Gregory-Laflamme instability of BTZ black hole in new massive gravity
aa r X i v : . [ h e p - t h ] O c t Gregory-Laflamme instability of BTZ black hole innew massive gravity
Taeyoon Moon ∗ and Yun Soo Myung † ,Institute of Basic Sciences and Department of Computer Simulation, Inje University,Gimhae 621-749, Korea Abstract
We find the Gregory-Laflamme s -mode instability of the non-rotating BTZ blackhole in new massive gravity. This instability shows that the BTZ black hole couldnot exist as a stable static solution to the new massive gravity. For non-rotating BTZblack string in four dimensions, however, it is demonstrated that the BTZ black stringcan be stable against the metric perturbation. PACS numbers:04.30.Nk, 04.70.Bw
Typeset Using L A TEX ∗ e-mail address: [email protected] † e-mail address: [email protected] Introduction
The dRGT gravity [1, 2, 3] is considered as a promising massive gravity model which yieldsEinstein gravity in the massless limit. Recently, it was shown that the stability of theSchwarzschild black hole in the four dimensional dRGT gravity could be determined bythe Gregory-Laflamme (GL) instability [4, 5] of a five-dimensional black string. The smallSchwarzschild black hole with mass M S in the dRGT gravity and its bi-gravity extension [6,7], and fourth-order gravity [8] is unstable against metric and Ricci tensor perturbationsfor m ′ ≤ O (1) M S and m ≤ M S , respectively. These results may indicate that static blackholes in massive gravity do not exist.Interestingly, it turned out that in a massive theory of the Einstein-Weyl gravity, thelinearized Einstein tensor perturbations exhibit unstable modes of the Schwarzschild-AdSblack hole featuring the GL instability of five-dimensional AdS black string, in contrast tothe stable Schwarzschild-AdS black hole in the Einstein gravity [9]. The linearized Ricci ten-sor perturbations were employed to exhibit unstable modes of the Schwarzschild-Tangherlini(higher dimensional Schwarzschild) black hole in higher-dimensional fourth order gravitywhich features the GL instability of higher dimensional black strings [10], in comparisonwith the stable Schwarzschild-Tangherlini black holes in higher-dimensional Einstein grav-ity. These imply that the GL instability of the black holes in the massive gravity originatesfrom the massiveness, but not a nature of the fourth-order gravity giving ghost states. Also,one could avoid the ghost problem arising from the metric perturbations in the fourth-ordergravity when using the linearized Einstein and Ricci tensors because their linearized equa-tions become the second-order tensor equations.On the other hand, it was shown that the four-dimensional BTZ black string in Einsteingravity is stable against metric perturbations regardless of the horizon size, which is alsosupported by a thermodynamic argument of Gubser-Mitra conjecture [11, 12]. Later on,however, it was argued that the BTZ black string is not always stable against metricperturbations [13]. In the literatures [11, 12], there exists a threshold value for µ > µ to avoid a confusion m here) which is related to thecompactification of the extra dimension of the tensor perturbation. It was shown in [13]that for µ ≥ /ℓ with ℓ AdS curvature radius, the BTZ black string is stable against s -mode metric perturbation, while for µ < /ℓ it is unstable. Therefore, it seems to benecessary to point out which one is correct. 2he new massive gravity has been introduced as a fourth-order gravity with a healthymassive spin-2 mode and a massless spin-2 ghost mode, which is pure gauge only in threedimensions [14]. This parity-even gravity describes two modes of helicity +2 and − m > / ℓ with m the mass of graviton, the three-dimensional BTZ black hole in the new massive gravity isshown to be stable against s -mode metric perturbation [16] when using the positivity of thepotential outside the horizon. To this direction, it was recently reported that the stability ofthe BTZ black hole was mainly determined by the asymptotes of black hole spacetime: thecondition of the s -mode stability is consistent with the generalized Breitenlohner-Freedman(BF) bound ( m ≥ − / ℓ ) for metric perturbations on asymptotically AdS spacetime [17].This result may imply that the stability condition is extended simply from m > / ℓ to m ≥ − / ℓ if m is allowed to be a negative quantity. However, one expects that twodifferent type of instabilities appears for the BTZ black hole in new massive gravity: oneis from the BF bound based on the tensor propagation on asymptotically AdS spacetime,while the other is the GL instability of a massive graviton propagating on the BTZ blackhole spacetime. This is similar to two instabilities of AdS black holes to trigger a holographicsuperconductor phase within the AdS/CFT correspondence [18]. In these models, the AdS d black hole becomes unstable to form non-trivial fields outside its horizon when being closeto extremality whose near-horizon geometry is AdS × M d − . For a massive scalar withmass m between − ( d − / ℓ and − / ℓ , two AdS spacetimes are unstable [19].Hence, it suggests strongly that the stability of BTZ black hole should be revisited innew massive gravity by observing the GL instability of four-dimensional black string. Wewill show that the instability of a massive graviton persists even in three-dimensional BTZblack hole. This establishes that the instability of the black holes in the D ≥ s -mode metric perturbation because its mass squared is positive( µ > Linearized perturbation equation
We start with the three-dimensional fourth order gravity defined as S fog = Z d x √− g " R − λ S κ + αR + βR µν R µν (2.1)with κ the three-dimensional gravitational coupling constant. From the action (2.1), theEinstein equation is derived to be1 κ (cid:16) R µν − Rg µν + λ S g µν (cid:17) + E µν = 0 , (2.2)where E µν takes the form E µν = 2 β (cid:16) R µρνσ R ρσ − R ρσ R ρσ g µν (cid:17) + 2 αR (cid:16) R µν − Rg µν (cid:17) + β (cid:16) ∇ R µν + 12 ∇ Rg µν − ∇ µ ∇ ν R (cid:17) + 2 α (cid:16) g µν ∇ R − ∇ µ ∇ ν R (cid:17) . (2.3)For Λ = λ S + 2 κ (3 α + β )Λ with ¯ R µν = 2Λ¯ g µν , the non-rotating BTZ black hole solutionis given by ds = ¯ g µν dx µ dx ν = − V ( r ) dt + dr V ( r ) + r dφ , (2.4)where the metric function takes the form V ( r ) = −M + r ℓ (2.5)with ℓ = − / Λ and M the ADM mass. From the condition of V ( r + ) = 0, the horizon islocated at r = r + .For a perturbation around the BTZ black hole g µν = ¯ g µν + h µν , (2.6)the linearized Einstein tensor, Ricci tensor, and Ricci scalar are given by δG µν ( h ) = δR µν −
12 ¯ g µν δR − h µν , (2.7) δR µν ( h ) = 12 (cid:16) ¯ ∇ ρ ¯ ∇ µ h ρν + ¯ ∇ ρ ¯ ∇ ν h ρµ − ¯ ∇ h µν − ¯ ∇ µ ¯ ∇ ν h (cid:17) , (2.8) δR ( h ) = ¯ ∇ α ¯ ∇ β h αβ − ¯ ∇ h − h, (2.9)4here the Einstein tensor G µν = R µν − Rg µν / g µν . With the help of the above quantities,the linearized Einstein equation can be written as h κ + (12 α + 2 β )Λ i δG µν +(2 α + β ) h ¯ g µν ¯ ∇ − ¯ ∇ µ ¯ ∇ ν + 2Λ¯ g µν i δR + β (cid:16) ¯ ∇ δG µν − Λ¯ g µν δR (cid:17) = 0 . (2.10)Taking the trace of (2.10) provides the linearized Ricci scalar equation h (8 α + 3 β ) ¯ ∇ − n κ − α + β )Λ oi δR = 0 , (2.11)which implies that for 8 α + 3 β = 0 (the new massive gravity), the d’Alembertian operatoris removed. In the new massive gravity [14] S NMG = Z d x √− g " R − λ S κ + β (cid:18) R µν R µν − R (cid:19) , (2.12) δR is constrained to vanish δR = 0 (2.13)provided that κβ Λ = −
2. This explains why we choose the new massive gravity. Plugging δR = 0 and α = − β/ (cid:16) ¯ ∇ − − M (cid:17) δG µν = 0 , (2.14)where the mass squared is given by M = − κβ + Λ2 ≡ m − ℓ . (2.15)Choosing the transverse-traceless gauge¯ ∇ µ h µν = 0 , h = 0 , (2.16)Eq. (2.14) leads to the fourth-order equation for the metric perturbation h µν (cid:16) ¯ ∇ − (cid:17)(cid:16) ¯ ∇ − − M (cid:17) h µν = 0 , (2.17)which might imply the two second-order linearized equations h ¯ ∇ − i h µν = 0 , (2.18) h ¯ ∇ − − M i h µν = 0 (2.19)5ff critical point ( M = 0 , m = 1 / ℓ ). Even though Eq. (2.19) may describe 2 DOFfor a massive graviton in three dimensions, it might not be a correct equation becausethe ‘ − ’ sign disappears when splitting (2.17) into (2.18) and (2.19). In order to see thisghost problem explicitly, we consider the three-dimensional flat spacetime. In the case ofΛ → λ S → A between two conserved sources T ′ µν and T µν is given by [20]4 A = − β T ′ µν p ( p + m ) T µν + 1 β T ′ p ( p + m ) T − κT ′ p T (2.20)with p = − ∂ . After partial fractions, this leads to4 A = 2 κT ′ µν (cid:16) p − p + m (cid:17) T µν − κT ′ (cid:16) p − p + m (cid:17) T. (2.21)In order not to have a tachyon, we have to choose κβ < m >
0) with β >
0. Also,we require κ < κ for a positive β in 3Dflat spacetime. However, we might miss the ghost problem when splitting (2.17) into (2.18)and (2.19) without imposing the sign correction. Hence, it would be better to use thesecond-order equation (2.14) for the linearized Einstein tensor if it could describe 2 DOF.For this purpose, we have two constraints δG = − δR = 0 , ¯ ∇ µ δG µν = 0 , (2.22)where the last one comes from contracting the linearized Bianchi identity ¯ ∇ [ τ δR µν ] ρσ = 0with ¯ g τρ ¯ g µσ . Then, we have 2 DOF because 6 − − m >
0) because of thenon-tachyonic condition but it is allowed to be negative in asymptotically AdS spacetime.Accordingly, the BF bound for a tensor field could be read off from Eqs. (2.14) and (2.19)as M ≥ − ℓ → m ≥ − ℓ . (2.23)However, in the case of 4D BTZ black string, one requires µ > Gregory-Laflamme s -mode instability In order to investigate the GL instability of the massive graviton propagating on the BTZblack hole spacetime, we first start with (2.19) for convenience, because (2.14) and (2.19)are the same equation. For s ( k = 0)-mode analysis, a metric perturbation takes the formwith four components H tt , H tr , H rr , and H as h µν ( t, φ, r ) = e Ω t e ikφ | k =0 H tt ( r ) H tr ( r ) 0 H tr ( r ) H rr ( r ) 00 0 H ( r ) . (3.1)Substituting Eq.(3.1) into Eq.(2.19) and after a tedious manipulation, we obtain the second-order equation for a single physical field H tr ( r ) [16] as (cid:8) ( m − / ℓ )( r /ℓ − M ) + r /ℓ − M /ℓ + Ω (cid:9) H ′′ tr + (cid:26) r /ℓ − M r ( r /ℓ − M ) Ω + 5 r /ℓ − M r ( m − / ℓ ) + 5 r /ℓ − M r /ℓ + 2 M ℓ r ( r /ℓ − M ) (cid:27) H ′ tr + (cid:26) r /ℓ − M r ( r /ℓ − M ) Ω + 2 r /ℓ − M r /ℓ − M r ( r /ℓ − M ) ( m − / ℓ ) + (3 r /ℓ − M )( r /ℓ − M r /ℓ − M ) ℓ r ( r /ℓ − M ) − (cid:18) m − / ℓ + Ω r /ℓ − M (cid:19) ) H tr = 0 (3.2)with M = r /ℓ . This implies that the s -mode perturbation is described by a single field H tr , even though we were starting with four components and the massive graviton has twoDOF for k = 0.It turns out that the second-order equation (3.2) can be reduced to two first-orderequations with a constraint when using the perturbation equation (2.19) together with theTT gauge condition (2.16) [5]. The two coupled first-order equations are given by H ′ = M − r /ℓ rV H − Ω2 V ( H + + H − ) , (3.3) H ′− = Ω M H + h r − (2 m + 1 /ℓ ) r M i H + + h − r + (2 m + 1 /ℓ ) r M + r Ω M V i H − . (3.4)A constraint equation can be written as r Ω h − M /ℓ + r /ℓ + 2 m V + 4Ω i H − − rV (2 m + 1 /ℓ )Ω H + +2 V (2 m M − M /ℓ + 2Ω ) H = 0 , (3.5)7here H ≡ − H tr , H ± ≡ H tt V ( r ) ± V ( r ) H rr . (3.6)At infinity of r → ∞ , asymptotic solutions to Eqs.(3.3) and (3.4) are H ( ∞ ) = C ( ∞ )1 r − − √ m ℓ +1 / + C ( ∞ )2 r − √ m ℓ +1 / ,H ( ∞ ) − = ˜ C ( ∞ )1 r − − √ m ℓ +1 / + ˜ C ( ∞ )2 r − √ m ℓ +1 / , (3.7)where ˜ C ( ∞ )1 , are˜ C ( ∞ )1 = 2 m − /ℓ Ω (cid:16) √ m ℓ + 2 (cid:17) C ( ∞ )1 , ˜ C ( ∞ )2 = 2 m − /ℓ Ω (cid:16) − √ m ℓ + 2 (cid:17) C ( ∞ )2 . (3.8)At the horizon of r + = ℓ √M , their solutions are given by H ( r + ) = C ( r + )1 ( r − r + ) − − Ω ℓ/ (2 √M ) + C ( r + )2 ( r − r + ) − ℓ/ (2 √M ) , (3.9) H ( r + ) − = ˜ C ( r + )1 ( r − r + ) − Ω ℓ/ (2 √M ) + ˜ C ( r + )2 ( r − r + ) Ω ℓ/ (2 √M ) , (3.10)where ˜ C ( r + )1 , take the forms˜ C ( r + )1 = M ( − m ℓ + 1) − ℓ + Ω ℓ (2 m ℓ + 1) √M Ω ℓ ( M − Ω ℓ ) C ( r + )1 , ˜ C ( r + )2 = M ( − m ℓ + 1) − ℓ − Ω ℓ (2 m ℓ + 1) √M Ω ℓ ( M − Ω ℓ ) C ( r + )2 . (3.11)Imposing two boundary conditions of the regular solutions at infinity and horizon corre-spond to choosing C ( ∞ )2 = 0 and C ( r + )1 = 0, respectively.Eliminating H + in Eqs. (3.3) and (3.4) by using the constraint (3.5) leads to the twocoupled equations with H and H − only. For fixed m and various values of Ω, we solve theseequations numerically which yields permitted values of Ω as a function of m . As a result,this shows clearly that there exist unstable modes (see Fig.1). Solving the first-order equations (3.3) and (3.4) numerically, we begin it by considering asymptoticsolutions (3.9) and (3.10) at the horizon. For given m , we find the consistent values of Ω to have theasymptotic behavior of H ( ∞ ) ∼ r − − √ m ℓ +1 / . For a complete analysis, we have already checked theresults given in Fig.3 of the literature [5]. - m W Figure 1: Ω graphs are depicted as function of m for three different horizon radii of r + = 1 , , ℓ = 1. The data range for m is between − / / curvature radius( ℓ = 1) by taking into account the scaling symmetry given in Eqs. (3.3)-(3.5) as r → αr, m → m/α, Ω → Ω /α, ℓ → α ℓ (3.12)with an arbitrary constant α . The second point is that the threshold mass for r + = 1 , , m ≈ .
5, which implies that GL instability exists for m < ℓ (3.13)when recovering the AdS curvature radius ℓ . This means that the BTZ black hole isunstable against the s -mode metric perturbation regardless of the horizon size. On theother hand, the threshold mass of m = 1 / ℓ can be read off approximately by taking thelimit of Ω → m > ℓ (3.14)which is exactly the same condition obtained from the positivity of the potential ( V Ψ >
9) [16]. The potential appears in the Schr¨odinger equation d Ψ dr ∗ + [ ω − V Ψ ]Ψ = 0 , ω = i Ω (3.15)which was derived from the second-order equation (3.2) by introducing a new field Ψ definedby Ψ = H tr /f ( r ). Combining it with the BF bound (2.23) for stability condition of tensorfield at asymptotically AdS spacetime, we find that the instability of the BTZ black holein new massive gravity is extended to − ℓ ≤ m < ℓ . (3.16)Up to now, we have made our instability analysis with (2.19) for the metric tensor. Sincetwo equations (2.14) and (2.19) take the same form when replacing δG µν by h µν and theydescribe 2 DOF with (2.16) and (2.22), the instability analysis for h µν persists in that of δG µν . One additional advantage when using (2.14) is to avoid the ghost problem.Finally, considering M = m − / ℓ (2.15), we rewrite the stability condition (3.14)as M > µ >
0. This implies that the 4D black string is stable under the s -mode metric perturbationregardless of the horizon size [12]. We have shown that the new massive gravity which is known to be a unitary gravity modelin three dimensions, could not accommodate the BTZ black hole by observing the GLinstability for the mass m between − / ℓ and 1 / ℓ . The GL instability has nothing to dowith the ghost issue arising from the fourth-order gravity of the new massive gravity becausewe have used two second-order equations (2.19) and (2.14) for h µν and δG µν , respectively.Also, this instability could not be explained in terms of the Gubser-Mitra conjecture becausethe heat capacity of the BTZ black hole is always positive. This instability arises from themassiveness ( M = m − / ℓ = 0) of the new massive gravity. In the massless case of M = 0, one has a massless graviton propagating on the BTZ black hole spacetime which10s gauge artefact in three dimensions. In this case, the stability issue of the BTZ black holeis meaningless.This establishes that the instability of the static black holes in the D ≥ s -modemetric perturbation in Einstein gravity [12], while the BTZ black hole is unstable in thenew massive gravity. This shows a newly interesting feature of low dimensional black stringand black hole when comparing with higher dimensional black string and black holes. Acknowledgement
T.M. would like to thank Dr. Miok Park for useful discussions. This work was supportedby the National Research Foundation of Korea (NRF) grant funded by the Korea govern-ment (MEST) (No.2012-R1A1A2A10040499). Y.M. was supported partly by the NationalResearch Foundation of Korea (NRF) grant funded by the Korea government (MEST)through the Center for Quantum Spacetime (CQUeST) of Sogang University with grantnumber 2005-0049409. 11 eferences [1] C. de Rham and G. Gabadadze, Phys. Rev. D , 044020 (2010) [arXiv:1007.0443[hep-th]].[2] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. , 231101 (2011)[arXiv:1011.1232 [hep-th]].[3] S. F. Hassan and R. A. Rosen, Phys. Rev. Lett. , 041101 (2012) [arXiv:1106.3344[hep-th]].[4] R. Gregory and R. Laflamme, Phys. Rev. Lett. , 2837 (1993) [hep-th/9301052].[5] R. Gregory and R. Laflamme, Nucl. Phys. B (1994) 399 [hep-th/9404071].[6] E. Babichev and A. Fabbri, Class. Quant. Grav. , 152001 (2013) [arXiv:1304.5992[gr-qc]].[7] R. Brito, V. Cardoso and P. Pani, Phys. Rev. D , 023514 (2013) [arXiv:1304.6725[gr-qc]].[8] Y. S. Myung, Phys. Rev. D , 024039 (2013) [arXiv:1306.3725 [gr-qc]].[9] Y. S. Myung, arXiv:1308.1455 [gr-qc].[10] Y. S. Myung, Phys. Rev. D , 084006 (2013) [arXiv:1308.3907 [gr-qc]].[11] G. Kang, hep-th/0202147.[12] G. Kang and Y. O. Lee, AIP Conf. Proc. , 358 (2006).[13] L. -h. Liu and B. Wang, Phys. Rev. D , 064001 (2008) [arXiv:0803.0455 [hep-th]].[14] E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett. , 201301 (2009)[arXiv:0901.1766 [hep-th]].[15] E. Bergshoeff, M. Kovacevic, L. Parra and T. Zojer, PoS Corfu , 053 (2013).[16] Y. S. Myung, Y. -W. Kim, T. Moon and Y. -J. Park, Phys. Rev. D , 024044 (2011)[arXiv:1105.4205 [hep-th]]. 1217] T. Moon and Y. S. Myung, Gen. Rel. Grav. (2013) [arXiv:1303.5893 [hep-th]].[18] S. A. Hartnoll, Class. Quant. Grav. , 224002 (2009) [arXiv:0903.3246 [hep-th]].[19] B. Hartmann, arXiv:1310.0300 [gr-qc].[20] I. Gullu and B. Tekin, Phys. Rev. D80