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The scalar perturbation of the higher-dimensional rotating black holes. 3 Daisuke Ida, Yuki Uchida 0 Department of Physics, Tokyo Institute of Technology, 0 2 Tokyo 152-8550, Japan n and a Yoshiyuki Morisawa J Department of Physics, Osaka City University, 3 1 Osaka 558-8585, Japan 2 February 4, 2008 v 5 3 0 Abstract 2 The massless scalar field in the higher-dimensional Kerr black hole 1 (Myers-Perry solution with a single rotation axis) has been investigated. 2 0 Ithasbeenshownthatthefieldequationisseparableinarbitrarydimen- / sions. The quasi-normal modes of the scalar field have been searched in c five dimensions using the continued fraction method. The numerical re- q sult shows theevidence for the stability of the scalar perturbation of the - r five-dimensional Kerr black holes. The time scale of the resonant oscilla- g tion in the rapidly rotating black hole, in which case the horizon radius v: becomessmall,ischaracterizedby(blackholemass)1/2(Planckmass)−3/2 i rather than the light-crossing time of thehorizon. X r a 1 Introduction The higher dimensional space-times have been more seriously considered in recent studies as possible grounds for the unified theory of elementary particles. Though there are no experimental evidence that we live in more than four- dimensional space-time, it is suggested that the existence of the extra dimen- sionsmightbeprovedbythesignatureofblackholeproductioninplannedhigh energy experiments. This argument follows from the brane-world scenario [1]. In such a model, theStandardModelparticlesareconfinedtothethree-brane,whichmeansthat the Kaluza-Kleinmodes ofthe SMparticles arenotexcited, andonly gravitons canpropagateinthe bulk,hencetherearerathermildconstraintsonthe sizeof theextradimensions,sothatthefundamental(higher-dimensional)Planckscale can reduce to TeV scale. This is a possible solution to the hierarchy problem. 1 Another type of the brane-world models has been proposed by Randall and Sundrum, which utilize the warped metric of the AdS space. IfthefundamentalscaleisgivenbyTeV,theblackholeswillbeproducedat the future colliders such as the CERN Large Hadron Collider (LHC), of which centerofmassenergyis14TeV[2,3]. Thoughtheclassicalapproximationmight be marginal for hypothetical black holes to be produced at LHC, we shall here discuss the higher-dimensionalclassicalblack holes,which is relevantfor future colliders beyond the LHC. Recent studies show that the higher-dimensional black holes might be qualitatively rather different from usual four-dimensional ones. Significantly,the stationaryblackholesolutionsareshowntobe nonunique, whiletheuniquenessofstaticblackholehasbeenestablished[5]. Therearefive- dimensionalKerrsolution,whileEmparanandReall[6]haverecentlyfoundthe rotating black ring solution with S1 S2 horizon, and there exists a parameter × rangeswhere Kerrblackhole andblackringhavethe commonmassM andthe angular momentum J. The dimensionless Kerr parameter a := a/r , where ∗ H a=3J/2M is the Kerrparameterand r the horizonradius,are characteristic H quantitydeterminingthepropertyofthefive-dimensionalKerrblackhole,which are equivalent to the another dimensionless quantity J2/GM3 via J2 32 a2 = ∗ . (1) GM3 27πa2+1 ∗ The five-dimensional Kerr black hole can have arbitrary a > 0 for fixed total ∗ mass. Consider a sequence of the solution with the mass being fixed. Starting with a = 0 (Schwarzschild black hole), spin up the black hole. At a = 2.3, ∗ ∗ there emerge two branches of black ring solutions with same J, both have less horizon area than the Kerr black hole. However, for a > 2.8, the horizon ∗ area of the larger black ring exceeds that of the Kerr black hole with same J. Since the horizon area is just the black hole entropy, it means that the five-dimensional Kerr black hole are entropically unfavourable in microcanoni- cal ensemble. Though this is a purely thermodynamical argument, one might wonder the stability of the rapidly rotating black holes, and this is the subject of the present paper. As a simplest case, we here investigate the massless scalar field in the five- dimensional Kerr background. We formulate in arbitrary dimensions for later convenience, and then concentrate on the five-dimensional case in actual computation. The separability in five-dimensional case has been shown in Ref [8]. 2 Kerr metric in higher dimensions Themetricofthe(4+n)-dimensionalKerrmetricwithonlyonenon-zeroangular momentum is given in Boyer-Lindquist-type coordinates by [4] ∆ a2sin2ϑ 2a(r2+a2 ∆) g = − dt2 − dtdϕ − Σ − Σ 2 (r2+a2)2 ∆a2sin2ϑ + − sin2ϑdϕ2 Σ Σ + dr2+Σdϑ2+r2cos2ϑdΩ2, (2) ∆ n where Σ = r2+a2cos2ϑ, (3) ∆ = r2+a2 µr1−n, (4) − anddΩ2 denotesthestandardmetricoftheunitn-sphere. Thismetricdescribes n a rotating black hole in asymptotically flat, empty space-time with mass and angular momentum proportional to µ and µa, respectively. Hereafter, µ,a >0 are assumed. The event horizon is located at r = r , such that ∆ = 0, which is H |r=rH homeomorphictoS2+n. Whenn=1,aneventhorizonexistsonlywhena<√µ, and the event horizon shrinks to zero-area in the extreme limit a √µ. On → the other hand, when n 2, ∆ = 0 has exactly one positive root for arbitrary ≥ a>0. Thus, there are no extreme Kerr black holes in higher dimensions. 3 Separation of variables of massless scalar field equations Here we consider the scalar perturbation on the background metric (2). For- tunately, the massless scalar equation turns out to be separable. We shall just commentthattheHamilton-Jacobiequationforatestparticleisalsoseparable, so that we have so-called the Carter constant in arbitrary dimensions. We put φ = eiωt−imϕR(r)S(ϑ)Y(Ω) into the massless scalar field equation 2φ = 0, where Y(Ω) is the hyperspherical harmonics on n-sphere, of which ∇ eigenvaluesgivenby j(j+n 1)(j =0,1,2, ). Thenweobtaintheseparated − − ··· equations 1 d dS sinϑcosnϑ + ω2a2cos2ϑ sinϑcosnϑ dϑ dϑ (cid:18) (cid:19) m2csc2ϑ j(j+n 1)sec2ϑ+A(cid:2) S =0, (5) − − − (cid:3) and d dR ω(r2+a2) ma 2 r−n rn∆ + − dr dr ∆ (cid:18) (cid:19) ((cid:2) (cid:3) j(j+n 1)a2 − λ R=0, (6) − r2 − (cid:27) where λ:=A 2mω+ω2a2. − 3 We considertheeigenvalueproblemofEqs.(5)and(6)subjecttothequasi- normal boundary condition. This boundary condition is given by R (r r )iσ, (r r ) H H ∼ − → R r−(n+2)/2e−iωr, (r + ) (7) ∼ → ∞ where (r2 +a2)ω ma σ := H − (8) (n 1)(r2 +a2)+2r2 − H H hasbeendeterminedbytheasymptoticbehavioroftheEq.(6). Inotherwords, the waves are purely ingoing at the horizonand purely outgoing at the infinity. 3.1 Possible Unstable Modes HereweshowalongthelineofDetweilerandIpser[12]thatthepossibleunstable mode is restricted to some region of the complex ω-plane. We assume that Eqs. (5) and (6) have an unstable eigenmode with frequency ω: ω =ω +iω , ω <0, (9) R I I whereω andω aretherealandimaginarypartofω,respectively. Multiplying R I scalar field equation 2φ = 0 by rnsinϑcosnϑΣφ∗, and integrating over the ∇ region outside the event horizon, we obtain Σ(r2+a2)+µa2r1−nsin2ϑ 2aµr1−n 2ω +m φ2rnsinϑcosnϑdrdϑ =(100), R − ∆ ∆ | | Z Z (cid:26) (cid:20) (cid:21) (cid:27) and a2 Σ(r2+a2)+µa2r1−nsin2ϑ j(j+n 1) sec2ϑ+ +(ω2 ω2) − r2 I − R ∆ Z Z (cid:26)(cid:20) (cid:18) (cid:19) (cid:20) (cid:21) 2aµr1−n a2 ∂φ 2 ∂φ 2 +mω +m2 csc2ϑ φ2+∆ + rnsinϑcosnϑdrdϑ=(110). R ∆ (cid:18) − ∆(cid:19)(cid:21)| | (cid:12)∂r(cid:12) (cid:12)∂ϑ(cid:12) ) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) The Equation (10) implies that mωR >0 holds when(cid:12) m(cid:12)=0,(cid:12)and(cid:12)ωR =0 when 6 m=0. Since ∆ 0 is monotonically increasing function for r >r , we have H ≥ rn−1 Σ(r2+a2)+µa2r1−nsin2ϑ =r(cid:2)n−1 (r2+a2)2 a2∆sin2ϑ(cid:3) rn−1 (r2+a2)2 a2∆ − ≥ − (cid:2) (cid:3) = rn+3(cid:2)+rn+1a2+µa2 (cid:3) > rn+3+rn+1a2+µa2 (12) H H Then, from Eq. (10), the real part of ω turns out to be bounded: maµ ma ω < | | = | | := mω . (13) | R| rn+3+a2rn+1+µa2 µr1−n | | H H H H 4 On the other hand, using a similar inequality rn−1[(r2+a2)2 a2∆sin2ϑ] rn+3+rn+1a2+µa2 − ≥ rn+3+rn+1a2+µa2, (14) ≥ H H Eq. (11) leads to the inequality m2a ω2 ω2 + ω , (n 1), (15) I ≤ R r2 H ≥ H ω2 (ω mω )2, (n=1). (16) I ≤ | R|−| | H In particular, it turns out that there no unstable mode in the axisymmetric (m = 0) and the Schwarzschild (a = 0) cases. In the non-axisymmetric case, when unstable modes are not ruled out, we rely on the numerical calculation. 4 Numerical Computation Here we employ the continued fraction method [9, 10, 14] for this problem, which can determine the resonant frequency ω and the separation constant A with very high accuracy. We assume the following series expansion for S ∞ S = (sinϑ)|m|(cosϑ)j a (cos2ϑ)k, (17) k k=0 X which automatically satisfies the regular boundary conditions at ϑ = 0,π/2 whenever converges. Substituting this into Eq. (5), we obtain the three-term recurrence relations α a +β a =0, (18) 0 1 0 0 α a +β a +γ a =0, (k =1,2, ) (19) k k+1 k k k k−1 ··· where α = 8(k+1)(2j+n+2k+1), (20) k − β = 2[(j+ m +2k)(j+n+ m +2k+1) A], (21) k | | | | − γ = ω2a2. (22) k − ∗ ∗ Here we have defined the dimensionless quantity ω := ωr and a := a/r , ∗ H ∗ h since the behavior of the system depends only on a . When a = 0, the eigen- ∗ ∗ value A is explicitly determinedfrom the requirementthatthe series expansion ends within finite terms, since otherwise divergent. Thus we have A = (2ℓ+j+ m)(2ℓ+j+ m +n+1)+O(ω a ), ∗ ∗ | | | | (ℓ=0,1,2, ) (23) ··· 5 and the 0th-order eigenfunctions are given in terms of the Jacobi polynomials: P = (sinϑ)|m|(cosϑ)j ℓjm n+1 n+1 F ℓ,ℓ+j+ m + ,j+ ;cos2ϑ . × − | | 2 2 (cid:18) (cid:19) (24) In a similar way, we expand the radial function R into the form iσ −(n+2)/2−iσ r r r+r R = e−iω∗r − H H r r (cid:18) H (cid:19) (cid:18) H (cid:19) ∞ r r k H b − , (25) k × r+r k (cid:18) H(cid:19) X which automatically satisfies the quasi-normal boundary condition, whenever convergent. When n = 1, substituting Eq. (25) into Eq. (6), we obtain the five-term recurrence relations α˜ b +β˜ b +γ˜ b +δ˜ b +ǫ˜ b =0, (26) k k+1 k k k k−1 k k−2 k k−3 (k =0,1, ) ··· where b =b = =0 and coefficients are given by −1 −2 ··· α˜ = 4(k+1)(k+1+2iσ), (27) k β˜ = 6 4k 4(k+1)2a2 4λ+32ω σ k − − − ∗− ∗ +i 2ma (16k+10+2a2)ω , (28) ∗− ∗ ∗ γ˜k = −8k(cid:2)2+4k−11+8j2a2∗−8λ+6(cid:3)4ω∗σ 4i(4k 1)(2ω +σ), (29) ∗ − − δ˜ = 8+4k 4j2a2 4λ+32ω σ k − − ∗− ∗ 4i[4(k 1)ω σ], (30) ∗ − − − ǫ˜ = +(2k 3)(2k 3+4iσ). (31) k − − Forn 2,therecurrencerelationshavemoreterms. Inwhatfollows,werestrict ≥ ourselveson the n=1 caseas a simple case. The five-termrecurrencerelations reduce to the four-term ones α˜′b +β˜′b +γ˜′b +δ˜′b =0, (k =0,1, ) (32) k k+1 k k k k−1 k k−2 ··· via the Gaussian elimination: α˜′ = α˜ , β˜′ =β˜ , γ˜′ =γ˜ , (33) 1 1 1 1 1 1 α˜′ = α˜ , β˜′ =β˜ , γ˜′ =γ˜ , δ˜′ =δ˜ , (34) 2 2 2 2 2 2 2 2 α˜′ = α˜ , β˜′ =β˜ α˜′ ǫ˜ /δ˜′ , k k k k− k−1 k k−1 γ˜′ = γ˜ β˜′ ˜ǫ /δ˜′ , δ˜′ =γ˜ β˜′ ǫ˜ /δ˜′ , k k− k−1 k k−1 k k− k−1 k k−1 (k =3,4, ) (35) ··· 6 The similar procedure α˜′′ = α˜′, β˜′′ =β˜′, γ˜′′ =γ˜′, (36) 1 1 1 1 1 1 α˜′′ = α˜′, β˜′′ =β˜′ α˜′′ δ˜′/γ˜′′ , k k k k− k−1 k k−1 γ˜′′ = γ˜′ β˜′′ δ˜′/γ˜′′ , (k =2,3, ) (37) k k− k−1 k k−1 ··· leads to the three-term relations α˜′′b +β˜′′b +γ˜′′b =0. (k =0,1, ) (38) k k+1 k k k k−1 ··· Given the three-term recurrence relations the formal expressions of the ratio of the successive coefficients are obtained in two ways: a β 1 γ α γ α γ α k k−1 k−1 k−2 2 1 1 0 = − − − a −α − α β + ··· β + β k−1 k−1 k−1 k−2 1 0 γ α γ α γ k k k+1 k+1 k+2 = − − , (39) −β + β + β + ··· k k−1 k−2 where (1/x+)(y/z) abbreviates 1/[x+(y/z)]. In a similar manner, we obtain bk′ = β˜k′′′−1 1 −γ˜k′′′−1α˜′k′′−2 −γ˜2′′α˜′1′−γ˜1′′α˜′0′ bk′−1 −α˜′k′′−1 − α˜′k′′−1 β˜k′′′−2+ ··· β˜1′′+ β˜0′′ = γ˜k′′′ −α˜′k′′γ˜k′′′+1−α˜′k′′+1γ˜k′′′+2 . (40) −β˜′′+ β˜′′ + β˜′′ + ··· k′ k′−1 k′−2 For fixed k and k′, we have simultaneous equations (39) and (40) for two un- known complex numbers ω and A. We perform the numerical evaluation of ∗ the solutions using a standard nonlinear root searchalgorithm provided by the MINPACK subroutine package. The equations for different sets of (k,k′) are used as a numerical check. Inthecaseofnon-rotatingblackhole,thefirsteightquasi-normalmodeswith several values of angular momentum obtained in our computation are plotted in Fig. 1. Typical examples of quasi-normal modes of the five-dimensional Kerr black holes are plotted in Figs 2–7, where the Kerr parameter a is continuously ∗ varied. Here the resonant frequencies are normalized by √µ rather than rH, i.e., we work on 1+a2ω -plane. Though the Kerr parameter a is taken ∗ ∗ ∗ to be sufficiently large values, each mode does not seem to pass through the p real axis of the frequency plane in any cases. The m > 0 branches seem to approach to some points on the real axis, which is a similar behaviour to the four-dimensional case [9, 11, 13, 16]. Though we have searched for unstable modes up to m,j 5, they are not found. | | ≤ 5 Summary Wehaveinvestigatedthemasslessscalarfieldequationinthehigher-dimensional Kerrblackholebackground. Thefieldequationisseparableinarbitrarydimen- 7 sions. Under the quasi-normal boundary condition, we have searched for the resonantfrequencies inthe complex ω -plane in the case ofthe five dimensions. ∗ In the Schwarzschild case (a = 0), the resonant frequencies are character- ∗ ized by ℓ and the excitation number N. For nonzero Kerr parameter, these resonant frequencies split for different sets of m and j. Continuously varying the Kerr parameter a , we keep track of resonant modes, of which trajectories ∗ arecharacterizedbythesetofwavenumbers: (ℓ,m,j,N). Eachtrajectorydoes not pass through the real axis of the complex frequency plane. For m > 0, however, these trajectories seem to approach to points on the real axis of the frequencyplane. Thecorrespondingtimescaleoftheresonantoscillationinthe limit a 0, in which case the horizon radius shrinks to zero, is characterized ∗ → by the mass scale µ1/2 rather than the light-crossing time r . H ∼ ∼ These resultsshowthe evidence ofthe stability ofthe scalarperturbationof the five-dimensional Kerr black holes. Acknowledgments TheauthorswouldliketothankR.Emparan,H.Ishihara,H.Kodama,T.Mishima, K.Nakamura,K. Nakao,K.Oda, H. Onozawa,M. SasakiandT. Shiromizu for useful discussions and comments. D.I. was supported by JSPS Research, and this researchwas supported in part by the Grant-in-Aid for Scientific Research Fund (No. 6499). References [1] N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, Phys. Lett. B429, 263 (1998); I. Antoniadis et al., Phys. Lett. B436, 257 (1998); L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). [2] P.C. Argyres, S. Dimopoulos and J. March-Russell, Phys. Lett. B441, 96 (1998). [3] S.B. Giddings and S. Thomas, hep-ph/0106219. [4] R.C. Myers and M.J. Perry,Ann. Phys. 172, 304 (1986). [5] G.W.Gibbons,D.IdaandT.Shiromizu,gr-qc/0203004;G.W.Gibbons,D. Ida and T.Shiromizu, Phys.Rev. Lett. 89, 041101(2002);G.W. Gibbons, D. Ida and T. Shiromizu, Phys. 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D 41, 374 (1990) 9 ω * 8 2l+j+|m|=0 2l+j+|m|=1 7 2l+j+|m|=2 2l+j+|m|=3 6 2l+j+|m|=4 2l+j+|m|=5 2l+j+|m|=6 5 2l+j+|m|=7 * 2l+j+|m|=8 ω m 4 2l+j+|m|=9 I 2l+j+|m|=10 3 2 1 0 0 1 2 3 4 5 6 7 8 Re ω * Figure 1: For non-rotating black hole, first eight quasi-normal frequencies for 2ℓ+j+ m =0, ,10 are plotted in the complex ω plane. ∗ | | ··· 10