aa r X i v : . [ h e p - t h ] M a r KEK-TH-1867
Black rings at large D Kentaro Tanabe
Theory Center, Institute of Particles and Nuclear Studies, KEK,Tsukuba, Ibaraki, 305-0801, Japan [email protected]
Abstract
We study the effective theory of slowly rotating black holes at the infinite limit of thespacetime dimension D . This large D effective theory is obtained by integrating theEinstein equation with respect to the radial direction. The effective theory gives equationsfor non-linear dynamical deformations of a slowly rotating black hole by effective equations.The effective equations contain the slowly rotating Myers-Perry black hole, slowly boostedblack string, non-uniform black string and black ring as stationary solutions. We obtainthe analytic solution of the black ring by solving effective equations. Furthermore, byperturbation analysis of effective equations, we find a quasinormal mode condition ofthe black ring in analytic way. As a result we confirm that thin black ring is unstableagainst non-axisymmetric perturbations. We also include 1 /D corrections to the effectiveequations and discuss the effects by 1 /D corrections. ontents /D corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 /D corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A.1 Slowly rotating Myers-Perry black hole . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2 Slowly boosted black string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
B Trivial perturbations in ring coordinate 34C /D corrections of effective equations 35 Introduction
Stationary asymptotically flat black holes have only spherical horizon topology in fourdimensions by the Hawking’s topology theorem [1]. This situation changes drastically inhigher dimensions, and higher dimensional black hole has various horizon topology. Asone concrete example of this fact, Emparan and Reall found the five dimensional blackhole solution, black ring , whose horizon topology is S × S [2]. This is the first asymp-totically flat black hole solution which has a non-spherical horizon topology. After thisdiscovery many new five dimensional black hole solutions with multiple horizons have beenconstructed by the solution generating technique such as the inverse scattering method ( e.g., see [3] for a review). In these solutions the topology of each horizon is S or S × S .We expect that black hole solutions can have much more various and non-trivial horizontopology in higher dimensions than five. To study this variety it would be the first stepto extend the Emparan-Reall’s discovery to higher dimensions, that is, the constructionof the black ring in higher dimensions. However the solution generating technique cannotbe applied to the asymptotically flat solution in higher dimensions than five. Then thereare some efforts to construct the black ring in higher dimensions by numerical methods[4, 5] and by analytical approximation methods such as the blackfold method [6, 7] .In this paper we consider the construction of the D dimensional black ring solution byusing another analytical approximation method, the large D expansion method [10, 11].Recently this large D expansion method was extended to construct the stationary [12, 13]and time-dependent black hole solutions [14, 15]. So it is interesting to see if we can obtainthe black ring solution by the large D expansion method. The paper [13] discussed theeffective theory and gave the effective equation for the large D stationary rotating blackholes. This stationary solution was assumed to have the O (1) horizon angular velocity atthe large D limit. On the other hand the analysis by the blackfold method [6] found thatthe horizon angular velocity of the D dimensional thin black ring becomesΩ H = 1 √ D − R , (1.1)where R is a ring radius. This means that the horizon angular velocity of the thin blackring is O (1 / √ D ), not O (1) at the large D limit. Thus the effective equation in [13] cannotbe straightforwardly applied to the (thin) black ring. Hence, to construct the black ringsolution by the large D expansion method, we should study the effective theory of theblack hole with O (1 / √ D ) horizon angular velocity anew. The purposes of this paperare to give the effective theory of black holes with O (1 / √ D ) horizon angular velocity byincluding time-dependence and to construct the large D black ring solution by solving theeffective equations. We call a black hole with O (1 / √ D ) horizon angular velocity slowlyrotating large D black hole in this paper. We expect that there is no black ring with O (1) The blackfold method has achieved a great success along this direction. In fact black holes withvarious horizon topology, including expected and unexpected ones, has been found [8, 9]. D the tension of a black string becomes smallcompared to its mass by O ( D − ) [11]. As a result the horizon angular velocity should bealso small at large D . Hence, generically, there is no black ring with O (1) horizon angularvelocity at large D . This is why we do not use the word “slowly rotating black ring” inour analysis.The large D effective theory for dynamical black holes has been considered in [14, 15],where it was shown that the large D expansion method gives simple effective equationsfor the dynamical black holes. The equations for dynamical black branes obtained in [15]can be solved very easily, and the numerical solution of the equations capture the non-linear evolution and endpoint of the Gregory-Laflamme instability of the black brane [16]in higher dimensions than the critical dimension [17]. We will study the large D effectiveequation for slowly rotating large D black holes in the similar manner with [15]. Then wewill see that the large D black ring is found as an analytic stationary solution of the effectiveequations. Furthermore, by perturbing effective equations, we can obtain a quasinormalmode condition of the black ring. As found numerically in [18], the quasinormal modesays that the thin black ring is unstable against non-axisymmetric perturbations, andrelatively fat black ring is stable. Our large D effective equations describe the non-linearevolution of such non-axisymmetric instabilities of the black ring. We can also include the1 /D corrections to the effective equations. We find that the 1 /D corrections give strikingfeatures to the instability modes of the black ring.This paper is organized as following. In section 2 we give the effective theory of slowlyrotating large D black holes by the effective equations. We also discuss some generalproperties of the stationary solution of the equations and give formula for thermodynamicquantities such as the mass and angular momentum of the solution with 1 /D corrections.In section 3 we construct the black ring solution analytically by solving the effectiveequations. Some physical properties such as quasinormal modes and phase diagram ofthe black ring will be also discussed there. We close this paper by giving discussion andoutlook of this work in section 4. The appendices contain technical details and some usefulbyproducts of the main results in this paper. Especially the slowly rotating Myers-Perryblack hole and slowly boosted black string will be rediscovered with their quasinormalmode frequencies analytically in Appendix A.Note that we perform the large D expansion of the Einstein equation to find D dimen-sional black hole solutions. In the expansion, defining n by D = n + 4 , (1.2)we use 1 /n as the expansion parameter instead of 1 /D in this paper. This definition of n is same with one of [6, 7] for black rings. 2 Effective equations
We consider the large D effective theory for slowly rotating large D black holes. Theeffective equations will be given as equations for the energy and momentum density ofa dynamical black hole. In this section we study general properties of solutions of theeffective equations without specifying an embedding of solutions. The effective equations describe non-linear dynamical deformations of black holes. Toobtain them, it is natural to use the ingoing Eddington-Finkelstein coordinate for themetric ansatz as ds = − Adv − u v dv + u a dX a ) dr + r G ab dX a dX b − C a dvdX a + r H d Ω n , (2.1)where X a = ( z, Φ). A , u v , G ab , C a are functions of ( v, r, X a ). u a and H are functionsof ( v, X a ). In this paper we use v as a time coordinate in this system and the effectivetheory obtained below, while t was used for it in [15]. This metric ansatz partially fixes thegauge of the radial coordinate r . The residual gauge can be fixed in solving the Einsteinequations. X a = ( z, Φ) are inhomogeneous spatial directions of black holes, and we takeΦ as the rotational direction. We give large D scalings and boundary conditions of themetric functions. Large D scalings To solve the Einstein equation by 1 /n expansion, we should specifybehaviors of each metric functions at large D . In the analysis of the blackfold [6] theyfound that the horizon angular velocity of the D = n + 4 dimensional thin black ring isgiven by Ω H = 1 √ n + 1 1 R + O ( R − ) , (2.2)where R is a ring radius. This result implies that the metric function C Φ in eq. (2.1)becomes O (1 / √ n ) at large D . From this observation we define the slowly rotating large D black hole by C Φ = O (1 / √ n ) . One may think that this assumption is restrictive fora black ring and describes the dynamics only of a thin black ring solution. However thisis not correct. As we will see, solutions obtained in this paper cover wider range thanone by the blackfold because we do not require the much larger ring radius than the ringthickness. In our analysis ring radius and thickness can be comparable, and our large D solution can describe the dynamics also of not-thin black ring.In our setting ∂ Φ is not a Killing vector. Thus the metric functions have Φ-dependencesin general. Then, to have consistent 1 /n expansions of the Einstein equations, we should The stationary solution considered in [13] was assumed to have C Φ = O (1). ∂ Φ = O ( √ n ) . If we take ∂ Φ = O (1), the Einstein equations have not only 1 /n series but also 1 / √ n series, and the analysis becomes a bit involved. To treat the scaling ∂ Φ = O ( √ n ) it is useful to introduce new coordinate φ defined byΦ = φ √ n . (2.3)Then ∂ Φ = O ( √ n ) is equivalent to ∂ φ = O (1). Summarizing, our metric ansatz for theslowly rotating large D black holes is ds = − Adv − u v dv + u a dx a ) dr + r G ab dx a dx b − C a dvdx a + r H d Ω n , (2.4)where x a = ( z, φ ). Each metric functions have following large D expansions A = X k ≥ A ( k ) ( v, r, x a ) n k , C z = X k ≥ C ( k ) z ( v, r, x a ) n k +1 , C φ = X k ≥ C ( k ) φ ( v, r, x a ) n k +1 , (2.5) u v = X k ≥ u ( k ) v ( v, r, x a ) n k , u z = X k ≥ u ( k ) z ( v, x a ) n k +1 , u φ = X k ≥ u ( k ) φ ( v, x a ) n k +1 , (2.6) G zz = 1 + X k ≥ G ( k ) zz ( v, r, x a ) n k +1 , G zφ = X k ≥ G ( k ) zφ ( v, r, x a ) n k +2 , (2.7)and G φφ = G ( z ) n X k ≥ G ( k ) φφ ( v, r, x a ) n k +1 , H = H ( z ) (2.8)Under these scaling assumptions, the Einstein equation can be consistently expanded in1 /n . φ is the rotational direction of the black hole, and we assume ∂ v and ∂ φ become theKilling vectors in a background geometry. This assumption implies that the asymptoticmetric at far from the horizon does not have v and φ dependences. So the metric function H ( z ) and the leading order of G φφ has only z dependences as seen in eq. (2.8). Finallywe introduce new radial coordinate R to realize ∂ r = O ( n ) defined by R = (cid:18) rr (cid:19) n , (2.9)where a constant r is a fiducial horizon size, and we set to r = 1. We consider the decoupled mode excitation [19]. So we assume ∂ v = O (1) and ∂ r = O ( D ). oundary conditions We give boundary conditions for metric functions. In this paperwe consider non-linear dynamics of the decoupled mode excitations [19]. The decoupledmode condition demands following boundary conditions in the asymptotic region R ≫ A = 1 + O ( R − ) , C a = O ( R − ) , G zφ = O ( R − ) . (2.10)The dynamical solutions have the horizon-like surface where A vanishes at the leadingorder in 1 /n expansion . We regard the surface as a horizon of the dynamical blackhole. The boundary condition on the horizon is the regularity condition of the each metricfunctions. We solve the Einstein equation by the large D expansion. The leading order equations ofthe Einstein equations contain only R -derivatives, so we can integrate them easily. Then,the leading order solutions after imposing the boundary conditions are obtained as A (0) = 1 − p v ( v, x a ) R , C (0) a = p a ( v, x a ) R , u (0) v = − H ( z ) p − H ′ ( z ) , (2.11) u (0) a = u (0) a ( z ) , G (0) zz = 0 , G (0) zφ = p z p φ p v R , (2.12)and G (0) φφ = − (cid:18) H ( z ) G ′ ( z ) H ′ ( z ) G ( z )(1 − H ′ ( z ) ) (cid:19) log R + p φ G ( z ) p v R . (2.13) p v ( v, x a ) and p a ( v, x a ) are mass and momentum density of the solution. They are intro-duced as integration functions of R -integrations of the Einstein equations. We can see thatthe horizon position of this dynamical solution is at R = p v ( v, x a ). Furthermore we foundthat the function H ( z ) should satisfy the following condition1 − H ′ ( z ) + H ( z ) H ′′ ( z ) = 0 , (2.14)from the boundary condition of G (0) zφ at asymptotic region R ≫
1. In eq. (2.12) we alreadyimposed the condition (2.14). This condition is equivalent to the constant condition of themean curvature of r = const. surface in the asymptotic region for static solutions [12] . Inthis paper we consider small rotation Ω H = O (1 / √ n ), so the leading order solution has noeffects of rotations and gives same equations with static solutions. The condition (2.14) isintegrated as p − H ′ ( z ) H ( z ) = 2ˆ κ, (2.15) In this paper we consider the slowly rotating black holes. Thus the horizon position is determinedmainly by g uu = − A . If we perform ( D −
1) + 1 decomposition of the Einstein equation on R = const. surface, this conditionis obtained from the momentum constraint as shown in [12]. κ is a constant related with the surface gravity of the horizon as we will see later.So we call ˆ κ a reduced surface gravity. At the leading order the integration functions p v and p a are arbitrary functions of ( v, x a ), and there is no constraint among them. u (0) a isalso arbitrary function of z at the leading order. At higher order in 1 /n , constraints for u (0) a appear, and we can determine their explicit forms.Solving the next-to leading order Einstein equations we obtain non-trivial conditionswhich p v ( v, x a ) and p a ( v, x a ) should satisfy. The non-trivial conditions can be obtainedalso from the momentum constraints on R = const . surface. These conditions give effectiveequations for the slowly rotating large D black holes. Their forms are ∂ v p v − H ′ ( z )2ˆ κH ( z ) ∂ z p v − ∂ φ p v κG ( z ) + ∂ φ p φ G ( z ) + H ′ ( z ) H ( z ) p z = 0 , (2.16) ∂ v p φ − H ′ ( z )2ˆ κH ( z ) ∂ z p φ − ∂ φ p φ κG ( z ) + 1 G ( z ) ∂ φ " p φ p v − κ G ( z ) H ( z ) + 2 G ′ ( z ) H ( z ) H ′ ( z )4ˆ κ G ( z ) H ( z ) ∂ φ p v + H ′ ( z ) H ( z ) p z p φ p v + G ′ ( z ) H ′ ( z )ˆ κG ( z ) H ( z ) p φ = 0 , (2.17)and ∂ v p z − H ′ ( z )2ˆ κH ( z ) ∂ z p z − ∂ φ p z κG ( z ) + ∂ z p v + 1 G ( z ) ∂ φ " p φ p z p v + H ′ ( z ) H ( z ) p z p v − G ′ ( z ) G ( z ) p φ p v + G ′ ( z )ˆ κG ( z ) ∂ φ p φ + H ( z ) G ′ ( z ) H ′ ( z ) + G ( z )( G ′ ( z ) − H ( z ) H ′ ( z ) G ′′ ( z ))4ˆ κ G ( z ) H ( z ) p v − − H ′ ( z ) κH ( z ) p z = 0 . (2.18)These equations describe dynamical non-linear deformations of mass and momentum den-sity of the dynamical black hole. To solve these equations we should specify the embeddingfunctions G ( z ) and H ( z ). Before doing that, we study some general properties of solutionsof these effective equations. Note that the effective equations for dynamical black stringsobtained in [15] can be reproduced from eqs. (2.16), (2.17) and (2.18) as one example.As one simple solution of the effective equations, we consider a stationary solution p v = p v ( z ) , p a = p a ( z ) . (2.19)This solution has two Killing vectors ∂ v and ∂ φ . The effective equations can be solvedunder this ansatz. Eqs. (2.16) and (2.17) give p z = p ′ v ( z )2ˆ κ , p φ ( z ) = ˆΩ H G ( z ) p v ( z ) . (2.20)6Ω H is an integration constant of z -integration of eq. (2.17). To see the physical meaningof ˆΩ H , we rewrite the ( v, φ ) part of the leading order metric of this stationary solution as ds v,φ ) = − (cid:18) − p v ( z ) R (cid:19) dv + G ( z ) n (cid:18) dφ − ˆΩ H p v ( z ) R dv (cid:19) + O (1 /n ) . (2.21)From this expression we can see that Ω H = ˆΩ H / √ n gives the horizon angular velocity.Actually horizon generating Killing vector ξ is ξ = ∂∂v + ˆΩ H ∂∂φ = ∂∂v + Ω H ∂∂ Φ . (2.22)The Killing vector ξ becomes null at the horizon R = p v ( z ). Furthermore we can calculatethe surface gravity κ of the black hole by using ξ as κ = − ∂ r ( ξ µ ξ µ )2 ξ r (cid:12)(cid:12)(cid:12) R = p v ( z ) = n p − H ′ ( z ) H ( z )= n ˆ κ, (2.23)where x µ = ( v, x a ). Here we omit O (1 /n ) terms for simplification. Thus the integrationconstant ˆ κ represents the surface gravity of the black hole at the leading order in 1 /n expansions. Finally, substituting the solutions (2.20) into eq. (2.18), we obtain an equationfor p v ( z ) = e P ( z ) as P ′′ ( z ) − H ′ ( z ) H ( z ) P ( z ) − h G ′ ( z ) G ( z ) − G ′′ ( z ) G ( z ) + G ′ ( z ) G ( z ) H ( z ) H ′ ( z ) − ˆΩ G ( z ) G ′ ( z )(1 − H ′ ( z ) ) H ( z ) H ′ ( z ) i = 0 . (2.24)To solve this equation for P ( z ) we should specify the functions G ( z ) and H ( z ). Thefunctions can be determined by embedding the solution into a background geometry. Innext section we will embed the leading order solution into a flat background in the ringcoordinate. Such solution describes dynamical black rings. In Appendix A we show othersolutions such as the Myers-Perry black hole and boosted black strings by considering theembeddings into other backgrounds. Next we study physical quantities of the solution. The mass and angular momentum ofdynamical black holes can be evaluated by the effective energy-momentum tensor. Theeffective energy momentum tensor, T µν , is defined by T µν = − π G D (cid:16)h K µν i Σ − ¯ g µν h K i Σ (cid:17) , (2.25) The effective energy momentum tensor is originally defined on r = const . surface. Thus its componentruns over ( v, x a , x I ) where x I is a coordinate on S n in eq. (2.4). However the metric components of g IJ have been taken as non-dynamical ones by the gauge choice. So we consider only x µ = ( v, x a ) componentsof the effective energy momentum tensor. x µ = ( v, x a ) in eq. (2.4). Σ is a r = const. surface in the asymptotic region R ≫ G D is the gravitational constant in D dimensions. K µν is the extrinsic curvatureof r = const. surface. The square bracket represents the background subtraction at Σ.¯ g µν is the background induced metric on Σ. The background metric ¯ g µν in the asymptoticregion R ≫ g µν dx µ dx ν = − dv + r dz + G ( z ) n (cid:18) − H ( z ) H ′ ( z ) G ′ ( z ) log R nG ( z )(1 − H ′ ( z ) ) + O ( n − ) (cid:19) dφ . (2.26)The background extrinsic curvature, say ¯ K µν , is calculated as¯ K vv = O (1 /n ) , ¯ K zz = 2ˆ κ + O (1 /n ) , ¯ K φφ = − G ( z ) G ′ ( z ) H ′ ( z )2ˆ κH ( z ) + O (1 /n ) , (2.27)in our gauge choice for the radial coordinate r . The trace part of the background extrinsiccurvature, ¯ K , contains contributions from ¯ K IJ where x I is a coordinate on S n in eq. (2.4).It is obtained as¯ K = n " κ + 1 n (cid:18) κ − G ′ ( z ) H ′ ( z )2ˆ κG ( z ) H ( z ) − κ log R (cid:19) + O ( n − ) . (2.28)By definition, the background extrinsic curvature ¯ K µν satisfies K µν − ¯ K µν = O (1 / R ) , K − ¯ K = O (1 / R ) . (2.29)Then we can compute the effective energy-momentum tensor T µν . The results are T vv = n ˆ κp v π G D R (cid:0) O ( n − , R − ) (cid:1) , T va = 2ˆ κp a − ∂ a p v π G D R (cid:0) O ( n − , R − ) (cid:1) , (2.30)and T zz = − G ( z ) ( H ′ ( z ) p z + H ( z ) ∂ z p v ) + H ( z ) ∂ φ p φ π G D G ( z ) H ( z ) R (cid:0) O ( n − , R − ) (cid:1) ,T zφ = − n G ( z )(2ˆ κp z p φ − p v ( ∂ z p φ + ∂ φ p z )) + 2 G ′ ( z ) p φ p v π G D G ( z ) p v R (cid:0) O ( n − , R − ) (cid:1) ,T φφ = 1 n π G D ˆ κH ( z ) p v R h κH ( z )(2ˆ κp φ + p v ( ∂ φ p φ + G ( z ) ( ∂ v p v − κp v ))) − G ( z ) H ′ ( z ) p v ( G ′ ( z ) p v − κG ( z ) p z ) i (cid:0) O ( n − , R − ) (cid:1) . (2.31)Then we can define the mass M and angular momentum J Φ of a dynamical black holefrom this T µν as M = n ˆ κ Ω n π G D Z dz dφ √ n G ( z ) H ( z ) n p v ( v, x a ) , (2.32) J Φ = √ n ˆ κ Ω n π G D Z dz dφ √ n G ( z ) H ( z ) n p φ ( v, x a ) , (2.33)8here H ( z ) n term comes from the volume factor on S n . Ω n is the volume of unit S n . The √ n factor in J Φ comes from the relation between φ and Φ in eq. (2.3). For stationarysolutions we can define the mass and angular momentum of the black hole by the Komarintegral of each Killing vector. It can be shown easily that their definitions are equivalent.The horizon area of the solution, A H , becomes A H = Ω n Z dz dφ √ n G ( z ) H ( z ) n p v ( v, x a ) , (2.34)where we used the fact that the horizon is at R = p v . Then we find that the stationarysolution satisfies the Smarr formula given by n + 1 n + 2 M = κ π G D A + Ω H J Φ , (2.35)at the leading order in 1 /n expansions. Note that since Ω H = O (1 / √ n ) and κ = O ( n ),Ω H J Φ term does not contribute to the Smarr formula at the leading order. The massand area of the solution can be calculated also for non-stationary solutions p v = p v ( v, x a ).For this non-stationary solution, we can see that the Smarr formula (2.35) holds at theleading order in 1 /n expansion. But this formula does not hold at the higher order fornon-stationary solutions. /D corrections Solving the next-to-next-to leading order of the Einstein equations, we can obtain 1 /n corrections to the effective equations (2.16), (2.17) and (2.18). The explicit forms of the1 /n corrections to the effective equations are not simple, so we show them in AppendixC. Here we show only physical effects by 1 /n corrections to the solution.At first we define the mass and momentum density, p v and p a , up to 1 /n corrections.The mass and momentum density are introduced as integration functions of R − integrationsof the Einstein equations. Thus we should specify the normalization of the integrationfunctions. Expanding A and C a up to 1 /n corrections we define the mass and momentumdensity, p v and p a , by coefficients of 1 / R as A = 1 − p v ( v, x a ) R + O ( n − , R − ) , C a = p a ( v, x a ) R + O ( n − , R − ) . (2.36)This definition normalizes the integration functions up to O (1 /n ) by the asymptotic be-havior of the metric functions. Using this definition and the next-to-next-to leading orderstationary solutions, the surface gravity up to 1 /n correction is obtained as κn = ˆ κ − n " G ′ ( z ) H ′ ( z )4ˆ κG ( z ) H ( z ) + ˆ κ log p v + H ′ ( z )2 H ( z ) p z p v + ˆ κ G ( z ) p φ p v + O ( n − ) . (2.37)One may think that this expression is strange since the surface gravity κ seems not to beconstant when the solution is stationary where p v = p v ( z ) and p a = p a ( z ). However we9an show that ddz " G ′ ( z ) H ′ ( z )4ˆ κG ( z ) H ( z ) + ˆ κ log p v + H ′ ( z )2 H ( z ) p z p v + ˆ κ G ( z ) p φ p v = 0 , (2.38)by using the effective equations (2.16), (2.17) and (2.18) for the stationary solution. Hencethe surface gravity becomes constant for the stationary solution as expected. The horizonangular velocity Ω H = ˆΩ H / √ n is also obtained asˆΩ H = p φ G ( z ) p v + 1 n G ′ ( z ) H ′ ( z )2ˆ κ G ( z ) H ( z ) p φ log p v p v + O ( n − ) . (2.39)The first term in r.h.s. is equivalent to ˆΩ H at the leading order as seen in eq. (2.20). But,at higher order in 1 /n expansion, they are different. We can show that, when the solutionis stationary, the horizon angular velocity is constant up to 1 /n correction by using theeffective equations with 1 /n corrections given in Appendix C.We can calculate the mass, angular momentum and area of the solution up to O (1 /n ).The results are complicate, so we show the results only for the stationary solutions p v = p v ( z ) and p a = p a ( z ). The mass and angular momentum formula up to O (1 /n ) become M = n ˆ κ Ω n π G D Z dz dφ √ n G ( z ) H ( z ) n M ( z ) , (2.40) J Φ = √ n ˆ κ Ω n π G D Z dz dφ √ n G ( z ) H ( z ) n J φ ( z ) , (2.41)where M ( z ) = p v + 1 n " p v − p z κ H ′ ( z ) H ( z ) − p v κ G ′ ( z ) H ′ ( z ) G ( z ) H ( z ) , (2.42) J φ ( z ) = p φ − n " p z p φ κp v H ′ ( z ) H ( z ) + p φ κ G ′ ( z ) H ′ ( z ) G ( z ) H ( z ) . (2.43)The mass and angular momentum can be obtained from the Komar integral for the sta-tionary solution or from the effective energy momentum tensor for general time-dependentsolution. These definitions coincide for the stationary solution. The formula of the horizonarea also has the 1 /n correction as A H = Ω n Z dz dφ √ n G ( z ) H ( z ) n A H ( v, x a ) (2.44)where A H = p v + 1 n " p v log p v − G ( z ) p φ p v . (2.45)Then we can see that the Smarr formula, n + 1 n + 2 M = κ π G D A H + Ω H J Φ , (2.46)can be satisfied up to 1 /n corrections. 10 Black ring and its physical properties
Next we solve the effective equations in an explicit embedding. Especially we consider anembedding into a flat background in the ring coordinate and find a stationary solution,that is, the black ring solution analytically.
The leading order metric has the following asymptotic form at R ≫ ds | R ≫ = − dv + 2 H ( z ) p − H ′ ( z ) dv − u (0) a ( z ) dx a n ! dr + r dz + r G ( z ) d Φ + r H ( z ) d Ω n . (3.1)We embed this leading order metric into a flat background in the ring coordinate. The D = n + 4 dimensional flat metric in the ring coordinate is [20] ds = − dt + R ( R + r cos θ ) " R dr R − r + ( R − r ) d Φ + r ( dθ + sin θd Ω n ) , (3.2)where 0 ≤ r ≤ R , 0 ≤ θ ≤ π and 0 ≤ Φ ≤ π . R is a ring radius. r = 0 is the originof the ring coordinate. The asymptotic infinity and the axis of Φ-rotation are at r = R .In this ring coordinate r = const. surface has a topology of S × S n +1 . θ = 0 is an innerequatorial plane, and θ = π is an outer equatorial plane. Remembering the definition of R = ( r/r ) n , we can embed the leading order induced metric on R = const. surface intothe flat background in this ring coordinate by r = r where r is a constant. As done inthe previous section we set to r = 1. Then, comparing eqs. (3.1) and (3.2) on r = 1surface, we find that the embedding gives following identifications H ( z ) = R sin θR + cos θ , G ( z ) = R √ R − R + cos θ , dθdz = R + cos θR . (3.3)It is easy to confirm that this identification actually satisfies eq. (2.14). We can calculatethe surface gravity, κ = n ˆ κ , by these identifications asˆ κ = p − H ′ ( z ) H ( z ) = √ R − R . (3.4)The embedded solution is the black ring solution because the horizon topology is now S × S n +1 . From R ≥ r , the ring radius should be larger than unity R ≥ R ≃ R ≫ In this note we use the ring coordinate by ( r, θ ) in [20]. Of course the following analysis can be donealso by using ( x, y ) coordinate in [20]. ∂ v p v + ( R + y )(1 + Ry ) R √ R − ∂ y p v − ( R + y ) R ( R − / ∂ φ p v + ( R + y ) R ( R − ∂ φ p φ + 1 + RyR p − y p z = 0 , (3.5) ∂ v p φ + ( R + y )(1 + Ry ) R √ R − ∂ y p φ − ( R + y ) R ( R − / ∂ φ p φ + ( R + y ) R ( R − ∂ φ " p φ p v − Ry + R R − ∂ φ p v + 1 + RyR p − y p z p φ p v + 2(1 + Ry ) R √ R − p φ = 0 (3.6)and ∂ v p z + ( R + y )(1 + Ry ) R √ R − ∂ y p z − ( R + y ) R ( R − / ∂ φ p z − ( R + y ) p − y R ∂ y p v + ( R + y ) R ( R − ∂ φ " p z p φ p v + 1 + RyR p − y p z p v − ( R + y ) p − y R ( R − p φ p v + 2( R + y ) p − y R ( R − / ∂ φ p φ + ( R + y ) p − y R − p v + 2 + 2 Ry − y + R (2 y − R √ R − − y ) p z = 0 . (3.7)Here we introduced a coordinate y defined by y = cos θ. (3.8)Our effective equations (3.5), (3.6) and (3.7) describe non-linear dynamical deformationsof the black ring from thin R ≫ R >
The black ring solution is obtained as the stationary solution of the effective equations(3.5), (3.6) and (3.7). The stationary solution is given by p v = e P ( y ) , p a = p a ( y ) . (3.9)As done in eq. (2.20), we find from eqs. (3.5) and (3.6) p z ( y ) = − ( R + y ) p − y √ R − p ′ v ( y ) , p φ = ˆΩ H R ( R − R + y ) p v ( y ) . (3.10)12urthermore eq. (3.7) gives an equation for p v = e P ( y ) as eq. (2.24) P ′′ ( y ) + 2 R + y P ′ ( y ) − R ( R + y )(1 + Ry ) + ˆΩ R ( R − ( R + y ) (1 + Ry ) = 0 . (3.11)This equation contains a pole at y = − /R in the source term. The solution can have asingular behavior by the source term at y = − /R , too. To obtain a regular solution at y = − /R , ˆΩ H should be ˆΩ H = √ R − R . (3.12)If ˆΩ H does not take this value, the function P ( y ) has a logarithmic divergence at y = − /R .Thus, as seen in five dimensional black ring [2] and higher dimensional black ring by theblackfold [6, 7], the regularity condition determines the horizon angular velocity of theblack ring. The solution of eq. (3.11) under the condition (3.12) is obtained analyticallyas P ( y ) = p + d R + y + (1 + Ry )(1 + Ry + 2 R ( R + y ) log ( R + y ))2 R ( R + y ) , (3.13)where p and d are integration constants. They are degree of freedom associated withtrivial deformations and do not affect physical properties of black rings. p is the 1 /n redefinition of r , and d comes from the redefinition of φ coordinate of the ring coordinateas discussed in Appendix B. Summarizing above results, we found the black ring solutionat the leading order of large D expansion by the metric ds = − (cid:18) − p v ( z ) R (cid:19) dv + 2 (cid:18) dv κ − u (0) a ( z ) dx a n (cid:19) dr − (cid:18) R + y κR p ′ v ( z ) R dzn + ˆΩ H R ( R − R + cos θ ) p v ( z ) R dφn (cid:19) dv + r dz + ˆΩ H κ RR + cos θ p ′ v ( z ) R dzdφn + r R ( R − R + cos θ ) (cid:18) − R ( R + cos θ ) R − R n + ˆΩ R ( R − R + cos θ ) p v ( z ) n R (cid:19) dφ n + r R sin θ ( R + cos θ ) d Ω n , (3.14)where R = ( r/r ) n with r = 1. The coordinates z and θ are related by eq. (3.3). ˆ κ and ˆΩ H are given in eqs. (3.4) and (3.12). The function p v = e P ( y ) is the solution of theequation (3.11), and it is obtained in eq. (3.13) with y = cos θ . Our black ring solutionbreaks down at R = 1, and the very fat black ring cannot be captured by our large D solution . However, the black ring with not so large ring radius can be described by eq.(3.14). So we can study the properties of not only the thin black ring but also not-thinblack ring R > A very fat black ring means the solution with R = 1 + O ( n − ). .3 Quasinormal modes We investigate quasinormal modes of the black ring solution. The quasinormal modes areobtained by perturbation analysis of the effective equations around the black ring solution.The perturbation ansatz is p v ( v, y, φ ) = e P ( y ) (cid:16) ǫe − iωv e imφ F v ( y ) (cid:17) , (3.15) p z ( v, y, φ ) = − ( R + y ) p − y κR p ′ v ( y ) (cid:16) ǫe − iωv e imφ F z ( y ) (cid:17) , (3.16) p φ ( v, y, φ ) = R ( R − / ( R + y ) p v ( y ) (cid:16) ǫe − iωv e imφ F φ ( y ) (cid:17) , (3.17)where we used eq. (3.12). There is one remark on the quantum number m associated with ∂ φ . In the ring coordinate (3.2) the coordinate Φ has the range of 0 ≤ Φ ≤ π . Thus thequantum number m Φ associated with ∂ Φ is quantized as m Φ = 0 , ± , ± , ... . From therelation (2.3), these quantum numbers are related by m = m Φ √ n . (3.18)So m in the perturbations can take non-integer values in general.Perturbing the effective equations (3.5), (3.6) and (3.7) with respect to ǫ , we obtainperturbation equations for F v ( y ), F z ( y ) and F φ ( y ). The perturbation equations have apole at y = − /R again. To solve the perturbation equations we should impose regularityconditions. If we specify the behavior of the perturbation fields at the pole y = − /R asthe regularity condition by F v ( y ) ∝ (1 + Ry ) ℓ (1 + O (1 + Ry )) , (3.19)where ℓ is a non-negative integer, we get one non-trivial condition for the frequency ω as1 √ R − m + imR + ℓR ) − iR ω h R ω + iR p R − (cid:16) m + 3 imR + (3 ℓ − R (cid:17) ω − R ( R − (cid:16) m + 6 im R + 2(3 ℓ − m R + 2 i (3 ℓ − mR + 3( ℓ − ℓR (cid:17) ω − i ( R − / (cid:16) m + 3 im R + 3( ℓ − m R + 6 i ( ℓ − m R − (4 − ℓ + 3 ℓ ) m R + 3 iℓ ( ℓ − mR + ℓ ( ℓ − R (cid:17)i = 0 . (3.20)This is the quasinormal mode condition for the black ring. One may feel strange aboutthis derivation since we derive the quasinormal mode condition from the local condition(3.19) at y = − /R . If the condition (3.20) is satisfied, the perturbation actually becomesregular at y = − /R , but its global structure, e.g. , of F v ( y ), is still unknown. Furthermorethe quantum number ℓ associated with the harmonics in the ring coordinate should begiven by the global solution of F v ( y ). But the quantum number is introduced by the local14igure 1: Plots of the leading order quasinormal modes with ℓ = 0 and m = 2, ω ( ℓ =0) ± ,of the black ring. The left panel shows the real part of the frequency normalized by thereduced surface gravity ˆ κ . The right panel shows the imaginary part. The black and grayline represent ω ( ℓ =0)+ and ω ( ℓ =0) − respectively. The real part of ω ( ℓ =0)+ and ω ( ℓ =0) − are same. ω ( ℓ =0)+ is unstable for R > D limit. This localization of the quantumnumber associated with the harmonics at the large D limit has been observed in thespherical harmonics and spheroidal harmonics in [13]. So we expect that the same featurewould appear in the harmonics in the ring coordinate. It is interesting to investigate thisproperty in detail, although we do not pursue this structure in this paper. In the followingwe give results derived from eq. (3.20) for non-axisymmetric and axisymmetric modesseparately. Non-axisymmetric modes ( m = 0) The quasinormal mode condition (3.20) can besolved in a simple form for ℓ = 0 by ω ( ℓ =0) ± = √ R − R h ˆ m ± i ˆ m (1 ∓ ˆ m ) i , ω ( ℓ =0)0 = √ R − R h ˆ m − i ( ˆ m − i , (3.21)where ˆ m = m/R . ω ( ℓ =0)+ is an instability mode when R > m . ω ( ℓ =0) ± can be understoodas the quasinormal mode of the boosted black string as we will see below. We regard ω ( ℓ =0)0 in eq. (3.21) as a gauge mode in this paper . At large radius limit R ≫ ω ( ℓ =0) ± has corresponding modes, ω ( ℓ =0)0 does not have. So it might be naturalto consider that ω ( ℓ =0)0 is a gauge mode. In Figure 1 we show plots of the quasinormalmode ω ( ℓ =0) ± with m = 2. ω ( ℓ =0)+ becomes stable in relatively fat region R < m . Thisplot reproduces the behavior of the numerical results in [18] . We could not find any We have not still studied residual gauge of ( v, z, φ ) coordinate in detail. The gauge of r -coordinatewas fixed in eq. (2.1). So we should check the gauge transformation of perturbations to check which modesare physical. Actually there are residual gauge. In Appendix B we study stationary one of them. It is abit involved task to eliminate whole gauge modes. So instead we use another simple method to identifyphysical modes here. The perturbation considered in [18] is the mode with m Φ = 2, not m = 2 as seen in eq. (3.18). Thusour result for m = 2 does not exactly correspond to the mode in [18]. However, in D = 5 ( n = 1), we can ℓ = 0 and m = 0 mode perturbations.Note that the instability mode ω ( ℓ =0)+ always saturates the superradiant condition Re h ω ( ℓ =0)+ − m ˆΩ H i = 0 . (3.22)The nature of this saturation at the large D limit is unclear. The dynamically unstablemode of the Myers-Perry black hole also shows the peculiar relation with the superra-diant condition at the large D limit [13, 21]. For equally spin Myers-Perry black hole,the coincidence of the onset of the superradiant and dynamically unstable regime wasconfirmed up to 1 /D corrections [21], and it is consistent with the numerical result [22].For singly rotating Myers-Perry black hole the coincidence was also found at the large D limit [13]. However it is known numerically that their onsets are at different rotationparameters for the singly rotating Myers-Perry black hole in finite dimensions [23, 24].Thus the coincidence of the onsets of superradiant and dynamically unstable regime isonly the feature at the large D limit for the singly rotating Myers-Perry black hole. Aswe will see below, the saturation of the superradiant condition of the black ring does notalways hold at higher order in 1 /n expansion, and we find that onsets of the superradiantand dynamically unstable regime are same also for the black ring at the large D limit. Endpoints of instability?
We found non-axisymmetric instabilities of the blackring solution. One may ask what its endpoint is. In [25] they discussed that the instabilityof the five dimensional thin black ring leads to the fragmentation into black holes byfollowing two reasons. One is that the dynamical timescale to release the inhomogeneity bygravitational wave radiations is much longer than the timescale of the black ring instability.Thus the inhomogeneity by the instability of the thin black ring will grow in time. Anotherreason is that the fragmenting solution can be more entropic than the black ring. Thiscan be understood by the fact that the non-uniform black string is less entropic thanlocalized black holes in five dimensions. Then we expect that going to the fragmentingsolution is preferable as the endpoint of the instability than going back to the (fatter)stable black ring. So the growth of the inhomogeneity of the black ring would not stop,and the fragmentation occurs. In our case this second reason of the discussion cannot hold.The non-uniform black string can be the endpoint of the Gregory-Laflamme instability asseen in [15] since the non-uniform black string becomes more entropic solution in higherdimensions than the critical dimension [17]. Then, also for the black ring instability,we can say same thing. In enough higher dimensions the non-uniform solution is moreentropic than the fragmenting solutions. So we expect that the black ring would not go expect that m Φ = 2 and m = 2 are not different so much. The quantum numbers m and m Φ has the relation (3.18). Then the superradiant factor isRe h ω − m Φ Ω H i = Re h ω − m ˆΩ H i .
16o the fragmentation by the instability. On the other hand the first reason can be appliedto our case. Actually the inhomogeneity by the black ring instability and its rotationgive time-dependent quadrupole moments to gravitational fields. Then the timescale ofgravitational wave radiations by time-dependent quadrupole moments, t GW , is estimatedas done in [25] by t GW ∼ H R n +1 G D M ∼ R n +2 G D M ∼ e n/ R (cid:18) Rr (cid:19) n , (3.23)where the exponential factor e n/ comes from Ω n in G D M as seen in eq. (2.32). We re-stored the black ring thickness r for usefulness. The timescale of the black ring instability, t BR , is estimated by the quasinormal mode frequency as t BR = (cid:16) Im h ω ( ℓ =0)+ i(cid:17) − ∼ r . (3.24)So t BR is O (1) quantity in 1 /n expansion. Then, remembering that our large D black ringsolution satisfies R > r , the ratio of timescales is exponentially small in n at large Dt BR t GW ∼ e − n/ (cid:18) Rr (cid:19) − n − ≪ O (1) . (3.25)This implies that inhomogeneity by the black ring instability cannot be dissipated by thegravitational wave radiation to the infinity in 1 /n expansion. But the inhomogeneousblack ring solution would not fragment into small black holes unlike in five dimensionsas discussed above. Thus the inhomogeneity would stop at some point like the non-uniform black string observed in [15]. Then we reach the conclusion for the endpoint ofthe black ring instability that the black ring evolves to a non-uniform black ring, NUBR,as a metastable solution at large D . The metastable solution at large D means thatthe solution is stationary in 1 /n expansion, but it is not stationary in O ( e − n/ ) by thegravitational wave emissions. The stationary solution with inhomogeneities along rotatingdirection is prohibited by the rigidity theorem [1, 26]. Our statement for the endpointof the black ring instability is consistent with the rigidity theorem since NUBR is notstationary in O ( e − n/ ). Such exponentially suppressed evolutions cannot be describedby our effective equations (3.5), (3.6) and (3.7) by 1 /n expansions. It is interesting toinvestigate this possibility by solving the effective equations for the black ring (3.5), (3.6)and (3.7) directly as the investigation of the endpoint of the Gregory-Laflamme instabilityin [15]. If there is a metastable solution such as NUBR, we can find a stationary andnon-axisymmetric solution of the effective equations.This statement for the endpoint of the black ring instability is only for the leadingorder result in 1 /n expansion. If we include 1 /n corrections to the effective equations,the results show the dimensional dependences, and we can observe the critical dimensionin which the stability of NUBR would change. Actually we have investigated the critical The author thanks Roberto Emparan for the suggestion of this possibility. /n corrections [29]. So ourconjecture that the endpoint of the black ring instability is a stable NUBR is valid onlyat the leading order in 1 /n expansion. Axisymmetric modes ( m = 0) For m = 0 modes, we can also solve eq. (3.20) in asimple form as ω ( m =0) ± = ±√ ℓ − − i ( ℓ − . (3.26)Note that the numerator of eq. (3.20) is a cubic algebraic equation of ω , but one solution ofthe numerator equation is canceled with the denominator of eq. (3.20) for m = 0 modes.So there are only two modes as eq. (3.26). The quasinormal mode frequencies (3.26)are same with one of the Schwarzschild black hole at large D [19]. This is because weconsider the decoupled mode excitations. In the decoupled mode excitation, the dynamicsof perturbations is determined almost locally on the horizon. For m = 0 modes, theperturbation does not have interactions along φ direction. Thus the perturbation feels thehorizon as one of the D − /n expansion. Thus the instability found in [25, 27] for the fat black ring againstaxisymmetric perturbations is not contained in our setup. Such instability occurs due tointeractions between horizons of the black ring, so it may be non-decoupled mode insta-bility. However, by including 1 /n corrections, we find a suggestion for the axisymmetricinstabilities of the black ring as we will see later. As one useful observation let us see the large ring radius limit R ≫ P ( y ) in eq. (3.13) to P ( y ) = O (1 /R ) , (3.27)by p = 0 and d = 0. Then we get the large radius limit of the black ring solution as ds = − (cid:18) − R (cid:19) dv + 2 dvdr − √ n R dvdx + (cid:18) − n R (cid:19) dx + r d Ω n , (3.28)where we defined the black string direction dx by dx = Rdφ/ √ n = Rd Φ. This large radiuslimit solution is actually regarded as the large D limit metric of the boosted string withthe boost velocity sinh α = 1 √ n , (3.29)18s found in [6]. Using this boost relation we can reproduce the large radius limit of thequasinormal modes, ω ( ℓ =0) ± , of the black ring from the quasinormal modes of the blackstring. The quasinormal modes of the black string, ω BS ± was obtained in [11] by large D expansion. For the perturbation ∼ e − iωv e ikx , the leading order result is ω BS ± = ± i ˆ k (1 ∓ ˆ k ) , (3.30)where ˆ k = k/ √ n . Now the boost transformation on the black string is acting by dv → cosh αdv − sinh αdx, dx → cosh αdx − sinh αdv. (3.31)This transformation on ω BS ± gives quasinormal modes of the boosted black string, ω bBS ± ,as ω BS ± → ω bBS ± = k sinh α + ω BS ± cosh α. (3.32)Using eq. (3.29) and ˆ k = k/ √ n , we obtain the quasinormal modes of the boosted blackstring with the boost velocity (3.29) as ω bBS ± = ˆ k ± i ˆ k (1 ∓ ˆ k ) . (3.33)This quasinormal modes precisely reproduce the large radius limit of quasinormal modesof the black ring by identifying ˆ k = ˆ m . In Appendix A we give the direct derivation ofthe quasinormal modes of the boosted black string from the effective equations. /D corrections We can obtain 1 /n corrections to results obtained above by solving the effective equationsup to O (1 /n ). In the following we set to p = 0 and d = 0 for P ( y ) in eq. (3.13). p v ( z )and p φ ( z ) for the stationary solution are p v ( y ) = e P ( y ) " P (1) ( y ) n , (3.34) p φ ( y ) = R ( R − / ( R + y ) e P ( y ) × " n (cid:16) P (1) ( y ) − Ry ) R − P ( y ) + log ( R − R − ) R (cid:17) , (3.35) The boost transformation is one by the exact symmetry of the black string. Thus we can obtainthe quasinormal mode of the boosted black string from the black string. As for the singly rotting Myers-Perry black hole, the boost transformation comes from an approximation symmetry appearing only at theleading order of large D limit. Thus quasinormal modes of the Myers-Perry black hole do not follow boosttransformation rules. P (1) ( y ) is given by P (1) ( y ) = p + d R + y + ( R − R ( R + y ) log ( R − R − ) − Ry ) R ( R + y ) Li (cid:18) RyR − (cid:19) + ( R − y − R (1 − y ))2 R ( R + y ) (log ( R + y )) − R ( R − R + y ) h Ry + 2 y + 2 R (4 − y ) + 2 R y (5 − y ) − R (5 − y ) − R y (1 + y ) − R (12 − y + 5 y ) − R y (10 + y )+ R (4 − y − y ) i + log ( R + y ) R ( R − R + y ) h − R y − y + 3 R (1 − y ) + R (2 + y )+ 3 R y (2 + y ) − Ry (3 + y ) + 3 R y (1 + y ) − R (2 − y + y )+ 2(1 + Ry )( R − R + y ) log ( R − R − ) i . (3.36) p and d are integration constants of 1 /n corrections. Li ( x ) is the polylogarithm function.This solution has the surface gravity κ = n √ R − R " − n + O ( n − ) , (3.37)and the horizon angular velocityΩ H = 1 √ n √ R − R " − n R − R − R − )2 R + O ( n − ) . (3.38)The horizon angular velocity up to O (1 /n ) can be obtained by imposing the regularity of p v ( z ) at the pole y = − /R up to O (1 /n ). We can see that this horizon angular velocityreproduces eq. (2.2) by the blackfold method at the large radius limit up to O (1 /n ). Thesurface gravity has the following large radius limitlim R →∞ κ = n h − n + O ( n − ) i = n α (cid:0) O ( n − ) (cid:1) , (3.39)Then this expression implies that the surface gravity is reproduced by the boosted blackstring with the boost velocity (3.29) up to 1 /n corrections. We can also compare the 1 /n corrections of our results with the 1 /R corrections by the blackfold method [7] to thehorizon angular velocity. However, in this comparison, there are some subtle things suchas a definition of R , so the comparison is not clear. The fact that 1 /n corrections do notcontain 1 /R contribution to Ω H is consistent with the result in [6].Finally we give 1 /n corrections to quasinormal modes of the black ring. ω ( ℓ =0) ± up to1 /n corrections is ω ( ℓ =0) ± = √ R − R " ˆ m ± i ˆ m (1 ∓ ˆ m ) + δ ˆ ω ( ℓ =0) ± n , (3.40)20igure 2: Plots of the quasinormal mode ω ( ℓ =0)+ with ℓ = 0 and m = 2 in n = 10 of theblack ring up to O (1 /n ) corrections. The left panel shows the real part of the frequencynormalized by the reduced surface gravity. The right panel shows the imaginary part. Thedashed line represents the leading order result. The thick line is the result with O (1 /n )corrections.where δ ˆ ω ( ℓ =0) ± = 12 ˆ mR h ˆ m ( R − R ˆ m ) + i ˆ m (2 + ( R + 16) ˆ m + 8 ˆ m )+ 2 ˆ m (1 − i ˆ m ) log ( R − R − ) ∓ (cid:16) m (2 + (3 R −
4) ˆ m )+ i (2 + (3 R + 1) ˆ m − R −
10) ˆ m − m log ( R − R − )) (cid:17)i . (3.41)At the large radius limit R → ∞ with fixed ˆ m , this quasinormal mode reduces to one ofthe boosted black string with the boost parameter given in eq. (3.29) as seen in AppendixA. Thus we could confirm that the blackfold analysis by [6] is correct up to 1 /n corrections.From this 1 /n correction of ω ( ℓ =0)+ , we find that the black ring becomes unstable againstnon-axisymmetric perturbations for R > R D where R D = m " − n − m − m − )2 m + O ( n − ) . (3.42)At the large radius limit this threshold ring radius R D gives the marginally stable ”wavenumber ˆ m D ” as ˆ m D ≡ mR D = 1 + O ( n − , R − ) . (3.43)This corresponds to the Gregory-Laflamme mode of the boosted black string with the boostvelocity (3.29) as seen in Appendix A. In Figure 2 we show the plots of the quasinormalmode frequency ω ( ℓ =0)+ with m = 2 in n = 10 up to the 1 /n corrections. The plots in Figure2 show that the results become much closer to numerical results [18] by 1 /n corrections.At R = 1 our quasinormal mode formula breaks down due to the term log ( R − R − ).Thus the quasinormal mode frequency behavior around R = 1 is not reliable.One interesting observation on 1 /n corrections to quasinormal modes is the relation ofthe superradiant and dynamically unstable regime. At the leading order the real part of21igure 3: The plot of the superradiant factor is shown for the dynamically unstable mode ω ( ℓ =0)+ with m = 2 and n = 10. The superradiant factor can be negative and positive indynamically unstable regime R > m , while it is always positive in stable regime.the quasinormal mode frequency ω ( ℓ =0)+ is marginal in the superradiant conditionRe h ω ( ℓ =0)+ − m ˆΩ H i = O ( n − ) . (3.44)By including the 1 /n corrections, we see that the real part of the quasinormal modefrequency ω ( ℓ =0)+ deviates from the superradiant condition. Actually the real part of thefrequency does not always satisfy the superradiant conditionRe h ω ( ℓ =0)+ − m ˆΩ H i = 2ˆ κn ( ˆ m − − ( R −
4) ˆ m + 2 R ˆ m ) R + O ( n − ) . (3.45)In Figure 3 we show the plot of eq. (3.45) for n = 10 and m = 2. The superradiant regimecan be seen in the dynamically unstable regime R > m . But there is also the dynamicallyunstable mode which does not satisfy the superradiant condition at
R > R S > m , R S = m + 1 + √ m + m m . (3.46)Thus the black ring has dynamically unstable modes which satisfy the superradiant con-dition for R S > R > m , and the superradiant condition becomes not to be satisfied byunstable modes at R > R S . This behavior of the dynamically unstable modes is the strik-ing property of the black ring also seen in numerical results [18]. In fact the all knowndynamically unstable modes of black holes such as one of the Myers-Perry black holesalways satisfy the superradiant condition. The physics behind this feature is not stillclear.We can also obtain 1 /n corrections to the quasinormal modes with m = 0 in a simpleform. The result is ω ( m =0) ± = √ R − R " ±√ ℓ − − i ( ℓ −
1) + δ ˆ ω ( m =0) ± n , (3.47)22here δ ˆ ω ( m =0) ± = ± R √ ℓ − R − " ℓ − − ℓ − ℓ − R ( ℓ −
1) + 2(4 ℓ − R ( ℓ − − i R ( ℓ − R − " ℓ − − ℓ − ℓ − R ( ℓ −
1) + 2(2 ℓ − R ( ℓ − . (3.48)At the large radius limit R → ∞ the quasinormal mode frequency ω ( m =0) ± deviates fromone of Schwarzschild black hole [19] in 1 /n corrections. This deviation can be understoodas a boost effect of the black string as discussed in Appendix A. Around R = 1 theimaginary part of the 1 /n correction δ ˆ ω ( m =0) ± becomes positive. This might mean that theaxisymmetric perturbation becomes also unstable around R = 1. Actually we find thatthere is a marginally stable mode at R = R f as ω ( m =0) ± (cid:12)(cid:12)(cid:12) R = R f = − ℓ √ ℓ − O ( n − ) , (3.49)where R f is the marginal radius for the apparent axisymmetric instability of the blackring given by R f = 1 − n ℓ ℓ −
1) + O ( n − ) . (3.50)The imaginary part becomes positive at 1 > R > R f , and the axisymmetric perturbationmay be unstable there. As mentioned above our formula of the quasinormal mode breaksdown around R = 1. Thus we cannot say immediately that the axisymmetric instabilitymode of the black ring is found from our quasinormal mode formula. However R f < R = 1. This may be related with the instability of the fat black ring [25, 27] Let us draw the phase diagram of the black ring solution obtained by the large D expansionmethod above. To do it we collect the formula for thermodynamic quantities of the blackring. The mass M and the angular momentum J Φ formula are given in eqs. (2.32) and(2.33). For the black ring, using the ring coordinate embedding by eq. (3.3) and theleading order stationary solution (3.14), these formula become M = n Ω n G D ˆ M , J Φ = √ n Ω n G D ˆ J Φ , (3.51)where ˆ M = Z dy ( R − e P ( y ) R p − y R p − y R + y ! n , (3.52)and ˆ J Φ = Z dy ( R − / e P ( y ) R ( R + y ) p − y R p − y R + y ! n . (3.53)23ote that y = cos θ in the ring coordinate (3.2). The formula (2.34) gives the area of theblack ring as A H = 2 π Ω n ˆ A H , (3.54)where ˆ A H = Z dy √ R − e P ( y ) p − y R p − y R + y ! n . (3.55) P ( y ) is given in eq. (3.13). We set to p = d = 0 in eq. (3.13) . In these formula theblack ring thickness r is set to unity. Here we consider only the leading order contributionsto the mass, angular momentum and area. This is because it seems to be difficult to trackthe effect of 1 /n corrections precisely in M , J Φ and A H due to a term in integrandswith a power of n . At large n the integrations of such terms can be very large, and itis not clear how to control the size of such integrations in 1 /n expansion. In this paper,we take into account only the leading order contributions to observe general feature, notdetail numerical values, of the phase diagram of the large D black ring. Hence the phasediagrams for the large D black ring shown below have O (1 /n ) errors. On the other handthe temperature and horizon angular velocity do not have such troublesome terms in theirdefinitions, so we include 1 /n corrections. The temperature T H of the black ring is T H ≡ κ π = n π √ R − R " − n . (3.56)The horizon angular velocity of the black ring isΩ H = ˆΩ H √ n = 1 √ n √ R − R " − n R − R − R − )2 R . (3.57)To draw the phase diagram we define reduced quantities by the mass as following j n +1Φ = c j J n +1Φ G D M n +2 , a n +1H = c a A n +1H ( G D M ) n +2 , (3.58)and t H = c T T H ( G D M ) / ( n +1) , ω H = c ω Ω H ( G D M ) / ( n +1) . (3.59)The normalization numerical coefficients are taken from [6] as c j = Ω n +1 n +5 ( n + 2) n +2 ( n + 1) ( n +1) / , c a = Ω n +1 π ) n +1 n ( n +1) / ( n + 2) n +2 ( n + 1) ( n +1) / , (3.60)and c t = 4 π √ n + 1 √ n (cid:18) ( n + 2)Ω n +1 (cid:19) − / ( n +1) , c ω = √ n + 1 (cid:18) ( n + 2)Ω n +1 (cid:19) − / ( n +1) . (3.61) These values of p and d are chosen so that the solution becomes P ( y ) = O (1 /R ) at the large radiuslimit R ≫
1. These choices do not affect the essential feature of the phase diagram. j Φ , a H ) phase diagram of the black ring in n = 10 is shown. The curveby black dots is the numerical evaluation of the black ring solution obtained by the large D expansion method. The gray dashed line is the leading order result by the blackfoldmethod. The thick gray line is the result with O (1 /R ) corrections by [7]. The left panelshows the phase diagram in not-thin region, and right panel is for the thin region R ≫ r .At the large radius R ≫ r these results show similar behavior as expected. The differenceof values are within O (1 /n ) errors in thin region.Then we obtain j n +1Φ = 12 n +5 Ω n +1 Ω n (cid:18) n (cid:19) n +2 (cid:18) n (cid:19) − ( n +1) / ˆ J n +1Φ ˆ M n +2 , (3.62) a n +1H = 12 n Ω n +1 Ω n (cid:18) n (cid:19) n +2 (cid:18) n (cid:19) − ( n +1) / ˆ A n +1H ˆ M n +2 , (3.63) t H = 2 n/ ( n +1) (cid:18) n (cid:19) / (cid:18) Ω n Ω n +1 (cid:19) / ( n +1) (cid:18) n (cid:19) − / ( n +1) κ ˆ M / ( n +1) , (3.64)and ω H = 2 / ( n +1) (cid:18) n (cid:19) / (cid:18) Ω n Ω n +1 (cid:19) / ( n +1) (cid:18) n (cid:19) − / ( n +1) ˆΩ H ˆ M / ( n +1) . (3.65)We evaluate j Φ , a H , t H and ω H numerically for the large D black ring solution by usingeqs. (3.52) and (3.53) with P ( y ) given in eq. (3.13). As mentioned above, we set to p = d = 0. Figure 4 shows the phase diagram of ( j Φ , a H ) for the black ring solution bythe blackfold and large D expansion method from R/r = 1 . R/r = 20 in n = 10. R and r are a ring radius and ring thickness of the black ring. The curve by black dotsis the result by numerical evaluations of eqs. (3.62) and (3.63) for our large D black ringsolution. The gray dashed line is the leading order result by the blackfold method [6]. Thethick gray line is O (1 /R ) correction of the blackfold [7]. As expected we can see thatthese results show similar behavior at the large radius region R ≫
1. Two results by theblackfold and large D expansion method seems to have a difference by the constant offset25igure 5: The ( j Φ , t H ) (left) and ( j Φ , ω H ) (right) phase diagram in n = 10 are shown.The gray dashed and thick line corresponds to the leading order and O (1 /R ) correctedresults of the blackfold method. The curve by black dots is the numerical evaluations forthe large D black ring solution.at R ≫
1. This difference can be understood as O (1 /n ) errors in the large D expansionmethod. By taking the large radius limit R ≫
1, from P ( y ) = O (1 /R ), we find that M = n Ω n +1 G D Rr n (cid:0) O ( n − , R − ) (cid:1) , J Φ = √ n Ω n +1 G D R r n (cid:0) O ( n − , R − ) (cid:1) , (3.66)and A H = 2 π Ω n +1 Rr n +10 (cid:0) O ( n − , R − ) (cid:1) , (3.67)where we restored the black ring thickness r . On the other hand the leading order resultsby the blackfold [6] gives M = ( n + 2)Ω n +1 G D Rr n (cid:0) O ( R − ) (cid:1) , J Φ = √ n + 1Ω n +1 G D R r n (cid:0) O ( R − ) (cid:1) , (3.68)and A H = 2 π r n + 1 n Ω n +1 Rr n +10 (cid:0) O ( R − ) (cid:1) . (3.69)These results coincide at the leading order in 1 /n expansion. 1 /n corrections in eqs. (3.68)and (3.69) gives smaller a H and j Φ than the leading order results in 1 /n expansion. So1 /n correction would reduce the difference seen in Figure 4. This O (1 /n ) error would givethe difference of the onset at R ≫ O (1 /R ) corrections in the results by the large D expansion method. This is because thedefinitions of thermodynamic quantities contain the y -integrations of R n / ( R + y ) n . Theblackfold by the 1 /R expansion and the large D expansion method by 1 /n expansion donot give same results for such integrations in higher order corrections of O ( n − , R − ).Figure 5 are plots of the phase diagram of ( j Φ , t H ) and ( j Φ , ω H ) for the black ringby the blackfold (gray dashed and thick lines) and the large D expansion method (curveby black dots). In these diagrams we see that results by the blackfold and the large D R ≫ j Φ ∼
1. This might be becauseour results contain O (1 /n ) errors which become larger at fat ring region, and the blackfoldmethod also has not-small errors there . We constructed the effective theory for the slowly rotating large D black holes. The slowrotation is defined by O (1 / √ D ) horizon angular velocity at large D . This solution classof large D black holes contains the slowly rotating Myers-Perry black hole, slowly boostedblack string and black ring solution. The black ring should be slowly rotating at large D since the tension effect which determines the horizon angular velocity is small. So ouranalysis of the black ring in this paper by the large D effective theory is not restrictedone. We solved the effective equations and found the black ring solution analytically.Furthermore, by perturbation analysis, the quasinormal mode condition of the black ringwas obtained. The quasinormal mode says that the thin black ring has the Gregory-Laflamme type instability in the non-axisymmetric perturbations as found numerically.The black ring solution obtained in this paper describes not only thin black ring, but alsonot-thin black ring. For not-thin black ring, we found that the black ring becomes stableagainst non-axisymmetric perturbations. We gave some discussions on the endpoint ofthe non-axisymmetric instabilities of the black ring. At large D the gravitational waveemission becomes much less efficient by O ( e − D/ ) than the dynamical instability, and theinstability would not lead to the fragmentation of the horizon in enough higher dimensions.This fact suggests the existence of a metastable solution, non-uniform black ring, as theendpoint of the black ring instability at large D . We also studied 1 /D corrections to thequasinormal modes and phase diagram.In this paper we considered the construction of the black ring solution by the large D expansion method as the first step to reveal the variety of higher dimensional black holes.So we have some natural extensions of our current work. One is to consider the O (1)horizon angular velocity solution. In [13] the effective theory for the stationary solutionwith O (1) horizon angular velocity was constructed, but they did not include the timedependent deformations. It is possible to include the time dependent deformations in thework [13] by the same way in this paper. The non-liner dynamical deformations of thesingly rotating Myers-Perry black hole can be described by such framework. The singlyrotating Myers-Perry black hole is known to have the unstable modes, and it is interestingto study the instability mode and stationary deformed solution associated with the zeromode, so called bumpy black hole, as the endpoint of the instability by the large D effective theory. The second possible extension is to investigate the non-linear evolution In not so much higher dimension, it was observed that the blackfold method has a good accuracyeven in not thin region j Φ ∼
27f the instability of the black ring. The effective equations obtained in this paper describesuch evolutions. It is interesting to solve the effective equations numerically and studythe endpoint of the black ring instability. Our effective equations have much simplerform than original Einstein equations, and they are expected to be much more tractablenumerically. In that analysis our conjecture for the existence of the non-uniform black ringcan be also studied. The third is to include further angular momentum to the effectiveequations. The effective equations in this paper describe the dynamical black hole withessentially only one angular momentum. By adding further number of angular momentato the effective equations, we can describe much richer dynamics of the black holes, andit will be found that the variety of the black hole horizon becomes much richer than oneof the singly rotating black hole. These work would give new interesting insights to thehigher dimensional black hole physics.
Acknowledgments
The author was grateful to Roberto Emparan for useful discussions and comments on thedraft. This work was supported by JSPS Grant-in-Aid for Scientific Research No.26-3387.
A Other solutions
In this appendix we study other embeddings of the solution. In the asymptotic region R ≫ R = const. surface has the following form ds | R =const. = − dv + r (cid:0) dz + G ( z ) d Φ + H ( z ) d Ω n (cid:1) + O ( R − ) . (A.1) H ( z ) should satisfy the equation given by1 − H ′ ( z ) + H ( z ) H ′′ ( z ) = 0 . (A.2)In the following this metric is embedded into a flat background in spherical coordinates.Such solutions describes the slowly rotating Myers-Perry black hole and slowly boostedblack string solution. A.1 Slowly rotating Myers-Perry black hole
The first spherical coordinate is the spherical coordinate of D = n + 4 dimensional flatspacetime. The metric in the spherical coordinate is ds = − dt + dr + r ( dz + sin z d Φ + cos z d Ω n ) . (A.3)The embedding of the leading order metric is given by r = r in this flat backgroundmetric. Then the embedding says that H ( z ) and G ( z ) are identified by H ( z ) = cos z, G ( z ) = sin z. (A.4)28ote that this H ( z ) actually satisfies eq. (A.2). The surface gravity of this sphericalcoordinate embedding is κ = n ˆ κ = n . (A.5) r = const. surfaces in the metric (A.3) have the topology of S n +2 . So the topology of theblack hole horizon is S n +2 in this embedding. Effective equations
The effective equations (2.16), (2.17) and (2.18) in the sphericalcoordinate embedding are ∂ v p v + ∂ z p v tan z − ∂ φ p v sin z + ∂ φ p φ sin z − p z tan z = 0 , (A.6) ∂ v p φ + ∂ z p φ tan z − ∂ φ p φ sin z + ∂ φ p v + 1sin z ∂ φ " p φ p v − p φ − p φ p z p v tan z = 0 , (A.7)and ∂ v p z + ∂ z p z tan z − ∂ φ p z sin z + ∂ z p v + 1sin z ∂ z " p z p φ p v − p φ cot z + p z tan z sin zp v sin z + 2 cot z sin z ∂ φ p φ − cos 2 z cos z p z = 0 . (A.8) Stationary solution
We solve eqs. (A.6), (A.7) and (A.8) for the stationary solution.The stationary solution ansatz assumes that ∂ v and ∂ φ are the Killing vectors. Thus weassume p v = e P ( z ) , p φ = p φ ( z ) , p z = p z ( z ) . (A.9) p a ( z ) are given by p v ( z ) as derived in eq. (2.20) with the integration constant ˆΩ H . Theequation for P ( z ) given in eq. (2.24) in the spherical coordinate embedding is P ′′ ( z ) + P ′ ( z ) tan z − ˆ a cos z = 0 . (A.10)Here we rename ˆΩ H in eq. (2.24) by ˆ a just for usefulness. The solution of this equation is P ( z ) = p + d sin z −
12 ˆ a cos z. (A.11)Integration constants, p and d , describe trivial deformations of the solution, so we set to p = d = 0. Then, using eq. (2.20), we find that the stationary solutions of eqs. (A.6),(A.7) and (A.8) are p v ( z ) = e − ˆ a cos z/ , p φ ( z ) = ˆ ap v ( z ) sin z, p z ( z ) = p ′ v ( z ) . (A.12)This stationary solution describes the D = n + 4 dimensional slowly rotating Myers-Perryblack hole [28] with a = ˆ a/ √ n . This can be confirmed directly by performing a coordinatetransformation from the Myers-Perry black hole. As we can see below, quasinormal modesof the stationary solution also coincide to one of the Myers-Perry black hole obtained foundin [13]. 29 uasinormal modes By perturbing the equations (A.6), (A.7) and (A.8) around thestationary solution, we obtain quasinormal mode frequencies. The perturbations are p v ( v, z, φ ) = e − a cos z/ (cid:16) ǫF v ( z ) e − iωv e imφ (cid:17) ,p φ ( v, z, φ ) = a sin ze − a cos z/ (cid:16) ǫF φ ( z ) e − iωv e imφ (cid:17) ,p z ( v, z, φ ) = a cos z sin ze − a cos z/ (cid:16) ǫF z ( z ) e − iωv e imφ (cid:17) . (A.13)We impose the boundary condition on the perturbations at z = 0 as F v ( z ) ∝ z ℓ (1 + O ( z )) . (A.14)Then the frequency is discretized by quantum numbers, ℓ and m . In the following weassume m = O ( n − ). This assumption can be understood by observing the sphericalharmonics. At large D the spherical harmonics on S n +1 , Y jℓm Φ ( z ), in the metric (A.3) isreduced to [13] Y jℓm Φ ( z ) ∝ e im Φ Φ (sin z ) | m Φ | +2 k S cos j z (A.15)where j is the quantum number on S n . k S is the quantum number along z -direction.Comparing eqs. (A.14) and (A.15), we see that ℓ is given by ℓ = | m Φ | + 2 k S . (A.16)In this paper we do not consider the excitation on S n , so our boundary condition corre-sponds to one of j = 0 in [13]. Note that, since we rescaled φ coordinate by eq. (2.3), thequantum number m Φ associated with ∂ Φ is related with the quantum number m of ∂ φ by m Φ = √ nm . Thus the spherical harmonics has √ nm dependence, not m dependence. Ifwe assume m = O (1), the spherical harmonics is dominated by the quantum number m as Y jℓm Φ ( z ) ∝ (sin z ) √ nm . (A.17)The analysis of this function becomes involved since the z -derivative can be very large anddiverging in n . To avoid this complexity we assume m = O (1 /n ). Then the perturbationscan satisfy the boundary condition (A.14) without diverging terms in n . Thus, in theperturbation (A.13), we introduce new O (1) quantity ¯ m by m = ¯ mn . (A.18)Then the quasinormal mode frequencies are written by ℓ , ¯ m and ˆ a . We can extend the allcalculations up to O (1 /n ) by including 1 /n corrections. The obtained quasinormal modesup to O (1 /n ) is ω ± = ω (0) ± + ω (1) ± n + O ( n − ) , (A.19) This boundary condition was also used in [13]. ω (0) ± = ±√ ℓ − − i ( ℓ −
1) (A.20)and ω (1) ± = ± √ ℓ − ℓ − − ˆ a (2 ℓ − ℓ + 8)2 ℓ √ ℓ − i ˆ a ¯ m √ ℓ − ℓ ! − i ( ℓ − ℓ − − i ˆ a ( ℓ − ℓ + 8)2 ℓ − ˆ a ¯ m ( ℓ − ℓ ! . (A.21)Actually this quasinormal mode reproduces the quasinormal mode of the Schwarzschildblack hole up to 1 /n corrections for ˆ a = 0 [19] and one of Myers-Perry black hole [13] with a = ˆ a/ √ n and m = ¯ m/ √ n . A.2 Slowly boosted black string
We consider the embedding in another spherical coordinate, which is the coordinate of thespacetime with one compact direction. The D = n + 4 dimensional spacetime with onecompact direction has the following metric in the spherical coordinate ds = − dt + d Φ + dr + r (cid:0) dz + sin z d Ω n (cid:1) . (A.22)Φ is a coordinate of the compact direction. The embedding r = r of the leading ordermetric (A.1) into eq. (A.22) gives following identifications G ( z ) = 1 , H ( z ) = sin z, (A.23)where we set to r = 1. The embedded solution has S × S n +1 horizon topology in thespacetime with one compact dimension. So the solution of the effective equations describenon-linear dynamical deformations of the black string. Effective equations
The effective equations in this embedding become ∂ v p v − cot z ∂ z p v − ∂ φ p v + ∂ φ p φ + p z cot z = 0 , (A.24) ∂ v p φ − cot z ∂ z p φ − ∂ φ p φ − ∂ φ p v + ∂ φ " p φ p v − cot z p z p φ p v = 0 , (A.25)and ∂ v p z − cot z ∂ z p z − ∂ φ p z + ∂ z p v + ∂ z " p z p φ p v − cot z p z p v − cos 2 z sin z p z = 0 . (A.26)31 tationary solutions We have two stationary solutions of the effective equations. Oneis the boosted black string. The ansatz for the boosted black string is p v ( z ) = e P ( z ) , p φ = p φ ( z ) , p z = p z ( z ) . (A.27)Then the effective equations give the solution p v ( z ) = e p + d cos z , p φ = ˆ σp v ( z ) , p z = p ′ v ( z ) . (A.28)ˆ σ is an integration constant, and it describes the boost parameter of the boosted blackstring with the boost parameter sinh α = ˆ σ √ n , (A.29)where the boost transformation of the black string and definition of α are given in eq.(3.28). The integration constants p and d are horizon size and position of the blackstring origin respectively. Thus we can set to p = 0 and d = 0.Another stationary solution is the non-uniform black string. This solution is inhomo-geneous along Φ direction. So ∂ Φ is not the Killing vector. The ansatz for the non-uniformblack string is p v = p v ( φ ) , p φ = p φ ( φ ) , p z = 0 . (A.30)The condition p z = 0 reflects the fact that the Gregory-Laflamme mode of the black stringexists only in the S-wave sector. This ansatz gives the equation for the non-uniform blackstring at large D [29] as p φ = ∂ φ p v + P φ (A.31)and ∂ φ p v + ∂ φ p v − ∂ φ p v ∂ φ p v p v + ( ∂ φ p v ) p v = 0 . (A.32) P φ is an integration constant describing the momentum along φ direction, and we set to P φ = 0 in eq. (A.32). To solve this equation we need the numerical treatment. In thispaper we consider only the boosted black string solution. Quasinormal modes
Considering the perturbations around the stationary solution(A.28) p v ( v, z, φ ) = 1 + ǫF v ( z ) e − iωv e i ˆ kφ ,p φ ( v, z, φ ) = ˆ σ (cid:16) ǫF φ ( z ) e − iωv e i ˆ kφ (cid:17) ,p z ( v, z, φ ) = ǫF z ( z ) e − iωv e i ˆ kφ , (A.33)32nd boundary conditions at cos z = 0 given by F v ( z ) ∝ (cos z ) ℓ (1 + O (cos z )) , (A.34)we obtain the quasinormal mode condition as1ˆ k + ˆ σ ˆ k + ˆ σ ( iℓ + i ˆ k + ω ) h ω + ( − i + 3 iℓ + 3ˆ σ ˆ k + 3 i ˆ k ) ω + ( − ℓ + ℓ (3 + 6 i ˆ σ ˆ k − k ) + ˆ k (5ˆ k + 3ˆ σ ˆ k − k + 2 i ˆ σ (3ˆ k − ω − i ( ℓ + ℓ ( − − i ˆ σ ˆ k + 3ˆ k ) + kℓ (3 i ˆ σ − k − σ ˆ k − i ˆ σ ˆ k + 3ˆ k )+ ˆ k (2 + i ˆ σ ˆ k − k + ˆ k + ˆ σ (2 − k ) − i ˆ σ ˆ k (3ˆ k − i = 0 . (A.35)This condition can be solved for ℓ = 0 mode in a simple form by ω ( ℓ =0) ± = ˆ σ ˆ k ± i ˆ k (1 ∓ ˆ k ) . (A.36)This is the quasinormal modes of the S-wave sector of the scalar type gravitational per-turbation of the boosted black string. ω ( ℓ =0)+ shows the Gregory-Laflamme instability [16]for ˆ k <
1. For ˆ k = 0 mode the quasinormal mode condition gives ω (ˆ k =0) ± = ±√ ℓ − − i ( ℓ − . (A.37)Although, for general modes ℓ = 0 and ˆ k = 0, we cannot obtain a simple solution of eq.(A.35), it can be seen that the instability mode exists only in the S-wave ( ℓ = 0) sector.This is consistent with the result in [30] that the black string is unstable only for theS-wave sector.We can also find 1 /n corrections to the quasinormal modes. For the S-wave sector, thequasinormal modes are ω ( ℓ =0) ± = ˆ σ ˆ k ± i ˆ k (1 ∓ ˆ k )+ ˆ k n h i ( ∓ − k ± k ) − σ (1 ∓ k + 2ˆ k ) + i ˆ σ ( ∓ k ) i + O ( n − ) . (A.38)If we take ˆ σ = 0, the result reproduces the quasinormal modes of the black string obtainedin [11, 31]. The relation of the boosted black string and black ring become clear byobserving eq. (A.38). Actually, if we take the large radius limit of the quasinormal modes ω ( ℓ =0) ± of the black ring in eq. (3.41), we obtain ω ( ℓ =0) ± = ˆ m ± i ˆ m (1 ∓ ˆ m )+ ˆ m n h ∓ m + 4 ˆ m + i ( ∓ m ± m ) i . (A.39)Comparing eqs. (A.38) and (A.39) by identifying ˆ k = ˆ m , we can see that the large radiuslimit of the black ring corresponds to the boosted black string with the boost parametersinh α = ˆ σ √ n = 1 √ n + 1 , (A.40)33p to O (1 /n ). This boost parameter corresponds to eq. (3.29), and this is consistent withthe result in [6] . The Gregory-Laflamme mode ˆ k GL of the boosted black string, forwhich the imaginary part of ω ( ℓ =0)+ vanishes, is obtained asˆ k GL = 1 − − ˆ σ n + O ( n − ) . (A.41)This Gregory-Laflamme mode of ˆ σ = 1 reproduces the threshold wave number (3.43) forthe black ring.For ˆ k = 0 modes, the quasinormal modes up to 1 /n corrections are ω (ˆ k =0) ± = ±√ ℓ − − i ( ℓ − n h ± (3 ℓ − − ˆ σ ) √ ℓ − − i ( ℓ − ℓ − − ˆ σ ) i + O ( n − ) . (A.42)Using eq. (A.40), the quasinormal mode (A.42) reproduces eq. (3.48) at the large radiuslimit R = ∞ . B Trivial perturbations in ring coordinate
In this appendix we study trivial perturbations in the D = n + 4 dimensional ring coordi-nate given by ds = − dt + R ( R + r cos θ ) " R dr R − r + ( R − r ) d Φ + r ( dθ + sin θd Ω n ) . (B.1)There are two trivial perturbations, axisymmetric and non-axisymmetric trivial perturba-tions. Axisymmetric trivial perturbation
The axisymmetric trivial perturbation is the re-definition of the ring radius R . The perturbations R → R + δR, (B.2)and r → r + δR (cid:18) rR + R − r R cos θ (cid:19) , θ → θ − δR sin θr (B.3)do not change the form of the metric (B.1) up to O ( δR ).As another trivial axisymmetric perturbation, there is a perturbation which becomestrivial at the large D limit. If we consider the perturbation,Φ → Φ + δ Φ (B.4) In [6] they did not use 1 /n expansion, so the relation (3.29) obtained in [6] should be valid at all orderin 1 /n at the large radius limit. r → r + δ Φ ( R − r )( r + R cos θ ) R ( R + r cos θ ) , θ → θ − δ Φ ( R − r ) sin θr ( R + r cos θ ) , (B.5)the metric does change its form by δ ( ds ) = − δ Φ 2 R ( R − r )( R + r cos θ ) (cid:16) R cos θdr + r ( Rdr − ( R − r ) sin θdθ ) (cid:17) . (B.6)Thus this transformation is not a trivial perturbation. However, if we consider the large D limit, this transformation becomes trivial. Taking δ Φ = ˆ δ Φ /n and using R = ( r/r ) n ,the metric deformation (B.6) becomes δ ( ds ) = ˆ δ Φ n R ( R − R + cos θ ) dθ + O ( n − ) , (B.7)where we set to r = 1, and we neglect dr and drdz terms since dr ≃ d R / ( n R ) is higherorder in 1 /n . The deformation (B.7) can be absorbed into the 1 /n redefinition of θ . So thetransformation (B.5) becomes trivial at the large D limit. We can see that R = ( r/r ) n changes its form by the transformation (B.5) to R → R exp " ˆ δ Φ (cid:18) R − − ( R − R ( R + cos θ ) (cid:19) . (B.8)So, observing the solution eq. (3.13), we find that the integration constants p and d ineq. (3.13) represent the transformation (B.5) and the O (1 /n ) redefinition of r . So p and d in eq. (3.13) do not have physical degree of freedom. Non-axisymmetric trivial perturbations
Let us consider the perturbation given by r → r + ǫe im Φ Φ √ R − r R ( r + R cos θ ) , z → z + ǫe im Φ Φ √ R − r r sin θ. (B.9)For m Φ = 1, the metric does not change the form under this non-axisymmetric pertur-bation up to O ( ǫ ). So the stationary deformation of the black ring with m Φ = 1 is justa trivial deformation. Thus it might be reasonable to regard that the perturbations with m Φ = 1 and ℓ = 0 does not have physical degree of freedom. C /D corrections of effective equations In this appendix we show 1 /n corrections to the effective equations (2.16), (2.17) and(2.18). The effective equations up to O (1 /n ) are ∂ v p v − H ′ ( z )2ˆ κH ( z ) ∂ z p v − ∂ φ p v κG ( z ) + ∂ φ p φ G ( z ) + H ′ ( z ) H ( z ) p z + ∆ v n = 0 , (C.1)35 v p φ − H ′ ( z )2ˆ κH ( z ) ∂ z p φ − ∂ φ p φ κG ( z ) + 1 G ( z ) ∂ φ " p φ p v − κ G ( z ) H ( z ) + 2 G ′ ( z ) H ( z ) H ′ ( z )4ˆ κ G ( z ) H ( z ) ∂ φ p v + H ′ ( z ) H ( z ) p z p φ p v + G ′ ( z ) H ′ ( z )ˆ κG ( z ) H ( z ) p φ + ∆ φ n = 0 , (C.2)and ∂ v p z − H ′ ( z )2ˆ κH ( z ) ∂ z p z − ∂ φ p z κG ( z ) + ∂ z p v + 1 G ( z ) ∂ φ " p φ p z p v + H ′ ( z ) H ( z ) p z p v − G ′ ( z ) G ( z ) p φ p v + G ′ ( z )ˆ κG ( z ) ∂ φ p φ + H ( z ) G ′ ( z ) H ′ ( z ) + G ( z )( G ′ ( z ) − H ( z ) H ′ ( z ) G ′′ ( z ))4ˆ κ G ( z ) H ( z ) p v − − H ′ ( z ) κH ( z ) p z + ∆ z n = 0 . (C.3)The corrections terms ∆ v , ∆ φ and ∆ z are given by∆ v = − ∂ φ p v (cid:0) G ′ H ′ + 4ˆ κ GH (cid:1) κ G H − ∂ z p v (cid:0) G ′ − κ GHH ′ (cid:1) κ GH (C.4)+ 1 p v − G ′ H ′ p φ ˆ κG H + p φ H ′ ∂ z p φ κG H + H ′ ∂ φ p z ˆ κG H + 3 ∂ φ p φ κG ! + H ′ ∂ φ p φ p z ˆ κG H + 3( ∂ φ p φ ) κG + (cid:18) κH − κ (cid:19) p z ! + ∂ φ p φ (cid:16) G ′ H ′ ˆ κ H − G (cid:17) G + p z G ′ G + (cid:0) κ − H (cid:1) H ′ H ! − H ′ ∂ z ∂ φ p φ κ G H − H ′ ∂ φ p z κ G H + 3( ∂ φ p v ) p φ κG p v + − H ′ ∂ φ p v p φ p z ˆ κG H + p φ (cid:18) − H ′ ∂ z p v κG H − ∂ φ p v κG (cid:19) − ∂ φ p v ∂ φ p φ p φ ˆ κG p v + (cid:18) − κ H (cid:19) ∂ z p z − ˆ κG ∂ z p v + ∂ φ p φ κ G + H ′ p v (cid:0) H ( G ′ ) H ′ + G ( G ′ − HG ′′ H ′ ) (cid:1) κ G H , (C.5)36 φ = G ′′ ˆ κG p φ − GHG ′ H ′ ˆ κ + 8 GH G ′′ ˆ κ − (cid:0) κ H − (cid:1) ( G ′ ) κ G H ∂ φ p v − κ (cid:0) ( G ′ ) − GG ′′ (cid:1) H + 8ˆ κ GG ′ H ′ H − (cid:0) ( G ′ ) − GG ′′ (cid:1) H − GG ′ H ′ κ G H p φ log p v + 3 p φ log p v ( ∂ φ p v ) ˆ κG p v − p φ ( ∂ φ p v ) ˆ κG p v + 3 p φ log p v ( ∂ φ p v ) κ G p v + p φ ( ∂ φ p v ) κ G p v + (cid:16) H ′ H − G ′ G (cid:17) ∂ φ p z κ − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ φ p φ κ G H + G ′ ∂ z p φ κG − ∂ z p φ κ − G ′ H ′ ∂ v p φ κ GH + log p v (cid:0) κ (cid:0) ( G ′ ) − GG ′′ (cid:1) H + (cid:0) GG ′′ − ( G ′ ) (cid:1) H − GG ′ H ′ (cid:1) ∂ φ p v κ G H + (cid:0)(cid:0) κ H − (cid:1) G ′ − κ GHH ′ (cid:1) ∂ φ p z κ GH − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ φ p φ κ G H + H ′ ∂ z p φ κH ! + ∆ (3) φ p v + ∆ (2) φ p v + ∆ (1) φ p v , (C.6)and∆ z = (cid:18) − G ′ H ′ κ GH (cid:19) ∂ v p z − ∂ z p z κ + p z − G + H G ′′ G + H ( G ′ ) κG H + log p v (cid:0) − κ H ( H ′ ) G + (cid:0) G ′ H ′ − H ( H ′ ) G ′′ (cid:1) G + H ( H ′ ) ( G ′ ) (cid:1) κ G H ! + p v log p v G ′ (cid:0) − κ (cid:0) ( G ′ ) − GG ′′ (cid:1) H + (cid:0) ( G ′ ) − GG ′′ (cid:1) H + GG ′ H ′ (cid:1) κ G H + − κ (cid:0) G ′ ) − GG ′′ G ′ + G G (3) ( z ) (cid:1) H + (cid:0) ( G ′ ) − GG ′ G ′′ (cid:1) H + G ( G ′ ) H ′ κ G H ! + (cid:0) HH ′ ( G ′ ) + G (cid:0)(cid:0) κ H + 1 (cid:1) G ′ − HH ′ G ′′ (cid:1)(cid:1) ∂ φ p φ κ G H + ( GH ′ − HG ′ ) ∂ z p z κGH + (cid:0) GHG ′ H ′ ˆ κ − GH G ′′ ˆ κ + (cid:0) κ H − (cid:1) ( G ′ ) (cid:1) ∂ z p v κ G H − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ z ∂ φ p φ κ G H + log p v (cid:0) HH ′ ( G ′ ) + 3 G ( G ′ − HH ′ G ′′ ) (cid:1) ∂ φ p φ κ G H − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ φ p z κ G H + H ′ ∂ z p z ˆ κH + ∂ z ∂ φ p φ κG ! + ∆ (4) z p v + ∆ (3) z p v + ∆ (2) z p v + ∆ (1) z p v . (C.7)37he coefficients ∆ (1 , , φ and ∆ (1 , , , z have messy forms as∆ (3) φ = − log p v ∂ φ p v ˆ κG − H ′ ∂ z p v κG H ! p φ + ∂ φ p v ∂ φ p φ κG − ∂ φ p v κ G ! − ∂ φ p v ∂ v p v ˆ κG + log p v ∂ φ p v − ∂ φ p φ ˆ κG − ∂ φ p v κ G ! − H ′ ∂ φ p v ∂ z p v κ G H + ∂ φ p v ∂ v p v ˆ κG !! p φ + p z (cid:18) p φ (cid:18) H ′ ∂ φ p v ˆ κG H − p v H ′ ∂ φ p v ˆ κG H (cid:19) − log p v p φ H ′ ( ∂ φ p v ) κ G H (cid:19) + (cid:18) − p v ∂ φ p φ ( ∂ φ p v ) ˆ κ G − ∂ φ p φ ( ∂ φ p v ) κ G (cid:19) p φ , (C.8)∆ (2) φ = (cid:18) − log p v G ′ H ′ κG H − G ′ H ′ κG H (cid:19) p φ + (cid:16) G ′ H ′ ˆ κ H − G (cid:17) ∂ φ p v G − H ′ ∂ φ p z κG H + ∂ φ p v κ G + 3 H ′ ∂ z p φ κG H + H ′ ∂ z ∂ φ p v κ G H + 3 ∂ v ∂ φ p v κG + log p v − G ′ H ′ ∂ φ p v κ G H + H ′ ∂ φ p z ˆ κG H + 3 ∂ φ p φ κG + ∂ φ p v κ G + H ′ ∂ z ∂ φ p v κ G H − ∂ v ∂ φ p v κG !! p φ + κ + (cid:18) κH − κ (cid:19) log p v − H ˆ κ ! p z p φ + ∂ φ p φ ∂ φ p v κ G + ∂ φ p v ∂ φ p φ κ G + H ′ ∂ φ p φ ∂ z p v κ G H − H ′ ∂ φ p v ∂ z p φ κ G H + 3 ∂ φ p φ ∂ v p v κG + 5 ∂ φ p v ∂ v p φ κG + log p v ∂ φ p φ ) ˆ κG + 3 ∂ φ p v ∂ φ p φ κ G + H ′ ∂ z p v ∂ φ p φ κ G H H ′ ∂ φ p z κ G H + 3 ∂ φ p φ κ G ! − ∂ v p v ∂ φ p φ ˆ κG + ∂ φ p v + H ′ ∂ φ p v ∂ z p φ κ G H − ∂ φ p v ∂ v p φ ˆ κG !! p φ + 3 log p v ∂ φ p v ( ∂ φ p φ ) κ G + ∂ φ p v ( ∂ φ p φ ) κ G + p z log p v H ′ ∂ φ p v ∂ φ p φ κ G H + p φ − H ′ ∂ φ p φ κG H − ∂ z p v + H ′ ∂ v p v ˆ κH + log p v H ′ ∂ φ p φ ˆ κG H + H ′ ∂ φ p v κ G H + (cid:18) κ H − (cid:19) ∂ z p v − H ′ ∂ v p v κH !!! , (C.9)38 (1) φ = − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ φ p v ∂ φ p φ κ G H − ∂ φ p φ ∂ φ p φ κ G − H ′ ∂ z p φ ∂ φ p φ κ G H − ∂ v p φ ∂ φ p φ κG + log p v ∂ φ p φ − H ′ ∂ φ p z κ G H − ∂ φ p φ κ G ! − H ′ ∂ φ p φ ∂ z p φ κ G H + ∂ φ p φ ∂ v p φ ˆ κG ! + p z p φ (cid:0) κ H − (cid:1) G ′ κ GH + log p v (cid:0) − κ G ′ H − κ GH ′ H + 2 G ′ H + GH ′ (cid:1) κ GH ! + ∂ z p φ − H ′ ∂ v p φ ˆ κH + log p v − H ′ ∂ φ p φ κ G H + (cid:18) − κ H (cid:19) ∂ z p φ + H ′ ∂ v p φ κH !! + p φ (cid:16) G + G ′ H ′ ˆ κ H (cid:17) ∂ φ p φ G − ∂ φ p φ κ G + ∂ z p z − H ′ ∂ z ∂ φ p φ κ G H − H ′ ∂ v p z ˆ κH − ∂ v ∂ φ p φ κG + log p v G ′ H ′ ∂ φ p φ ˆ κ G H − H ′ ∂ φ p z κ G H − ∂ φ p φ κ G + (cid:18) − κ H (cid:19) ∂ z p z − H ′ ∂ z ∂ φ p φ κ G H + H ′ ∂ v p z κH + ∂ v ∂ φ p φ ˆ κG !! , (C.10)∆ (4) z = 3 ∂ φ p v ∂ z p v p φ κG + 3 ∂ φ p v ∂ z p v p φ κ G + p z (cid:18)(cid:18) p v ( ∂ φ p v ) ˆ κG − ∂ φ p v κG (cid:19) p φ + (cid:18) p v ( ∂ φ p v ) κ G − ( ∂ φ p v ) κ G (cid:19) p φ (cid:19) , (C.11)∆ (3) z = (cid:18) p v G ′ ∂ φ p v ˆ κG + G ′ ∂ φ p v κG − ∂ z ∂ φ p v κG (cid:19) p φ + G ′ ∂ φ p v κ G − ∂ z p φ ∂ φ p v κG + (cid:18) ∂ φ p z κG − ∂ z ∂ φ p v κ G (cid:19) ∂ φ p v − H ′ ∂ z p v κ G H + log p v (cid:18) G ′ ( ∂ φ p v ) κ G − ∂ φ p v ∂ φ p z ˆ κG (cid:19) + − ∂ φ p φ κG − ∂ φ p v κ G ! ∂ z p v + ∂ z p v ∂ v p v κG ! p φ + − p v ∂ φ p z ( ∂ φ p v ) κ G + ∂ φ p z ( ∂ φ p v ) κ G − ∂ z p φ ∂ φ p v κ G − ∂ φ p φ ∂ z p v ∂ φ p v ˆ κ G ! p φ + p z (cid:18) p φ (cid:18) H ′ ∂ φ p v κG H − p v H ′ ∂ φ p v ˆ κG H (cid:19) − log p v H ′ ( ∂ φ p v ) κ G H (cid:19) + p z − log p v ∂ φ p v ˆ κG + ∂ φ p v κG − H ′ ∂ z p v ˆ κG H ! p φ + ∂ φ p v ∂ φ p φ ˆ κG + ∂ φ p v κ G ! + H ′ ∂ φ p v ∂ z p v κ G H − ∂ φ p v ∂ v p v κG + log p v ∂ φ p v − ∂ φ p φ ˆ κG − ∂ φ p v κ G ! − H ′ ∂ φ p v ∂ z p v κ G H + ∂ φ p v ∂ v p v ˆ κG !! p φ − p v )( ∂ φ p v ) ∂ φ p φ κ G + ∂ φ p v ∂ φ p φ κ G ! , (C.12)39 (2) z = (cid:18) κ + (cid:18) κH − κ (cid:19) log p v − H ˆ κ (cid:19) p z + − H ′ ∂ φ p φ ˆ κG H − ∂ z p v + H ′ ∂ v p v ˆ κH + log p v H ′ ∂ φ p φ ˆ κG H + H ′ ∂ φ p v κ G H + (cid:18) κ H − (cid:19) ∂ z p v − H ′ ∂ v p v κH !! p z + G (cid:0) κ H − (cid:1) + HG ′ H ′ κG H − p v G ′ H ′ κG H ! p φ + 3 ∂ φ p v ∂ v p φ κG + (cid:0) − κ GH + 3 G ′ H ′ H + G (cid:1) ∂ φ p v κ G H − H ′ ∂ φ p z ˆ κG H − ∂ φ p φ κG + H ′ ∂ z p φ ˆ κG H + ∂ v ∂ φ p v ˆ κG + log p v − (cid:0) − κ GH + 6 G ′ H ′ H + G (cid:1) ∂ φ p v κ G H + 2 H ′ ∂ φ p z ˆ κG H + ∂ φ p φ ˆ κG + ∂ φ p v κ G + H ′ ∂ z ∂ φ p v κ G H − ∂ v ∂ φ p v κG !! p φ − ( ∂ φ p φ ) κG − ∂ φ p v ∂ φ p φ κ G − H ′ ∂ φ p v ∂ z p φ κ G H + ∂ φ p φ ∂ v p v ˆ κG + log p v ( ∂ φ p φ ) ˆ κG + 3 ∂ φ p v ∂ φ p φ κ G + H ′ ∂ z p v ∂ φ p φ κ G H − ∂ v p v ∂ φ p φ κG + ∂ φ p v H ′ ∂ φ p z ˆ κ G H + 3 ∂ φ p φ κ G ! + H ′ ∂ φ p v ∂ z p φ κ G H − ∂ φ p v ∂ v p φ κG !! p z + 3 log p v ∂ φ p v ∂ φ p φ ∂ φ p z κ G − ∂ φ p v ∂ φ p φ ∂ φ p z κ G + ( ∂ φ p φ ) ∂ z p v κ G + ∂ φ p v ∂ φ p φ ∂ z p φ κ G + p φ ∂ φ p φ κG + ∂ φ p v κ G ! ∂ z p φ + ∂ z p v ∂ φ p φ κ G + H ′ ∂ z p φ κ G H ! − H ′ ∂ φ p v ∂ z p z κ G H + ∂ φ p φ (cid:18) ∂ z ∂ φ p v κ G − ∂ φ p z ˆ κG (cid:19) + ∂ φ p v − G ′ ∂ φ p φ ˆ κ G − ∂ φ p z κ G + ∂ z ∂ φ p φ κ G ! + (cid:18) ∂ φ p z ˆ κG − ∂ z p φ κG (cid:19) ∂ v p v − ∂ z p v ∂ v p φ κG + 3 ∂ φ p v ∂ v p z κG + log p v ∂ φ p φ ∂ φ p z ˆ κG + 3 ∂ φ p v ∂ φ p z κ G + H ′ ∂ z p v ∂ φ p z κ G H − ∂ v p v ∂ φ p z κG + ∂ φ p v ∂ φ p z κ G − G ′ ∂ φ p φ ˆ κ G ! + H ′ ∂ φ p v ∂ z p z κ G H − ∂ φ p v ∂ v p z κG !! + p φ − G ′ ∂ φ p φ κG − G ′ ∂ φ p v κ G − ∂ φ p z κG + (cid:0) G (cid:0) H ′ + 6ˆ κ H − (cid:1) − κ H G ′ H ′ (cid:1) ∂ z p v κ G H + H ′ ∂ z p z κG H + 3 ∂ z ∂ φ p φ κG + H ′ ∂ z p v G + Hm (0 , , ( u, z, x )8ˆ κ G H − G ′ ∂ v p v κG + log p v − G ′ ∂ φ p φ ˆ κG − G ′ ∂ φ p v κ G + ∂ φ p z κG − G ′ H ′ ∂ z p v κ G H + G ′ ∂ v p v κG ! − ∂ v ∂ z p v κG ! , (C.13)40 (1) z = G (cid:0)(cid:0) κ H − (cid:1) G ′ + HH ′ G ′′ (cid:1) − H ( G ′ ) H ′ κ G H + log p v (cid:0) G (cid:0)(cid:0) κ H − (cid:1) G ′ + HH ′ G ′′ (cid:1) − H ( G ′ ) H ′ (cid:1) κ G H ! p φ + (cid:0) HH ′ ( G ′ ) + G (cid:0)(cid:0) − κ H (cid:1) G ′ − HH ′ G ′′ (cid:1)(cid:1) ∂ φ p v κ G H + (cid:16) G − G ′ H ′ ˆ κ H (cid:17) ∂ φ p z G + G ′ ∂ φ p φ ˆ κ G + (cid:0) − κ GH + 6 G ′ H ′ H + G (cid:1) ∂ z p φ κ G H − H ′ ∂ z p φ G + H∂ z ∂ φ p φ κ G H + G ′ ∂ v p φ ˆ κG − ∂ v ∂ φ p z ˆ κG + log p v (cid:0) − κ GH + 6 G ′ H ′ H + G (cid:1) ∂ φ p z κ G H + 3 G ′ ∂ φ p φ κ G − ∂ φ p z κ G + G ′ H ′ ∂ z p φ κ G H − H ′ ∂ z ∂ φ p z κ G H − G ′ ∂ v p φ ˆ κG + ∂ v ∂ φ p z κG ! + ∂ v ∂ z p φ κG ! p φ + G ′ ( ∂ φ p φ ) ˆ κ G − H ′ ( ∂ z p φ ) κ G H + p z (cid:0) − κ H (cid:1) G ′ G − (cid:0) κ H − (cid:1) log p v H ′ κ H + 2 H ′ H ! − (cid:0) GH ˆ κ + G ′ H ′ (cid:1) ∂ φ p v ∂ φ p z κ G H + ∂ φ p v κG − ∂ φ p φ κ G ! ∂ z p φ − ∂ φ p φ ∂ z ∂ φ p φ κ G + (cid:18) ∂ z p φ κG − ∂ φ p z ˆ κG (cid:19) ∂ v p φ − ∂ φ p φ ∂ v p z ˆ κG + log p v G ′ ( ∂ φ p φ ) κ G − ∂ φ p z ∂ φ p φ κ G − H ′ ∂ z p z ∂ φ p φ κ G H + ∂ v p z ∂ φ p φ κG − H ′ ( ∂ φ p z ) κ G H − ∂ φ p z ∂ φ p φ κ G − H ′ ∂ φ p z ∂ z p φ κ G H + ∂ φ p z ∂ v p φ κG ! + p z (cid:16) G − G ′ H ′ ˆ κ H (cid:17) ∂ φ p φ G + 2 ∂ z p z − H ′ ∂ v p z ˆ κH − ∂ v ∂ φ p φ ˆ κG + log p v (cid:0) − κ GH + 6 G ′ H ′ H + G (cid:1) ∂ φ p φ κ G H − H ′ ∂ φ p z κ G H − ∂ φ p φ κ G + (cid:18) − κ H (cid:19) ∂ z p z − H ′ ∂ z ∂ φ p φ κ G H + H ′ ∂ v p z ˆ κH + ∂ v ∂ φ p φ κG !! . (C.14) References [1] S. W. Hawking, “Black holes in general relativity,” Commun. Math. Phys. , 152(1972).[2] R. Emparan and H. S. Reall, “A Rotating black ring solution in five-dimensions,”Phys. Rev. Lett. , 101101 (2002) [hep-th/0110260].[3] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Rev. Rel. , 6 (2008) [arXiv:0801.3471 [hep-th]].[4] B. Kleihaus, J. Kunz and E. Radu, “Black rings in six dimensions,” Phys. Lett. B , 1073 (2013) [arXiv:1205.5437 [hep-th]].415] O. J. C. Dias, J. E. Santos and B. Way, “Rings, Ripples, and Rotation: ConnectingBlack Holes to Black Rings,” JHEP , 045 (2014) [arXiv:1402.6345 [hep-th]].[6] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodriguez, “The PhaseStructure of Higher-Dimensional Black Rings and Black Holes,” JHEP (2007)110 [arXiv:0708.2181 [hep-th]].[7] J. Armas and T. Harmark, “Black Holes and Biophysical (Mem)-branes,” Phys. Rev.D , no. 12, 124022 (2014) [arXiv:1402.6330 [hep-th]].[8] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “New Horizons forBlack Holes and Branes,” JHEP , 046 (2010) doi:10.1007/JHEP04(2010)046[arXiv:0912.2352 [hep-th]].[9] J. Armas and M. Blau, “New Geometries for Black Hole Horizons,” JHEP , 048(2015) doi:10.1007/JHEP07(2015)048 [arXiv:1504.01393 [hep-th]].[10] V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi and U. Miyamoto, “High and LowDimensions in The Black Hole Negative Mode,” Class. Quant. Grav. (2007) 5527[arXiv:0706.1555 [hep-th]].[11] R. Emparan, R. Suzuki and K. Tanabe, “The large D limit of General Relativity,”JHEP , 009 (2013) [arXiv:1302.6382 [hep-th]].[12] R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe and T. Tanaka, “Effective theoryof Black Holes in the 1/D expansion,” JHEP , 159 (2015) [arXiv:1504.06489[hep-th]].[13] R. Suzuki and K. Tanabe, “Stationary black holes: Large D analysis,”arXiv:1505.01282 [hep-th].[14] S. Bhattacharyya, A. De, S. Minwalla, R. Mohan and A. Saha, “A membraneparadigm at large D,” arXiv:1504.06613 [hep-th].[15] R. Emparan, R. Suzuki and K. Tanabe, “Evolution and End Point of the BlackString Instability: Large D Solution,” Phys. Rev. Lett. , no. 9, 091102 (2015)[arXiv:1506.06772 [hep-th]].[16] R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Phys. Rev.Lett. , 2837 (1993) [hep-th/9301052].[17] E. Sorkin, “A Critical dimension in the black string phase transition,” Phys. Rev.Lett. , 031601 (2004) [hep-th/0402216].[18] J. E. Santos and B. Way, “The Black Ring is Unstable,” Phys. Rev. Lett. , 221101(2015) [arXiv:1503.00721 [hep-th]]. 4219] R. Emparan, R. Suzuki and K. Tanabe, “Decoupling and non-decoupling dynamicsof large D black holes,” JHEP (2014) 113 [arXiv:1406.1258 [hep-th]].[20] R. Emparan and H. S. Reall, “Black Rings,” Class. Quant. Grav. , R169 (2006)[hep-th/0608012].[21] R. Emparan, R. Suzuki and K. Tanabe, “Instability of rotating black holes: large Danalysis,” JHEP (2014) 106 [arXiv:1402.6215 [hep-th]].[22] G. S. Hartnett and J. E. Santos, “Non-Axisymmetric Instability of Rotating BlackHoles in Higher Dimensions,” Phys. Rev. D , 041505 (2013)[23] M. Shibata and H. Yoshino, “Bar-mode instability of rapidly spinning black hole inhigher dimensions: Numerical simulation in general relativity,” Phys. Rev. D ,104035 (2010) [arXiv:1004.4970 [gr-qc]].[24] O. J. C. Dias, G. S. Hartnett and J. E. Santos, “Quasinormal modes of asymp-totically flat rotating black holes,” Class. Quant. Grav. , no. 24, 245011 (2014)[arXiv:1402.7047 [hep-th]].[25] H. Elvang, R. Emparan and A. Virmani, “Dynamics and stability of black rings,”JHEP , 074 (2006) [hep-th/0608076].[26] S. Hollands, A. Ishibashi and R. M. Wald, “A Higher dimensional stationary ro-tating black hole must be axisymmetric,” Commun. Math. Phys. , 699 (2007)[gr-qc/0605106].[27] P. Figueras, K. Murata and H. S. Reall, “Black hole instabilities and local Penroseinequalities,” Class. Quant. Grav. , 225030 (2011) [arXiv:1107.5785 [gr-qc]].[28] R. C. Myers and M. J. Perry, “Black Holes in Higher Dimensional Space-Times,”Annals Phys. , 304 (1986).[29] R. Suzuki and K. Tanabe, “Non-uniform black strings and the critical dimension inthe 1 /D expansion,” arXiv:1506.01890 [hep-th].[30] H. Kudoh, “Origin of black string instability,” Phys. Rev. D , 104034 (2006) [hep-th/0602001].[31] R. Emparan, R. Suzuki and K. Tanabe, “Quasinormal modes of (Anti-)de Sitter blackholes in the 1/D expansion,” JHEP1504