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Physics Letters B 621 (2005) 219–223

www.elsevier.com/locate/physlet

New instability for rotating black branes and strings

Vitor Cardosoa,b, José P.S. Lemosc

a McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, MO 63130, USAb Centro de Física Computacional, Universidade de Coimbra, P-3004-516 Coimbra, Portugal

c Centro Multidisciplinar de Astrofísica (CENTRA), Departamento de Física, Instituto Superior Técnico, Universidade Técnica de Lisboa,Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Received 20 May 2005; accepted 9 June 2005

Available online 20 June 2005

Editor: M. Cvetic

Abstract

The evolution of small perturbations around rotating black branes and strings, which are low energy solutions otheory, are investigated. For simplicity, we concentrate on the Kerr solution times transverse flat extra dimensions,compactified, but one can also treat other branes composed of any rotating black hole and extra transverse dimensioas analogue black hole models and rotating bodies in fluid mechanics systems. It is shown that such a rotating blis unstable against any massless (scalar, vectorial, tensorial or other) field perturbation for a wide range of wavelefrequencies in the transverse dimensions. Since it holds for any massless field it can be considered, in this sense,instability than the one studied by Gregory and Laflamme. Accordingly, it has also a totally different physical origiperturbations can be stabilized if the extra dimensions are compactified to a length smaller than the minimum wavelwhich the instability settles in, resembling in this connection the Gregory–Laflamme case. Likewise, this instability wno effect for astrophysical black holes. However, in the large extra dimensions scenario, where TeV scale black holeproduced, this instability should be important. It seems plausible that the endpoint of this instability is a static, or veryrotating, black brane and some outgoing radiation at infinity. 2005 Elsevier B.V. All rights reserved.

PACS: 04.50.+h; 04.70.Bw; 11.25.-w; 11.27.+d

asher

va-les.ex-

ingriosse-toess

1. Introduction

A consistent theory of quantum gravity, suchstring theory, seems to require the existence of hig

E-mail addresses: vcardoso@wugrav.wustl.edu(V. Cardoso),lemos@fisica.ist.utl.pt(J.P.S. Lemos).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.06.025

dimensions, which in order not to contradict obsertional evidence must be compactified on small scaString theory has made some important progress inplaining the entropy of certain black holes by countmicroscopic degrees of freedom, and thus scenawith extra, compactified dimensions must be takenriously. In turn, some of the most interesting objectsbe studied within string theory are those that poss

.

220 V. Cardoso, J.P.S. Lemos / Physics Letters B 621 (2005) 219–223

xtra

canoiesal

ified

esesti-ba-lyiser.t is

rlinins,r-

andd of

et-

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-

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, itill

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event horizons, possibly extended in the compact edimensions.

In d spacetime dimensions, an event horizonbe topologically a sphereSd−2, but when extended tp extra dimensions, it could naturally have topologeitherSd−2+p in which case it is a higher-dimensionblack hole, orSd−2 × Rp being then a blackp-braneor a black string in thep = 1 case[1]. TheRp topol-ogy of the transverse dimensions can be compactgiving flat toroidal topological spacesT p.

One of the first steps towards understanding thextended higher-dimensional solutions is to invegate their classical stability against small perturtions. If a solution is unstable, then it most certainwill not be found in nature (unless the instabilitysecular) and the solution looses most of its powOf course the next question one must ask is whathe final stable result of such instability. Now, indspacetime dimensions, the Schwarzschild–Tanghegeometry is stable against all kinds of perturbatiomassive or massless[2]. On the other hand, quite suprisingly, Gregory and Laflamme[3] showed that thisis not the case for higher-dimensional black branesblack strings. These objects are unstable. The kinblackp-branes originally studied in[3] were solutionsof ten-dimensional low energy string theory with mric of the form

(1)ds2 = ds2Schw+ dxi dxi,

whereds2Schw stands for thed-dimensional Schwarz

schild–Tangherlini line element, thexi are the coor-dinates of the compact dimensions, andi runs from1 to p. The total dimension of the spacetimeDobeys D = d + p. In string or supergravity theories one takesD = 10 or D = 11, but for gener-icity one can leave it as a free natural parameIn [3] it was shown that even though scalar and vtor perturbations of the black brane stay boundedtime, the tensorial sector of gravitational perturbtions displays an instability, with the perturbatiogrowing exponentially with time, possibly breakinthe extended black brane into several smaller blholes. The only way around this instability iscompactify the transverse directionsxi on a scalesmaller than the black hole radius[3] (for a fulldiscussion of this instability and its developmesee[4]).

Here we shall show that a related class of metrfor example, having the form

(2)ds2 = ds2Kerr + dxi dxi,

where the line elementds2Kerr stands for thed-dimen-

sional Kerr–Myers–Perry line element, is unstablescalar perturbations, as well as to vectorial, gravtional and other type of perturbations. In this sensis a stronger instability than the one studied in[3], be-cause it holds for any massless field. Accordinglyhas also a totally different physical origin, as we wsee.

2. Formulation of the problem and basicequations

2.1. The background metric

In four dimensions, there is only one possible rotion axis for a cylindrically symmetric spacetime, athere is therefore only one angular momentum pameter. In higher dimensions there are several chofor rotation axes and there is a multitude of angumomentum parameters, each referring to a particrotation axis. Here we shall concentrate on the splest case, for which there is only one angular mmentum parameter, which we shall denote bya. Thespecific metric we shall be interested in is givenBoyer–Lindquist-type coordinates by

ds2 = −∆ − a2 sin2 θ

Σdt2

− 2a(r2 + a2 − ∆)sin2 θ

Σdt dϕ

+ (r2 + a2)2 − ∆a2 sin2 θ

Σsin2 θ dϕ2

(3)

+ Σ

∆dr2 + Σ dθ2 + r2 cos2 θ dΩ2

n + dxi dxi,

where

(4)Σ = r2 + a2 cos2 θ,

(5)∆ = r2 + a2 − mr1−n,

and dΩ2n denotes the standard metric of the unitn-

sphere (n = d − 4), thexi are the coordinates of thcompact dimensions, andi runs from 1 top. This

V. Cardoso, J.P.S. Lemos / Physics Letters B 621 (2005) 219–223 221

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metric describes a rotating black brane in an asytotically flat, vacuum space–time with mass and anlar momentum proportional toµ andµa, respectively.Hereafter,µ,a > 0 are assumed.

The event horizon, homeomorphic toS2+n, is lo-cated atr = r+, such that∆|r=r+ = 0. Forn = 0, anevent horizon exists only fora < µ/2. Whenn = 1,an event horizon exists only whena <

õ, and the

event horizon shrinks to zero-area in the extreme lia → √

µ. On the other hand, whenn 2, ∆ = 0has exactly one positive root for arbitrarya > 0. Thismeans there is no bound ona, and thus there are nextreme Kerr black branes in higher dimensions.

2.2. Separation of variables and boundary conditions

Consider now the evolution of a massless scfield Ψ in the background described by(3). The evo-lution is governed by the curved space Klein–Gordequation

(6)∂

∂xµ

(√−g gµν ∂

∂xνΨ

)= 0,

whereg is the determinant of the metric. The metappearing in(6) should describe the geometry refering to both the black brane and the scalar field, buwe consider that the amplitude ofΨ is so small that itscontribution to the energy content can be neglecthen the metric(3) should be a good approximatioto gµν in (6). We shall thus work in this perturbativapproach. It turns out that it is possible to simplconsiderably equation(6) if we separate the angulavariables from the radial and time variables, as is din four dimensions[5]. For higher dimensions we folow [6]. In this connection see also[7] for a general(4 + n)-dimensional Kerr hole with several spin prameters. In the end our results agree with the resin [7], if we consider only one angular momentum prameter in their equations, and no extra compactidimensions.

We consider the ansatz

φ = eiωt−imϕ+ikixi

R(r)S(θ)Y (Ω),

and substitute this form in(6), whereY(Ω) are hyper-spherical harmonics on then-sphere, with eigenvaluegiven by−j (j +n−1) (j = 0,1,2, . . .). Then we ob-

tain the separated equations

1

sinθ cosn θ

(d

dθsinθ cosn θ

dS

dθ

)+ [

a2(ω2 − k2)cos2 θ − m2 csc2 θ

(7)− j (j + n − 1)sec2 θ + A]S = 0,

and

r−n d

dr

(rn∆

dR

dr

)

+ [ω(r2 + a2) − ma]2

∆

(8)− j (j + n − 1)a2

r2− λ − k2r2

R = 0,

whereA is a constant of separation,λ := A−2mωa +ω2a2, andk2 = ∑

k2i . Interestingly, note the importan

point that Eqs.(7) and (8)are just those that describthe evolution of a massive scalar field, with massk, inad-dimensional Kerr geometry.

Eqs.(7) and (8)must be supplemented by appropate boundary conditions, which are given by

(9)R ∼

(r − rH )iσ asr → rH ,

r−(n+2)/2e−i√

ω2−k2r asr → ∞,

where

(10)σ := [(r2H + a2)ω − ma]rH

(n − 1)(r2H + a2) + 2r2

H

has been determined by the asymptotic behavioEq. (8). In other words, the waves must be pureingoing at the horizon and purely outgoing at thefinity. For assigned values of the rotational paramea and of the angular indicesl, j,m there is a discrete(and infinite) set of frequencies called quasinormfrequencies, QN frequencies orωQN, satisfying thewave equation(8) with the boundary conditions jusspecified by Eq.(9).

3. The instability timescale

Now, whend = 4, analysis of the perturbationsa massive scalar field on the Kerr geometry has bdone[8]. Indeed, Eqs.(7) and (8)have been analytically solved by Zouros and Eardley in the limitlargek2 and by Detweiler in the limit of smallk2 [8].

222 V. Cardoso, J.P.S. Lemos / Physics Letters B 621 (2005) 219–223

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Detweiler’s results have recently been confirmedmerically by Furuhashi and Nambu[8]. Detweiler[8]shows that for smallk2, in particular,kM 1, thegeometry is unstable. In fact, he shows that the cacteristic frequenciesω satisfying the boundary conditions(9) are given by

(11)ω = σ + iγ,

with

(12)σ 2 = k2[1−

(kM

l + 1+ n

)2], n = 0,1,2, . . . ,

and

(13)γ = C

(k − am

2Mr+

),

with C = C(l,m,a,M,n) a positive constant, ashown in Detweiler[8]. So, form positive and

(14)k <am

2Mr+which is just a superradiance condition, the modethen unstable. In particular, the most unstable modthe l = m = 1 mode with ane-folding timeτ given byof

(15)τ

M= 24

1

ka

1

(kM)8.

Now, we know their results can be translated immdiately into our black brane geometry. Thus rotatblack branes of the form(3) with d = 4 andp extradimensions are unstable, with ane-folding time givenby (15). The results ford > 4 are not available in thliterature, but the instability should be present in thcases as well.

The only way around this instability is to make tsuperradiant factork − am

2Mr+ positive, which is thesame as havingkM > am

2r+ . Thus, for sufficiently smala/r+, this is still inside the range of validity of our approximations,kM 1. To escape this instability onmust compactify the transverse dimensions on a sscaleL, i.e., L

2πM<

2r+a

. Writing outr+ explicitly thisgives

(16)L

2πM< 2α,

whereα = M/a +√M2/a2 − 1∼ 2M/a, for smalla.

From (15) one obtains that the timescale for t

instability, in the approximations used isτ/M >

24(1/kM)9. SincekM 1 this is not a specially efficient instability, and it does not even appear to ovcome the Gregory–Laflamme instability timescaHowever, the only results available so far for tGregory–Laflamme instability refer to non-rotatinobjects. It is possible that as one increases the rotato large values the instability studied here becommore effective (as is apparent from Eq.(15)) and thatthe Gregory–Laflamme instability gets less effectiv

4. Conclusions

What is the physical interpretation of this instabity for rotating black branes? It is known that the Kegeometry displays superradiance[9]. This means thain a scattering experiment of a wave with frequenω < mΩ the scattered wave will have a larger amptude than the incident wave, the excess energy bwithdrawn from the object’s rotational energy. HeΩ is the horizon’s angular velocity (related toa, M

and r+ throughΩ = a/(2Mr+)) andm is again theazimuthal wave quantum number. Now supposeone encloses the rotating black hole inside a sphemirror. Any initial perturbation will get successiveamplified near the black hole event horizon andflected back at the mirror, thus creating an instabilThis is the black hole bomb, as devised in[10] andrecently improved in[11]. This instability is causedby the mirror, which is an artificial wall, but one cadevise natural mirrors if one considers massive fieIn this case, the mass of the field acts effectively amirror, and thus Kerr black holes are unstable agamassive field perturbations[8]. With this in mind, weexpect that black strings and branes of the form(2)will be unstable, because the compactified transvdirections work as an effective mass for the graviand for the scalar field. In fact, this simple reasonimplies that any rotating black brane (more genethan the one described by(2)) will be unstable. Thiswould be the case for the rotating black hole in strtheory found by Sen or other rotating black holes[12].Moreover, following Zel’dovich[9], it is known thatnot only the Kerr geometry, but any rotating absoing body for that matter displays superradiance. Ththis instability should appear in analogue black hmodels[13], and in rotating bodies in fluid mechanisystems.

V. Cardoso, J.P.S. Lemos / Physics Letters B 621 (2005) 219–223 223

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Here, we have derived the instability timescafor a scalar field, and not for geometry (metric) pturbations, since Teukolsky’s formalism for highedimensional rotating objects is not available. Still, targument presented above makes it clear that thestability should be present for metric perturbatioas well. We also expect that the instability will bstronger for metric modes, because of the folloing simple reasoning. Superradiance is the mecnism responsible for this instability, and thus the larthe superradiant effects, the stronger the instabiNow, we know that in the Kerr geometry ind = 4scalar fields have a maximum superradiant amplifition factor of about 2%, whereas gravitational modhave maximum superradiant amplification factorabout 138%[14]. Then it is expected the instabiity timescale to be almost two orders of magnitusmaller, and correspondingly, the instability shogrow much stronger for gravitational modes. For geral black branes, the Gregory–Laflamme instabiseems to be stronger than the one displayed hbut it is known that certain extremal solutions shonot exhibit the Gregory–Laflamme instability[15],whereas the instability dealt with here should gothe way to extremality. So, eventually it takes ovthe Gregory–Laflamme instability. Moreover, recestudies[16] seem to indicate that black strings inRandall–Sundrum inspired 2-brane model do nothibit the Gregory–Laflamme instability.

The endpoint of this rotating instability is noknown, and it can never be predicted with certaiby a linear analysis. However, it seems plausible tosume that the instability will keep growing until thenergy and angular momentum content of the fieldproaches that of the black brane, when back-reaceffects become important. The rotating brane will thbegin to spin down, and gravitational and scalar raation goes off to infinity carrying energy and angumomentum. The system will probably asymptotea static, or very slowly rotating, final state consistiof a non-rotating blackp-brane and some outgoinradiation at infinity. But we cannot discard the othpossibility that the horizon fragments.

Acknowledgements

We gratefully acknowledge stimulating correspodence on this problem with Donald Marolf. V.C. a

,

knowledges financial support from Fundaçao parCiência e Tecnologia (FCT), Portugal, through grSFRH/BPD/2003. This work was partially fundedFCT through project POCTI/FNU/44648/2002.

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See also: E. Berti, K.D. Kokkotas, E. Papantonopoulos, PRev. D 68 (2003) 064020.