Instability and new phases of higher-dimensional rotating black holes
Oscar J.C. Dias, Pau Figueras, Ricardo Monteiro, Jorge E. Santos, Roberto Emparan
IInstability and new phases of higher-dimensional rotating black holes ´Oscar J. C. Dias a , Pau Figueras b , Ricardo Monteiro a , Jorge E. Santos a , Roberto Emparan c ∗ a DAMTP, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road,Cambridge CB3 0WA, UK b Centre for Particle Theory & Department of Mathematical Sciences,Science Laboratories, South Road,Durham DH1 3LE, UK c Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),Passeig Llu´ıs Companys 23,E-08010 Barcelona, Spain, andDepartament de F´ısica Fonamental,Universitat de Barcelona, Marti i Franqu`es 1,E-08028 Barcelona, Spain
It has been conjectured that higher-dimensional rotating black holes become unstable at a suffi-ciently large value of the rotation, and that new black holes with pinched horizons appear at thethreshold of the instability. We search numerically, and find, the stationary axisymmetric pertur-bations of Myers-Perry black holes with a single spin that mark the onset of the instability andthe appearance of the new black hole phases. We also find new ultraspinning Gregory-Laflammeinstabilities of rotating black strings and branes.
Black holes are the most basic and fascinating objectsin General Relativity and the study of their propertiesis essential for a better understanding of the dynamicsof spacetime at its most extreme. In higher-dimensionalspacetimes a vast landscape of novel black holes has be-gun to be uncovered [1]. Its layout — i.e., the connec-tions between different classes of black holes in the spaceof solutions — hinges crucially on the analysis of theirclassical stability: most novel black hole phases are con-jectured to branch-off at the threshold of an instabilityof a known phase. Showing how this happens is an out-standing open problem that we address in this paper.The best known class of higher-dimensional blackholes, discovered by Myers and Perry (MP) in [2], ap-pear in many respects as natural generalizations of theKerr solution. In particular, their horizon is topologi-cally spherical. However, the actual shape of the horizoncan differ markedly from the four-dimensional one, whichis always approximately round with a radius parametri-cally ∼ GM . This is not so in d ≥
6. Considering forsimplicity the case where only one spin J is turned on (ofthe (cid:4) d − (cid:5) independent angular momenta available), it ispossible to have black holes with arbitrarily large J for agiven mass M . The horizon of these ultraspinning blackholes spreads along the rotation plane out to a radius a ∼ J/M much larger than the thickness transverse tothis plane, r + ∼ ( GM /J ) / ( d − . This fact was pickedout in [3] as an indication of an instability and a con-nection to novel black hole phases. In more detail, in thelimit a → ∞ with r + fixed, the geometry of the black holein the region close to the rotation axis approaches thatof a black membrane. Black branes are known to exhibit classical instabilities [4], at whose threshold a new branchof black branes with inhomogeneous horizons appears [5].Ref. [3] conjectured that this same phenomenon shouldbe present for MP black holes at finite but sufficientlylarge rotation: they should become unstable beyond acritical value of a/r + , and the marginally stable solu-tion should admit a stationary, axisymmetric perturba-tion signalling a new branch of black holes pinched alongthe rotation axis. Simple estimates suggested that in fact( a/r + ) crit should not be much larger than one. As a/r + increases, the MP solutions should admit a sequence ofstationary perturbations, with pinches at finite latitude,giving rise to an infinite sequence of branches of ‘pinchedblack holes’ (see fig. 1). Ref. [6] argued that this struc-ture is indeed required in order to establish connectionsbetween MP black holes and the black ring and black Sat-urn solutions more recently discovered. Our main resultis a numerical analysis that proves correct the conjectureillustrated in fig. 1.The solution for a MP black hole rotating in a singleplane in d dimensions is [2] ds = − dt + r d − m r d − Σ (cid:0) dt + a sin θ dφ (cid:1) + Σ (cid:18) dr ∆ + dθ (cid:19) +( r + a ) sin θ dφ + r cos θ d Ω d − , (1)Σ = r + a cos θ , ∆ = r + a − r d − m r d − . (2)The parameters here are the mass-radius r m and therotation-radius a , r d − m = 16 πGM ( d − d − , a = d − JM . (3) a r X i v : . [ h e p - t h ] O c t ( M f i x e d ) S J
A B C FIG. 1: Diagram of entropy vs. spin, at fixed mass, for MPblack holes in d ≥ A ). As the spingrows new of branches of black holes with further axisymmet-ric pinches ( B, C, . . . ) appear. We determine the points wherethe new branches appear, but it is not yet known in which di-rections they run. We also indicate that at the inflection point(0), where ∂ S/∂J = 0, there is a stationary perturbationthat should not correspond to an instability nor a new branchbut rather to a zero-mode that moves the solution along thecurve of MP black holes. The event horizon lies at the largest real root r = r + of∆.The linearized perturbation theory of the Kerr blackhole ( d = 4) was disentangled in [7] using the Newman-Penrose formalism. Attempts to extend this formalism todecouple a master equation for the gravitational pertur-bations of (1) in d ≥ h ab around the background (1). Choosing traceless-transverse (TT) gauge, h aa = 0 and ∇ a h ab = 0, theequations to solve are( (cid:52) L h ) ab = −∇ c ∇ c h ab − R c da b h cd = 0 , (4)where (cid:52) L is the Lichnerowicz operator in the TT gauge.Actually, we solve the more general eigenvalue problem( (cid:52) L h ) ab = − k h ab , (5)which is known to appear in two contexts: eqs. (5) de-termine the stationary perturbations of a black string in d + 1 dimensions (obtained by adding a flat direction z to(1)) with a profile e ikz h ab . Thus such modes with k > k Sch (cid:54) = 0)to finite rotation, with k growing as the rotation increasestowards the Kerr bound [10].Our reason to consider (5) instead of trying to solvedirectly for k = 0 is that there exist powerful numericalmethods for eigenvalue problems that give the eigenval-ues k together with the eigenvectors, i.e., the metric per-turbations. If the ultraspinning instability is present forMP black holes in d ≥
6, then, in addition to the analogueof the branch studied in [10], a new branch of negativemodes extending to k = 0 must appear. The eigenvalue k = 0 corresponds to a (perturbative) stationary solutionwith a slightly deformed horizon. In fact, as explainedabove, we expect an infinite sequence of such branchesthat reach k = 0 at increasing values of the rotation. Thesolutions for k > SO ( d − × SO (2)rotational symmetries of the MP solution and dependonly on the radial and polar coordinates, r and θ [3].Thus we take the ansatz ds = − e ν ( dt − ω dφ ) + e ν dφ + e η sin θ dθ + e γ ( dr − χ sin θdθ ) + e d Ω d − . (6)We decompose a given quantity Q = { ν , ν , η, γ, ω, χ } as Q = Q + δQ . The unperturbed contribution Q ( r, θ )describes (1). The perturbations δQ ( r, θ ) are determinedsolving the eigenvalue problem (5) subject to appropriateboundary conditions. After imposing TT gauge, eq. (5)reduces to four coupled PDEs for δη , δγ , δχ and δ Φ (theTT conditions then give δν , δν and δω ). The bound-ary conditions are that the perturbations are regular andfinite at the horizon, r = r + , at infinity, r = ∞ , and atthe poles θ = 0 , π/
2. In addition, we impose δχ ( r + ) = 0.It is important to ensure that the eigenmodes we findare not pure gauge, h ab = ∇ ( a ξ b ) . We can prove thatin the TT gauge, pure gauge perturbations within ouransatz necessarily diverge at either the horizon or infin-ity. Thus, with our boundary conditions, the eigenmodeswe obtain are never pure gauge.We use a numerical approach successfully applied tothe identification of the negative mode of Kerr and Kerr-AdS black holes [10]. It employs a Chebyshev spectralnumerical method (see [10] for further details). We havecarried out the calculations for d = 7 , ,
9. The cases d =5 (where the heuristics of [3] do not allow to predict anyinstability) and d = 6 present more difficult numerics.These, as well as a more detailed presentation of ournumerical approach, will be discussed elsewhere.The results for d = 7 are displayed in fig. 2, the othertwo cases being qualitatively very similar. We plot thenegative eigenvalue − k as a function of the rotation pa- (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
100 a (cid:144) r m (cid:45) k r m2 FIG. 2: Negative eigenvalues for the MP black hole in d = 7. rameter a . We normalize k and a relative to the mass-radius r m , which is equivalent to plotting their values forfixed mass (or mass per unit length, in the black stringinterpretation). As described above, the leftmost curve,which does not reach k = 0, is the higher-dimensionalcounterpart of the Kerr negative mode, and the eigen-values k are the wavenumbers of the Gregory-Laflammethreshold modes at rotation a . At larger rotation we findnew branches of negative modes that intersect k = 0 atfinite a/r m . We label these successive branches with aninteger (cid:96) = 1 , , , . . . , and refer to them as ‘harmonics’.The values of a/r m at which the stationary perturbationsappear are listed in table I. d ( a/r m ) | (cid:96) =1 ( a/r m ) | (cid:96) =2 ( a/r m ) | (cid:96) =3 .
075 1 .
714 2 . .
061 1 .
770 2 . .
051 1 .
792 2 . a/r m for the first three har-monics of stationary perturbation modes ( k = 0). The esti-mated numerical error is ± × − in d = 7 and ± × − in d = 8 , It is important to note that the k = 0 eigenmode ofthe harmonic (cid:96) = 1 does not correspond to a new sta-tionary solution. Instead it is a zero-mode that takes thesolution to a nearby one along the family of MP blackholes. The existence and location of this zero-mode is aconsequence of the fact that if the Hessian of the Gibbspotential G ( T, Ω i ) = M − T S − (cid:80) i J i Ω i , calculated alonga family of solutions, has a zero eigenvalue for some par-ticular solution, then there is a zero-mode perturbation ofthe gravitational (Euclidean) action I = G/T that takesthat solution to an infinitesimally nearby one along thatfamily. That is, perturbing the solution with that zero-mode does not correspond to branching-off into a newfamily of solutions.One can easily check that the determinant of the Hes-sian of the Gibbs potential is proportional to the deter-minant of the Hessian of the entropy with respect to only the angular momenta, i.e., to the determinant of H ij = (cid:18) ∂ S∂J i ∂J j (cid:19) M . (7)Therefore, for solutions with a single spin, there mustappear a stationary perturbation, in principle not asso-ciated to an instability of the black hole, at the inflectionpoint of the curve S ( J ) at fixed M (point 0 in fig. 1).For the MP solutions this happens at (cid:18) ar m (cid:19) d − = d − d − (cid:18) d − d − (cid:19) d − . (8)The values of ( a/r m ) mem for d = 7 , , a/r m ) for (cid:96) = 1 (first column in table I) up to the thirddecimal place. This is a very good check of the accuracyof our numerical methods.The k = 0 eigenmodes of the higher harmonics, (cid:96) ≥ (cid:96) = 2 signals the onset of the instability conjec-tured in [3]. The k = 0 eigenmodes for higher harmonicsconfirm the appearance of the sequence of new black holephases as the rotation grows.To visualize the effect on the horizon of the perturba-tions that give new solutions, and provide further con-firmation of our interpretation, we draw an embeddingdiagram of the unperturbed MP horizon and compare itwith the deformations induced by the ultraspinning har-monics (cid:96) ≥
2. This is best done using the embeddingproposed in [12], which has the advantage of allowing toembed the horizon along the entire range 0 ≤ θ ≤ π/ (cid:96) = 2 , , (cid:96) = 2 modes create a pinchcentered on the rotation axis θ = 0; (cid:96) = 3 modes havea pinch centered at finite latitude θ ; (cid:96) = 4 modes pinchthe horizon twice: around the rotation axis and at finitelatitude. These are the kind of deformations depictedin fig. 1. To better identify the number of times thatthe perturbed horizon crosses the unperturbed solution,in these figures we also plot the logarithmic differencebetween the two embeddings.Ref. [3] gave several arguments to the effect that criti-cal values a/r m close to 1 were to be expected. In partic-ular, it was pointed out that the change in the behavior ofthe black hole from ‘Kerr-like’ to ‘black membrane-like’could be pinpointed to the value of the spin where the (cid:61) r m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
14 X
Log (cid:200) Z (cid:45) Z (cid:123) (cid:61) (cid:200) a (cid:61) r m FIG. 3: Embedding diagram at ( a/r m ) crit of the d = 7 blackhole horizon, unperturbed (solid), and with the first unstableharmonic perturbation ( (cid:96) = 2, k = 0) (dashed). The em-bedding Cartesian coordinates Z and X lie resp. along therotation axis θ = 0 and the rotation plane θ = π/
2. We alsoshow the logarithmic difference between the embeddings ofthe perturbed ( Z (cid:96) =2 ) and unperturbed ( Z ) horizons. Thespikes represent the points where the two embeddings inter-sect. The perturbation has two nodes, so the horizon squeezesaround the rotation axis, then bulges out, and squeezes againat the equator, as in the conjectured shape A in fig. 1. (cid:61) r m (cid:45) (cid:45) (cid:45)
10 X
Log (cid:200) Z (cid:45) Z (cid:123) (cid:61) (cid:200) a (cid:61) r m FIG. 4: Like fig. 3, for (cid:96) = 3: between the first two nodes ofthe perturbation the horizon has a pinch (shape B in fig. 1). (cid:61) r m (cid:45) (cid:45) (cid:45) (cid:45)
12 X
Log (cid:200) Z (cid:45) Z (cid:123) (cid:61) (cid:200) a (cid:61) r m FIG. 5: Like fig. 3, for (cid:96) = 4: the four nodes deform thehorizon into shape C of fig. 1. temperature ( i.e., surface gravity) reaches a minimumfor fixed mass, which is the same, for solutions with asingle spin, as the inflection point of S ( J ). As we haveargued, the zero-mode at this solution should not signalan instability. The (cid:96) = 2 mode at the threshold of theactual instability instead appears at larger rotation, wellwithin the regime of membrane-like behavior as conjec-tured in [6]. We expect this to be true in general: theultraspinning instability of MP black holes should appearfor angular momenta strictly beyond the (codimension 1)locus in the space of angular momenta where the Hessian H ij has a zero eigenvalue.In particular, in d = 5 this criterion does not allowany ultraspinning instability for any J , J , and in d ≥ N = (cid:4) d − (cid:5) angular momenta J i equal itpredicts that the instability should appear at a/r m > − N/ ( d − . However we cannot predict the precise valuesof the rotation where the instability appears.We have identified the points in the phase diagramwhere the new branches must appear, but we cannot de-termine in which direction these run. This requires calcu-lating the area, mass and spin of the perturbed solutions.However, for any k (cid:54) = 0 — and numerically we can neverobtain an exact zero — the linear perturbations decay ex-ponentially in the radial direction, and so the mass andspin, measured at asymptotic infinity, are not corrected.It seems that in order to obtain the directions of the newbranches one has to go beyond our level of approximationor adopt a different approach.The new (cid:96) ≥ k . These imply a new ultraspinning Gregory-Laflammeinstability for black strings, in which the horizon is de-formed not only along the direction of the string, butalso along the polar direction of the transverse sphere.Observe that, even if the (cid:96) = 1, k = 0 mode is notan instability of the MP black hole, the modes (cid:96) = 1, k > (cid:96) have longer wavelength k − and sothe branch (cid:96) = 1 is expected to dominate the instability.The growth of k with a can be understood heuristically,since as a grows the horizon becomes thinner in direc-tions transverse to the rotation plane and hence it can fitinto a shorter compact circle.To finish, we mention that pinched phases of rotat-ing plasma balls, dual to pinched black holes in Scherk-Schwarz compactifications of AdS, have been found [13],as well as new kinds of deformations of rotating plasmatubes [14] and rotating plasma ball instabilities [15]. Therelation of our results to these and other phenomena ofrotating fluids will be discussed elsewhere. Acknowledgments.
We thank Troels Harmark, KeijuMurata, Malcolm Perry and especially Harvey Reall fordiscussions. We were supported by: Marie Curie contractPIEF-GA-2008-220197, and by PTDC/FIS/64175/2006,CERN/FP/83508/2008 (OJCD); STFC Rollinggrant (PF); Funda¸c˜ao para a Ciˆencia e Tecnolo-gia (Portugal) grants SFRH/BD/22211/2005 (RM),SFRH/BD/22058/2005 (JES); and by MEC FPA 2007-66665-C02 and CPAN CSD2007-00042 Consolider-Ingenio 2010 (RE). This is preprint DCPT-09/47. ∗ Electronic address: [email protected],pau.fi[email protected],[email protected],[email protected], [email protected][1] R. Emparan and H. S. Reall, Living Rev. Rel. (2008)6.[2] R. C. Myers and M. J. Perry, Annals Phys. , 304(1986).[3] R. Emparan and R. C. Myers, JHEP (2003) 025.[4] R. Gregory and R. Laflamme, Phys. Rev. Lett. (1993)2837.[5] S. S. Gubser, Class. Quant. Grav. , 4825 (2002).T. Wiseman, Class. Quant. Grav. , 1137 (2003).[6] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodriguez, JHEP (2007) 110.[7] S. A. Teukolsky, Astrophys. J. , 635 (1973).[8] See, for instance: A. Ishibashi and H. Kodama, Prog.Theor. Phys. (2003) 901. H. K. Kunduri, J. Luciettiand H. S. Reall, Phys. Rev. D (2006) 084021. K. Mu-rata and J. Soda, Prog. Theor. Phys. (2008) 561.T. Oota and Y. Yasui, arXiv:0812.1623 [hep-th]. H. Ko-dama, R. A. Konoplya and A. Zhidenko, arXiv:0904.2154[gr-qc].[9] D. J. Gross, M. J. Perry and L. G. Yaffe, Phys. Rev. D (1982) 330.[10] R. Monteiro, M. J. Perry and J. E. Santos,arXiv:0905.2334 [gr-qc]; ibidem, arXiv:0903.3256 [gr-qc].[11] B. Kleihaus, J. Kunz and E. Radu, JHEP (2007)058.[12] V. P. Frolov, Phys. Rev. D (2006) 064021.[13] S. Lahiri and S. Minwalla, JHEP (2008) 001.S. Bhardwaj and J. Bhattacharya, JHEP (2009)101.[14] M. M. Caldarelli, O. J. C. Dias, R. Emparan andD. Klemm, JHEP (2009) 024.[15] V. Cardoso and O. J. C. Dias, JHEP0904