ξ k δ Tab 0. (46) ZH 2π ZH ZH ≥ Again the null energy conditions for the second order perturbed matter ﬁelds stress- energy tensors have been used in (46). Then, (42) reduces to

δ2M Ω δ2J Φ δ2Q E (Φ,δΦ). (47) − H − H ≥ Σ1 E What remains now is to calculate Σ1 (Φ,δΦ). We adopt the trick in Ref. 21, and E E F F write Σ1 (Φ,δΦ) = Σ1 (Φ,δΦ ), where Φ (α) denotes a ﬁeld conﬁguration of another one-parameter family of EMDA black hole solutions with parameters given by

M F (α)=M + αδM, J F (α)=J + αδJ, QF (α)=Q + αδQ, (48)

where δM, δJ and δQ are chosen to agree with the corresponding values for our ﬁrst 2 2 2 order perturbation of Φ(λ). Then, for this family we have δ M = δ J = δ QB = 2 F δE = δ Cξ = EH(Φ,δΦ ) = 0. According to Eq.(11) we have

E F 2 F κ 2 F Σ1 (Φ,δΦ )= δ Qξ ξ δΘ(Φ,δΦ ) = δ A , (49) − Z − · −8π B B thus we obtain the second order perturbation inequality κ δ2M Ω δ2J Φ δ2Q δ2AF . (50) − H − H ≥−8π B 2 2 Taking two variations of the area formula AB =4π(r 2Dr + a ), we obtain + − + π δ2AF = 2ω Q6 2ωM 2Q4(3 + ǫ) 4ω2Q2(J 2 M 4(3+2ǫ)) B − ω4M 6ǫ3 − − − 8ω3M 2(M 4(1 + ǫ) J 2(3 + ǫ)) (δM)2 Q6 2ωM 2Q4(3 + ǫ) − − − − +4ω2Q2( 3J 2 + M 4(3+2ǫ))+8ω3M 2(J 2 M 4)(1 + ǫ) (δQ)2 − − +2ω2(2ωM 2 Q2)2(δJ)2 +8ω2JQ(2ωM 2 Q2)δQδJ − − 32ω3MQJ 2δMδQ + 16ω3MJ(2ωM 2 Q2)δMδJ , (51) − − in which (2ωM 2 Q2)2 4ω2J 2 ǫ = − − . (52) p 2ωM 2 December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

Examining the weak cosmic censorship conjecture by gedanken experiments for an EMDA black hole 11

For the near-extremal black hole, ǫ is a small parameter, then the surface gravity of EMDA black hole can be expressed as ωMǫ κ = . (53) 2ωM 2(1 + ǫ) Q2 − Expanding the right side of (50) to lowest order in ǫ, we obtain 2 2 Q (δQ) 2 δ2M Ω δ2J Φ δ2Q Q2 +2ωM 2ωM 2 , (54) − H − H ≥−4ω3M 5ǫ2 − where we have used the Eq.(39) to eliminate δM from the expression.

4.2. Near-Extremal EMDA Black Hole Cannot be Over-Charged or Over-Spun With the above preparation, we now investigate the new version of the gedanken experiments to over-charge or over-spin a near-extremal EMDA black hole. We consider a one-parameter family Φ(λ), and the background spacetime Φ(0) is a near-extremal EMDA black hole, ǫ 1. Deﬁne a function of λ as ≪ 2 h(λ)= 2ωM(λ)2 Q(λ)2 4ω2J(λ)2. (55) − − The WCCC is violated if there exists a solution Φ(λ) such that h(λ) < 0. Expanding h(λ) to second order in λ, we have h(λ)=(2ωM 2 Q2)2 4ω2J 2 − − + λ 8ωM(2ωM 2 Q2)δM 4Q(2ωM 2 Q2)δQ 8ω2JδJ − − − − 1 + λ2 (8ω(2ωM 2 Q2)+32ω2M 2)(δM)2 +8ωM(2ωM 2 Q2)δ2M 2 − − 8ω2(δJ)2 8ω2Jδ2J 32ωMQδMδQ 4Q(2ωM 2 Q2)δ2Q − − − − − + (8Q2 4(2ωM 2 Q2))(δQ)2 + O(λ3). (56) − − If we consider only the linear term of λ in (56), then by using the inequality (38), h(λ) is constrained by h(λ) 4ω2M 4ǫ2 + 4(Q2 +2ωM 2ωM 2)QδQλ + O(λ2), (57) ≥ − which implies that it is possible to make h(λ) < 0, i.e., the black hole could be over-charged or over-spun. Next, we include the O(λ2) term in (56). By using inequality (54) and for optimal choice (39), we have h(λ) 4ω2M 4ǫ2 + 4(Q2 +2ωM 2ωM 2)QδQλ ≥ − (Q2 +2ωM 2ωM 2)2Q2(δQ)2 + − λ2 + O(λ3) ω2M 4ǫ2 2 (Q2 +2ωM 2ωM 2)QδQ = 2ωM 2ǫ + − λ + O(λ3). (58) ωM 2ǫ December 14, 2020 1:37 WSPC/INSTRUCTION FILE ws-mpla

12 H-F Ding, X-H Zhai

Thus, no violation of (2ωM 2 Q2)2 4ω2J 2 0 can occur when the second order − − ≥ correction of the perturbation is taken into account, i.e., this near-extremal EMDA black hole cannot be over-charged or over-spun. In our case the parameter ω can be any nonzero number, which characterizes the black hole hairs (Dilaton, Axion, etc.). So, it is implied that the validity of WCCC is not relevant to the black hole hairs. Note that, the EMDA solutions contain the Kerr-Sen black hole40 as a special case, which can be seen from the metric (27) when ω = 1 and D = msinh2(α/2). − In this case, our result reduced to that obtained in Ref. 27 that the near-extremal Kerr-Sen black hole cannot be over-charged or over-spun on the level of the second order approximation.

5. Conclusions and remarks In this paper, we have used the new version of the gedanken experiments to examine the WCCC for an EMDA black hole. We derived the first order and second order inequalities relating the mass, angular momentum and electric charge in this framework. We show that no violations of WCCC can occur with the increase of the background solution parameters for a near-extremal EMDA black hole when the second order correction of the perturbation was taken into account. The result implies that once an EMDA black hole is formed, it will never be destroyed by being over-charged or over-spun. When the parameters ω = 1 and D = msinh2(α/2) − are taken, the EMDA metric becomes the Kerr-Sen one, and our conclusion reduces to that in Ref. 27 that the WCCC is preserved for a Kerr-Sen black hole. For EMDA black hole we have shown that the validity of WCCC is not affected by the increase of the background solution parameters when the perturbation matter contains only the electromagnetic matter source. In Ref. 31, it is shown that the WCCC is restored in the Einstein-Maxwell gravity with scalar hairs when the scalar perturbation is considered but the background conserved scalar charge41 is not included in the perturbation inequalities. It is accessible that a complete analysis of WCCC in the new version of the gedanken experiments may contain both the background conserved scalar charge in the perturbation inequalities and the perturbation matter with scalar field when the black holes have some scalar hairs. This will be an important issue to study the WCCC in the future work.

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