aa r X i v : . [ h e p - t h ] N ov HRI/ST/1010PUPT-2354
October 2010
New Near Horizon Limit in Kerr/CFT
Yoshinori Matsuo a † and Tatsuma Nishioka b ‡ a Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India b Department of Physics, Princeton University, Princeton, NJ 08544, USA
Abstract
The extremal Kerr black hole with the angular momentum J is conjectured tobe dual to CFT with central charges c L = c R = 12 J . However, the central chargein the right sector remains to be explicitly derived so far. In order to investigatethis issue, we introduce new near horizon limits of (near) extremal Kerr and five-dimensional Myers-Perry black holes. We obtain Virasoro algebras as asymptoticsymmetries and calculate the central charges associated with them. One of themis equivalent to that of the previous studies, and the other is non-zero, but still theorder of near extremal parameter. Redefining the algebras to take the standardform, we obtain a finite value as expected by the Kerr/CFT correspondence. † [email protected] ‡ [email protected] Introduction and overview
The origin of the black hole entropy has been of great interest in the past few decadesand remains to be fully understood. In string theory, some of black holes and branescan be described in terms of the microscopic degrees of freedom, and the entropy isreproduced by counting their microstates. In the case of anti-de Sitter (AdS) spacetime,these microstates are described by the conformal field theory (CFT). In this case, a lot ofinformation can be obtained by using symmetries. In the case of AdS , the Bekenstein-Hawking entropy of the BTZ black hole was accounted without specifying details of CFT.The asymptotic symmetry of the geometry is identified with the conformal symmetryof the dual field theory and the entropy is calculated by using the Cardy formula [1, 2].Recently, it was conjectured that the extremal Kerr black hole in four dimensionscorresponds to a two-dimensional CFT [3]. They investigated its near horizon geometrywhich has the SL (2 , R ) × U (1) isometry [4]. The U (1) symmetry is enhanced to anasymptotic symmetry and identified with the Virasoro algebra for the chiral half ofCFT. The Bekenstein-Hawking entropy is reproduced by using the Cardy formula inthe extremal case. This is called as the Kerr/CFT correspondence and this Virasoroalgebra is regarded as that of left mover. This analysis is generalized to many cases [5] and it is known that another Virasoro algebra can be obtained from extending the SL (2 , R ) part of the isometry [8, 9]. This symmetry is considered as that of rightmover which describes the non-extremal excitation from the extremality. Therefore, theextremal Kerr black hole with an angular momentum J is expected to be dual to anon-chiral CFT with c L = c R = 12 J . The recent discussion about the hidden conformalsymmetry of the non-extremal Kerr black hole also provides another evidence for thiscorrespondence [10].However, the asymptotic Virasoro algebras can be found only in the near horizonlimit of (near) extremal black holes. Moreover, the central charge for right mover is zerowhen it is evaluated by the conventional method. Therefore, a cut-off was introducedinto the spacetime to take the near-extremal correction into account, and it turned outto be not finite but infinitesimally small [8]. Although an indirect argument is presentedin [9], it is fair to say that the central charge c R = 12 J has not been explicitly derivedso far.In this paper, we investigate this issue in more detail. First, we study the five-dimensional extremal Myers-Perry black hole whose near horizon geometry includes See also [6, 7] for related works. if it has only a single spin [4] . In this case, we can find dual two-dimensionalCFT with central charges c L = c R similar to the one given by Brown and Henneaux [1].The CFT lives on light-cone coordinates which consist of the time and angular directions.On the other hand, we know the Kerr/CFT description where dual CFT is defined on thetime and angular directions with central charges c L = c R . The coordinates in Kerr/CFTdescription are almost equivalent to the light-cone coordinates near the horizon up to ascaling factor. This scaling factor makes central charges different from each other.To resolve this discrepancy, we reconsider the definitions of the algebras, centralcharges and temperatures. Usually, the Frolov-Thorne temperature is changed underthe rescaling of the coordinates, while the central charges are fixed because it representsa number of degrees of freedom. If we use these definitions of the central chargesand temperatures, the Cardy formula gives scale dependent quantity. However, theentropy should be independent of the scale since it counts a number of states. Thereforewe need to introduce “covariant” definition of the central charge to make the Cardyformula invariant under the scaling. The covariant central charge c ( cov ) is related to thescale-invariant central charge c as c ( cov ) = βc , where β is the period of the coordinate.Then, the covariant central charge depends on the scaling factor via the period β . Inthe Kerr/CFT correspondence, the Virasoro algebra for right mover has been studied byusing the quasi-local charges [13, 14]. In these studies generators are defined in the scalecovariant form, and the central charge depends on the definition of the time coordinatesin the near horizon geometry. In the near horizon limit, these covariant central chargefor left and right movers are not generally equal to each other, c ( cov ) L = c ( cov ) R . In thispaper, we show that the central charges satisfy the relation c L = c R even in Kerr/CFT,if we use the scale-invariant definition of c .We also consider general five-dimensional Myers-Perry black hole with two rotations,where the near horizon geometry is AdS , not AdS . Thus there is no natural light-cone coordinates, but we introduce new coordinates which agree with the light-conecoordinates in the single rotation limit. The resulting near horizon geometry is thesame as the usual one once replacing the time and angular directions with the light-cone directions. We evaluate the central charge for our new near horizon geometry on thetimeslice with respect to the original time. By using this coordinates, we can evaluatethe central charge associated with the Virasoro algebra for the right mover using theconventional covariant phase space method [15,16] and obtain the scale-invariant centralcharges satisfying c L = c R . We apply this analysis to the (four-dimensional) Kerr black The structure of AdS sometimes appears in certain limits of extremal rotating black holes. It hasbeen recently studied to investigate the Kerr/CFT correspondence in [11, 12]. c L = c R = 12 J .It is worth noting that the combinations of coordinates in our new near horizonlimits are the same as those in the analyses of the hidden conformal symmetry [10].In these studies our covariant central charges are the equivalent to the invariant one.Then, the Cardy formula can be used with the Frolov-Thorne temperature and thesecentral charges to calculate the entropy. Therefore the central charges c L = c R = 12 J is naturally obtained in the original coordinates before taking the near horizon limit.This paper is organized as follows. In Section 2, we introduce the five-dimensionalMyers-Perry black hole and its near horizon limit. Then, we review on the Kerr/CFTcorrespondence for left mover. In Section 3, we show the calculation of the central chargefor right mover following the previous study [8]. We introduce a cut-off to evaluate thecentral charge. In Section 4, we consider the special case in which the near horizongeometry has the structure of AdS . Then, we discuss the relation between this specialcase and the general case described in Section 2 and 3. In Section 5, we introducea new definition of the near horizon limit. By using this definition, we calculate thecentral charge for right mover without introducing the cut-off into the spacetime. InSection 6, we apply the new definition of the near horizon limit to the Kerr black hole,and calculate the central charge. Section 7 is devoted to the conclusion. In this section, we describe the five-dimensional Myers-Perry black hole and brieflyreview on the Kerr/CFT correspondence for it. The metric is expressed by using theBoyer-Lindquist coordinates as ds = − ∆ r ρ (cid:0) dt − a sin θ dφ − b cos θ dψ (cid:1) + sin θρ (cid:2) ( r + a ) dφ − a dt (cid:3) + cos θρ (cid:2) ( r + b ) dψ − b dt (cid:3) + 1 r ρ (cid:2) b ( r + a ) sin θ dφ + a ( r + b ) cos θ dψ − ab dt (cid:3) + r ρ ∆ dr + ρ dθ , (2.1)where ∆ and ρ are given by∆ = ( r + a )( r + b ) − µr , ρ = r + a cos θ + b sin θ . (2.2)In five-dimensional spacetime, we have two independent angular momenta in φ - and ψ -directions, respectively. Hence, this geometry has three parameters µ , a , b , which are3elated to the ADM mass M ADM and two angular momenta J φ and J ψ : M ADM = 3 πµ G N , J φ = πµa G N , J ψ = πµb G N , (2.3)where G N is the Newton constant. The outer and inner horizons are located at the radii r + and r − , which are expressed as r ± = 12 (cid:16) µ − a − b ± p ( µ − ( a + b ) ) ( µ − ( a − b ) ) (cid:17) , (2.4)and the angular velocities on the (outer) horizon areΩ φ = ar + a , Ω ψ = br + b . (2.5)The Hawking temperature T H and the Bekenstein-Hawking entropy S are given by T H = r − r − πµr + , S = π G N µr + , (2.6)respectively.Now, we consider the near horizon limit of this geometry [4]. We focus on thenear-extremal case in which the non-extremality is infinitesimally small. We define thisnon-extremality parameter by µ = µ (1 + ǫ ˆ µ ) , µ = ( a + b ) , (2.7)where the extremal condition is given by µ = µ and ˆ µ parametrizes the non-extremality.We introduce the near horizon coordinates as t = ǫ − √ µ t , r = r + ǫ √ µ r , (2.8a) φ = ˆ φ + ar + a t , ψ = ˆ ψ + br + b t , (2.8b)where r is the horizon radius in the extremal case and given by r = ab . (2.9)Then, the near horizon limit is obtained as the ǫ → ds = − ρ ˆ∆4 d ˆ t + ρ d ˆ r + ρ dθ + a µ sin θρ (cid:16) d ˆ φ + k ˆ φ ˆ rd ˆ t (cid:17) + b µ cos θρ (cid:16) d ˆ ψ + k ˆ ψ ˆ rd ˆ t (cid:17) + µ r ρ h sin θ (cid:16) d ˆ φ + k ˆ φ ˆ rd ˆ t (cid:17) + cos θ (cid:16) d ˆ ψ + k ˆ ψ ˆ rd ˆ t (cid:17)i , (2.10)4here ˆ∆ = (ˆ r − ˆ µ ) , ρ = r + a cos θ + b sin θ , (2.11) k ˆ φ = 12 r ba , k ˆ ψ = 12 r ab . (2.12)A correspondence between this near horizon geometry and its dual CFT is studiedin [18]. It was shown that the asymptotic symmetry gives the Virasoro algebra forleft mover if we take an appropriate boundary condition. The asymptotic symmetry isdefined as a symmetry which preserves a boundary condition: £ ξ ( g µν + O ( χ µν )) = O ( χ µν ) , (2.13)where χ µν defines the boundary condition. If the geometry takes the following form: ds = f ( θ ) (cid:18) − ˆ r d ˆ t + d ˆ r ˆ r (cid:19) + γ ij ( θ ) (cid:0) dx i + k i rdt (cid:1) (cid:0) dx j + k j rdt (cid:1) + f θ ( θ ) dθ , (2.14)where x = ˆ φ and x = ˆ ψ , we can obtain the following asymptotic symmetry groups ξ ( ˆ φ ) = ǫ ˆ φ ( ˆ φ ) ∂ ˆ φ − ˆ rǫ ′ ˆ φ ( ˆ φ ) ∂ ˆ r , (2.15a) ξ ( ˆ ψ ) = ǫ ˆ ψ ( ˆ ψ ) ∂ ˆ ψ − ˆ rǫ ′ ˆ ψ ( ˆ ψ ) ∂ ˆ r , (2.15b)by taking appropriate boundary conditions, respectively. These vectors form the Vira-soro algebra as we will see below. We define ξ ( ˆ φ ) n and ξ ( ˆ ψ ) n by ξ ( ˆ φ ) and ξ ( ˆ ψ ) with ǫ ˆ φ ( ˆ φ ) = e in ˆ φ , ǫ ˆ ψ ( ˆ ψ ) = e in ˆ ψ , (2.16)respectively. Then these vectors obey[ ξ ( ˆ φ ) n , ξ ( ˆ φ ) m ] = − i ( n − m ) ξ ( ˆ φ ) n + m , [ ξ ( ˆ ψ ) n , ξ ( ˆ ψ ) m ] = − i ( n − m ) ξ ( ˆ ψ ) n + m , (2.17)respectively.In order to calculate the central charge, we consider the conserved charge associatedwith the Virasoro algebras. A definition of the conserved charge is given by [15, 16].The conserved charge is expressed in terms of the background metric ¯ g µν and its smallperturbation h µν , and given by Q ξ [ h ] = 18 πG N Z ∂ Σ k ξ [ h, ¯ g ] , (2.18)5here Σ is a timeslice and the integration is taken over its boundary ∂ Σ. The three-form k ξ is defined by k ξ [ h, ¯ g ] = ˜ k µνξ [ h, ¯ g ] (cid:0) d x (cid:1) µν , (2.19)where d x is the Hodge dual of the two-form d x µ ∧ d x ν , and the two-form ˜ k ξ [ h, ¯ g ] isgiven by ˜ k µνξ [ h, ¯ g ] = 12 h ξ µ D ν h − ξ µ D λ h λν + (cid:0) D µ h νλ (cid:1) ξ λ + 12 hD µ ξ ν − h µλ D λ ξ ν + 12 h µλ ( D ν ξ λ + D λ ξ ν ) − ( µ ↔ ν ) i . (2.20)The central charge c can be read off from the anomalous transformation of the charge:18 πG N Z ∂ Σ k ξ n [ £ ξ m ¯ g, ¯ g ] = δ n + m, n c . (2.21)For the metric (2.14), and asymptotic symmetry groups (2.15), we obtain c ˆ φ = 6 πk ˆ φ G N Z dθ p γ ( θ ) f θ ( θ ) , c ˆ ψ = 6 πk ˆ ψ G N Z dθ p γ ( θ ) f θ ( θ ) . (2.22)By using the explicit form of the metric (2.10), we obtain c ˆ φ = 3 πbµ G N , c ˆ ψ = 3 πaµ G N . (2.23)Since the Frolov-Thorne temperatures are given by T ˆ φ = ( a + b )( r + + r − )2 π ( r + b ) → r πb , (2.24) T ˆ ψ = ( a + b )( r + + r − )2 π ( r + a ) → r πa , (2.25)the Cardy formula reproduces the Bekenstein-Hawking entropy at the extremality: S = π c ˆ φ T ˆ φ = π c ˆ ψ T ˆ ψ = π µ r . (2.26)It should be noted that both CFTs corresponding to each asymptotic symmetry canreproduce the entropy. In this section, we consider the Kerr/CFT correspondence for the right mover inthe five-dimensional Myers-Perry black hole. For the Kerr black hole, the asymptotic6ymmetry for the right mover is obtained by introducing a different boundary conditionto that for the left mover [8]. For the five-dimensional Myers-Perry black hole, we canobtain a similar asymptotic symmetry group with an analogous boundary condition.For metrics which has the form of (2.14), we impose the following boundary condition: O ( χ µν ) = ˆ t ˆ r ˆ φ ˆ ψ θ ˆ t O ( r ) O ( r − ) O ( r − ) O ( r − ) O ( r − )ˆ r O ( r − ) O ( r − ) O ( r − ) O ( r − )ˆ φ O ( r − ) O ( r − ) O ( r − )ˆ ψ O ( r − ) O ( r − ) θ O ( r − ) . (3.1)Then, the asymptotic symmetry group ξ = (cid:16) ǫ ξ (ˆ t ) + ǫ ′′ ξ (ˆ t )2ˆ r (cid:17) ∂ ˆ t + (cid:16) − ˆ rǫ ′ ξ (ˆ t ) + ǫ ′′′ ξ (ˆ t )2ˆ r (cid:17) ∂ ˆ r + (cid:16) C ˆ φ − k ˆ φ ǫ ′′ ξ (ˆ t )ˆ r (cid:17) ∂ ˆ φ + (cid:16) C ˆ ψ − k ˆ ψ ǫ ′′ ξ (ˆ t )ˆ r (cid:17) ∂ ˆ ψ + O (ˆ r − ) , (3.2)preserves the boundary condition for the metric g µν + O ( χ µν ) → g µν + O ( χ µν ) . (3.3)For the asymptotic symmetry group (3.2) the central charge vanishes, and hence, wehave to introduce a cut-off. Here, we calculate the central charge by using the quasi-localcharge [13,14]. This charge is defined as an integration of the surface energy-momentumtensor on the boundary. The surface energy-momentum tensor is given by the conjugatemomentum of the induced metric on the surface, and can be expressed in terms of theextrinsic curvature as T µν = 2 √− γ π µν = 18 πG N ( K µν − γ µν K ) , (3.4)where γ µν is the induced metric on the boundary and π µν is its conjugate momentum,and K µν is the extrinsic curvature. Here, we consider the metric with small perturbation h µν , and take the difference of the surface energy-momentum tensor from that of thebackground ¯ g µν : τ µν [ h ] = T µν (cid:12)(cid:12)(cid:12) g =¯ g + h − T µν (cid:12)(cid:12)(cid:12) g =¯ g . (3.5) Instead of taking difference from the background, we can introduce a counter term such that thecharge Q ξ becomes finite (see Appendix A). Q QL ξ = Z ∂ Σ d x √ σ u µ τ µν ξ ν . (3.6)where u µ is a timelike unit normal to a timeslice Σ and σ is an induced metric onthe timeslice at the boundary ∂ Σ. This quasi-local charge corresponds to the energy-momentum tensor for the right mover in CFT as, Q QL ξ ∼ ¯ T (¯ z )¯ ǫ (¯ z ) . (3.7)The central extension can be read off from the anomalous transformation of this charge: δ ξ Q QL ζ = Z ∂ Σ d x √ σ u µ τ µν [ £ ξ ¯ g ] ζ ν . (3.8)For two-dimensional CFT, the central extension of the Virasoro algebra can be read offfrom the anomalous transformation of the energy-momentum tensor. In an analogousfashion to this, we estimate the central extension from the anomalous transformation ofthe ADM mass, which is δ ξ Q QL ζ with ǫ ζ ( t ) = 1. For the metric (2.14) and the asymptoticsymmetry group (3.2), we obtain δ ξ Q QL ∂ ˆ t = 18 πG N Z dφ dψ dθ k i k j γ ij ( θ ) p γ ( θ ) f θ ( θ )2Λ f ( θ ) ǫ ′′′ ξ (ˆ t ) , (3.9)where we have introduced a cut-off by putting the boundary at ˆ r = Λ. By using theexplicit form of the near horizon metric (2.10), it turns out that δ ξ Q QL ∂ ˆ t = πr µ G N Λ ǫ ′′′ ξ (ˆ t ) . (3.10)From the definition of the near horizon coordinate of ˆ r , it must satisfyˆ r ≪ r √ µ ǫ − , (3.11)in order for the expansion in ǫ to be valid. Therefore, we put the boundary of the nearhorizon geometry at Λ = 2 r √ µ ǫ − . (3.12)The central charge is related to the anomalous transformation of the charge as δ ξ Q = c ( QL ) ǫ ′′′ ξ (ˆ t ) + (non-anomalous terms) . (3.13)8hen, the central charge can be evaluated as c ( QL ) = 3 πµ / G N ǫ . (3.14)The Frolov-Thorne temperature for right mover can be read off from the Boltzmannfactor with respect to the charge Q ∂ ˆ t , and given by T = √ ˆ µ π . (3.15)Using the Cardy formula, we obtain S = π c ( QL ) T = π µ / √ ˆ µ G N ǫ . (3.16)Since the Bekenstein-Hawking entropy is expanded in the near-extremal case as S = π G N µr + = π G N µ (cid:18) r + 12 ǫ p µ ˆ µ + O ( ǫ ) (cid:19) , (3.17)the expression (3.16) agrees with the leading non-extremal correction.Even though the Bekenstein-Hawking entropy is correctly reproduced, the identifi-cation of the cut-off (3.12) is just a rough estimation. In order to justify this choiceof the cut-off, we compare this result with the AdS /CFT correspondence in the nextsection. /CFT correspondence When one of the angular momenta vanishes, the near horizon geometry of the five-dimensional Myers-Perry black hole has the structure of AdS . If this momentum isnot exactly zero but infinitesimally small in the near-extremal case, this AdS partbecomes the BTZ black hole. In this case, we can simply apply the ordinary AdS/CFTcorrespondence for the Myers-Perry black hole. In this section, we consider such a caseand compare it with the previous two results.In the previous sections, we have considered the near-extremal case in which the non-extremality is infinitesimally small. Here, we assume that one of the angular momentais also infinitesimally small and of the same order to the non-extremality. We define theparameters ˜ µ and ˜ b by the following relations, µ = a + b + 2 aǫ ˜ m , b = ǫ ˜ b , (4.1)9nd redefine the near horizon coordinates as t = ǫ − ˜ t , r = ǫ ˜ r , (4.2) φ = ˜ φ + ar + a t , ψ = ǫ − ˜ ψ . (4.3)By taking the near horizon limit ǫ →
0, the metric becomes ds = − cos θa ˜∆˜ r d ˜ t + a cos θ ˜ r ˜∆ d ˜ r + cos θ ˜ r (cid:18) d ˜ ψ − b ˜ r d ˜ t (cid:19) + a sin θ cos θ d ˜ φ + a cos θ dθ, (4.4)where ˜∆ is given by ˜∆ = ˜ r − a ˜ m ˜ r + a ˜ b = (˜ r − ˜ r )(˜ r − ˜ r − ) , (4.5)and ˜ r ± is the positions of the outer and inner horizons in terms of ˜ r , which is expressedas ˜ r ± = ǫ − r ± = a (cid:16) ˜ m ± p ˜ m − ˜ b (cid:17) . (4.6)Then, this geometry has the structure of the BTZ black hole. Strictly speaking, ˜ ψ hasan infinitesimal period 2 πǫ .The analysis of the asymptotic symmetry can be applied to this geometry straight-forwardly. For simplicity, we introduce the light-corn coordinates x ± = ˜ ψ ± ˜ ta . (4.7)By imposing the boundary condition: O ( χ µν ) = x + ˜ r x − ˜ φ θx + O ( r ) O ( r − ) O ( r ) O ( r − ) O ( r − )˜ r O ( r − ) O ( r − ) O ( r − ) O ( r − ) x − O ( r ) O ( r − ) O ( r − )˜ φ O ( r − ) O ( r − ) θ O ( r − ) , (4.8)which is the same as the original work by Brown and Henneaux [1] for AdS part, weobtain the following asymptotic symmetry groups: ξ (+) = ǫ + ( x + ) ∂ + −
12 ˜ rǫ ′ + ( x + ) ∂ ˜ r − a r ǫ ′′ + ( x + ) ∂ − , (4.9a) ξ ( − ) = ǫ − ( x − ) ∂ − −
12 ˜ rǫ ′− ( x − ) ∂ ˜ r − a r ǫ ′′− ( x − ) ∂ + . (4.9b)10he coordinate x + and x − parametrize the almost same directions to ˆ t and ˆ φ , andhence, these asymptotic symmetry groups are almost equivalent to those studied in theprevious sections. In order to see this, we take the limit of b → t, ˆ ψ ) with ( x + , x − ) weobtain the following relations: x + = ˆ t + O ( ǫ ) , x − = ǫ ˆ ψ . (4.10)Then, by taking ǫ → b = ǫ ˜ b , the near horizon metric(2.10) becomes ds = a cos θ µ ( dx + ) + p a ˜ b ˆ r dx + dx − + a ˜ b cos θ ( dx − ) + a cos θ d ˆ r + a sin θ cos θ d ˆ φ + a cos θ dθ . (4.11)Since the parameters ˆ µ and ˜ m are related as ˆ µ/ a ( ˜ m − ˜ b ), this expression agreeswith (4.4) in the extremal limit of ˜ m → ˜ b if we identify ˜ r ∼ a ˜ b + p a ˜ b ˆ r. (4.12)Then, excluding the last terms in (4.9), which are asymptotically subleading contribu-tions, we obtain ξ (+) ∼ ǫ + (ˆ t ) ∂ ˆ t − ˆ rǫ ′ + (ˆ t ) ∂ r + O ( ǫ ) , (4.13) ξ ( − ) ∼ ǫ − (cid:16) ˆ ǫ − ( ˆ ψ ) ∂ ˆ ψ − ˆ r ˆ ǫ ′− ( ˆ ψ ) ∂ ˆ r (cid:17) + O ( ǫ ) , (4.14)where ˆ ǫ − ( ˆ ψ ) = ǫ − ( ǫ ˆ ψ ). Therefore, ξ (+) and ξ ( − ) correspond to the asymptotic symmetrygroups for the right and left movers, respectively.These two sets of vectors form the Virasoro algebras as in the ordinary AdS case,but have slightly different structures. Since the coordinate ˜ ψ has an infinitesimal periodof 2 πǫ , light-cone coordinates x + and x − must satisfy the following periodicities x + ∼ x + + 2 πnǫ , x − ∼ x − + 2 πnǫ . (4.15)Then, the functions ǫ + ( x + ) and ǫ − ( x − ) can be expanded with the following forms: ǫ + ( x + ) = e inx + /ǫ , ǫ − ( x − ) = e inx − /ǫ , (4.16a) Strictly speaking, here we take the near-extremal limit and the small b limit separately, and hence,the parameter ǫ here and that in (2.7), (2.8) should be distinguished. The metric (4.11) agrees with(4.4) only in the near-extremal case, because we first take the near-extremal limit for (4.11). This relation is consistent with the definitions of ˆ r and ˜ r , since the definition of ˆ r can be rewrittenas r = r + ǫr √ µ ˆ r in ǫ → n . Now we define ξ (+) n and ξ ( − ) n by ξ (+) and ξ ( − ) with (4.16).Then, these vectors form the following algebras:[ ξ (+) n , ξ (+) m ] = − i n − mǫ ξ (+) n + m , (4.17)[ ξ ( − ) n , ξ ( − ) m ] = − i n − mǫ ξ ( − ) n + m , (4.18)respectively. Here we have an additional factor ǫ − which comes from the period of ˜ ψ .Before calculating the central charge, we discuss the general property of the Virasoroalgebra with an additional factor. In general, the following vector forms the Virasoroalgebra: ξ = f ( x ) ∂ x − rf ′ ( x ) ∂ r . (4.19)If the coordinate x has the period of 2 πβ , the function f ( x ) must respect this periodicity.We define ξ n by ξ with f n ( x ) = e inx/β . (4.20)Then, this vector forms the following algebra[ ξ n , ξ m ] = n − mβ ξ n + m . (4.21)This is the Virasoro algebra but has an additional factor of 1 /β . This factor appearsbecause the vector ξ is not dimensionless and hence the algebra depends on the choiceof the coordinate x . The factor β can be absorbed by taking the coordinate x to havethe period of 2 π , or equivalently, redefining ξ → ξ ′ = βξ .The Noether charges L n associated with these vector can have the central extension,and obeys the following algebra:[ L n , L m ] = − i ( n − m ) L n + m − iδ n + m, n c , (4.22)where we have rescaled the vector such that the algebra takes the standard form. In theoriginal definition of the asymptotic symmetry groups, we have chosen the normalizationsuch that ξ gives a conjugate momentum of x , namely ξ = ∂ x . Then, the algebrabecomes [ L n , L m ] = − i n − mβ L n + m − iδ n + m, n c ( cov ) β . (4.23)12ere, we have chosen the “covariant” definition of the central charge such that thecentral extension is related to the anomalous transformation of the charge as δQ ξ ∼ c ( QL ) f ′′′ ( x ) , (4.24)where Q ξ ∼ P L n f n ( x ). Therefore, the central charge we have derived in the previoussection is not c but c ( cov ) . By using this covariant definition, the generators L n , centralcharge c ( cov ) and the Frolov-Thorne temperature T have the following scaling properties: x → λx , L n → λ − L n ,c ( cov ) → λc ( cov ) , T → λ − T , (4.25)where the Frolov-Thorne temperature T is the weight for the charge L . These behav-iors imply that we can also apply the Cardy formula in these definitions. These twodefinitions, the standard scale-invariant and covariant one, are related with each otherby L n = βL n , c = c ( cov ) β , T = βT , (4.26)where T is the scale-invariant Frolov-Thorne temperature which is the weight for L .Obviously the Cardy formula takes the same form for both of these two definitions.These two sets of definitions are exactly same when the period of the coordinate x is2 π . Now, we evaluate the central charge. By using the definition of [15, 16], the centralextensions of the algebras for ξ (+) and ξ ( − ) are obtained as18 πG N Z k ξ ( ± ) n [ £ ξ ( ± ) m ¯ g, ¯ g ] = − i π δ n + m, a n ǫ . (4.27)Since the period of x + and x − are 2 πǫ , the scale-covariant central charges for ξ (+) and ξ ( − ) are c ( cov ) ± = ǫ πa G N . (4.28)The Frolov-Thorne temperatures associated with ∂ + and ∂ − are given by T ± = ˜ r + ± ˜ r − πa . (4.29)Then, the Cardy formula reproduces the Bekenstein-Hawking entropy of (4.4): S CF T = π c ( cov )+ T + + π c ( cov ) − T − = π G N a ˜ r + ǫ = S BH . (4.30)13he central charges (4.28) are actually equivalent to those derived in the previous sec-tions. Since the coordinates are rescaled asˆ ψ → x − = ǫ ˆ ψ , (4.31)the central charges before and after the rescaling are related with each other as c ( cov ) − = ǫ c ˆ ψ . (4.32)From (2.23) and (4.28), it is clear that c ˆ ψ and c ( cov ) − satisfy this relation in b → t and x + are equivalent in ǫ → c ( QL ) in (3.14) equals to c ( cov )+ in b → at the leading order of ǫ , there are higher order corrections. The near horizoncoordinate ˆ t is the time coordinate in AdS and not exactly equivalent to the light-cone coordinate x + . We can also introduce an asymptotic symmetry group of the timedirection for AdS by imposing a suitable boundary condition. However it is rathernatural to define a new near horizon limit to obtain asymptotic symmetry groups forthe general extremal Myers-Perry black holes. We will discuss it in the next section. In this section, we consider another definition of the near horizon limit. We havedefined the near horizon coordinates by (2.8). However, the central charge of right movercannot be calculated by the covariant phase space method given by [15, 16], and hencewe used the quasi-local charge [13, 14]. In the new coordinates, the asymptotic Virasorosymmetries are realized along the light-cone coordinates ( x + , x − ) similarly to the AdS spacetime, while those in the usual coordinates are associated with (ˆ t, ˆ ψ ) directions. Wewill show that the definition of [15, 16] also gives the central charge and reproduces theentropy via the Cardy formula.We define the new coordinates x + and x − , and redefine ˆ r as follows: x + = ǫ (cid:18) ψ + a − bµ t (cid:19) , x − = ψ − a + bµ t , ˆ r = r + ǫ a r . (5.1)In the near-extremal case of (2.7), the near horizon geometry takes the same form as(2.10), but the coordinates ˆ t and ˆ ψ are replaced with x + and x − . Namely, taking ǫ → ds = − ρ ˆ∆4 ( dx + ) + ρ d ˆ r + ρ dθ + a µ sin θρ (cid:16) d ˆ φ + k ˆ φ ˆ rdx + (cid:17) + b µ cos θρ (cid:16) dx − + k ˆ ψ ˆ rdx + (cid:17) + µ r ρ h sin θ (cid:16) d ˆ φ + k ˆ φ ˆ rdx + (cid:17) + cos θ (cid:16) dx − + k ˆ ψ ˆ rdx + (cid:17)i , (5.2)where we have also redefined ˆ∆ by ˆ∆ = ˆ r − µ a ˆ µ. (5.3)Then, we can obtain the asymptotic symmetry group in the same fashion as the previoussection (but the coordinate ˆ t and ˆ ψ are replaced with x + and x − ). By using the sameboundary condition as (3.1), we obtain the asymptotic symmetry groups, ξ = (cid:16) ǫ + ( x + ) + ǫ ′′ ξ ( x + )2ˆ r (cid:17) ∂ + + (cid:16) − ˆ rǫ ′ ξ ( x + ) + ǫ ′′′ ξ ( x + )2ˆ r (cid:17) ∂ ˆ r + (cid:16) C φ − k ˆ φ ǫ ′′ ξ ( x + )ˆ r (cid:17) ∂ ˆ φ + (cid:16) C ˆ ψ − k ˆ ψ ǫ ′′ ξ ( x + )ˆ r (cid:17) ∂ − + O (ˆ r − ) . (5.4)Since ψ ∼ ψ + 2 π , the coordinates x + and x − have the periodicity of x + ∼ x + + 2 πnǫ , x − ∼ x − + 2 πn . (5.5)Then, the function ǫ ξ ( x + ) must have the following form: ǫ ξ ( x + ) = e inx + /ǫ . (5.6)We define ξ n by ξ with (5.6). Then, ξ n forms the following algebra:[ ξ n , ξ m ] = − i n − mǫ ξ n + m . (5.7)Now let us consider the conserved charge defined by (2.18) in our new coordinates.In the previous definition of the near horizon coordinates, ˆ t is equivalent to the timeof the original coordinates up to the scaling factor. Hence, the timeslice in the nearhorizon geometry is also defined on ˆ t = const. plane. However, in the new definition, thecoordinate x + is not equivalent to the original time. We should perform the integrationon the original timeslice, hence the timeslice Σ (and its boundary ∂ Σ) is not x + = const.plane. 15or the asymptotic symmetry groups (5.4), we obtain˜ k + rζ [ £ ξ ¯ g, ¯ g ] = 0 , ˜ k − rζ [ £ ξ ¯ g, ¯ g ] = − k ˆ ψ f ( θ ) ǫ ζ ( x + ) ǫ ′′′ ξ ( x + ) , (5.8)where f ( θ ), k ψ , etc. are the same as those in the previous section. Since the centralextension becomes 18 πG N Z ∂ Σ k ξ m [ £ ξ n ¯ g, ¯ g ] = πk ˆ ψ G N ǫ Z dθ p γ ( θ ) f θ ( θ ) , (5.9)the scale-covariant central charge is evaluated as c ( cov )+ = 6 πk ˆ ψ ǫG N Z dθ p γ ( θ ) f θ ( θ ) . (5.10)Using the explicit form of the metric (5.2), we obtain c ( cov )+ = 3 πaµ G N ǫ . (5.11)The central charge for left mover can be calculated straightforwardly. Since the addi-tional term gives only the O ( ǫ ) corrections, the central charge for left mover equals to(2.23). It should be noted that the scale-invariant central charge is given by c + = c ( cov )+ /ǫ ,and equals to the value for left mover c − = c ˆ ψ .The Frolov-Thorne temperatures associated with ∂ + and ∂ − are given by T + = ǫ − r + − r − πa → √ µ ˆ µ πa , (5.12) T − = r + + r − πa → bπ . (5.13)Then the Cardy formula reproduces the Bekenstein-Hawking entropy up to O ( ǫ ): S = π c ( cov )+ T + + π c ( cov ) − T − = π G N µ (cid:18) r + 12 ǫ p µ ˆ µ + O ( ǫ ) (cid:19) . (5.14)Before closing this section, we would like to comment on the definition of the newnear horizon coordinates. First, the definitions of x ± is the same as those of the hiddenconformal symmetry [17] in the near-extremal limit. When one of the angular velocities( b ) is very small, these coordinates, x ± , are equivalent to the light-cone coordinates x + and x − defined in the previous section up to the factor of ǫ for x − . In this limit, thecentral charges c ( cov ) ± and the Frolov-Thorne temperatures agree with those in Section 4when we take into account the factor of ǫ or use the scale-invariant definitions. Sincethe coordinate x + and ˆ t are related as x + = aa + b ˆ t + O ( ǫ ), the central charge (3.14) canbe reproduced from c ( cov )+ by using (4.25). 16 Kerr/CFT revisited
In the previous section, we have defined the new near horizon limit of the five-dimensional Myers-Perry black hole. By using this, the central charges for right movercan be calculated by using the definition of [15, 16]. In this section, we consider such anew near horizon limit for the four-dimensional Kerr black hole.By using the Boyer-Lindquist coordinates, the Kerr geometry can be expressed as ds = − ∆ ρ ( dt − a sin θ dφ ) + sin θρ (cid:2) ( r + a ) dφ − adt (cid:3) + ρ ∆ dr + ρ dθ , (6.1)where ∆ and ρ are given by∆ = r − M r + a , ρ = r + a cos θ . (6.2)The Kerr geometry are characterized by two parameters M and a which are related tothe ADM mass and angular momentum as M ADM = MG N , J = aMG N . (6.3)The inner and outer horizons are given by r ± = M ± √ M − a , (6.4)and the angular velocity at the outer horizon r + isΩ H = ar + a . (6.5)The Hawking temperature and the Bekenstein-Hawking entropy are given by T H = r + − r − πM r + , S = 2 πM r + G N . (6.6)We consider the near-extremal case, and define a non-extremality parameter ˆ r H as M = a (1 + ǫ ˆ r H . (6.7)Then, the geometry is parametrized by a and ˆ r H . New near horizon coordinates x ± andˆ r are defined by the following relations, x + = ǫφ , x − = φ − a t M , r = a (1 + ǫ ˆ r ) . (6.8)17ere, the combination of t and φ in the definition of x + and x − are the same as thoseappeared in analysis of the hidden conformal symmetry [10] (in the definition of w ± ).The geometry is expressed in the near horizon limit of ǫ → ds = − (ˆ r − ˆ r H ) f ( θ )( dx + ) + f φ ( θ )( dx − + ˆ rdx + ) + f ( θ ) d ˆ r ˆ r − ˆ r H + f ( θ ) dθ , (6.9)where f ( θ ) and f φ ( θ ) are given by f ( θ ) = a (1 + cos θ ) , f φ ( θ ) = 4 a sin θ θ . (6.10)This geometry has the same form to the ordinary so-called NHEK geometry, but t and φ of the near horizon coordinates are replaced with x + and x − , respectively.The boundary condition for right mover is obtained in [8]. We use this boundarycondition by replacing ˆ t and ˆ φ with x + and x − : O ( χ µν ) = x + ˆ r x − θx + O ( r ) O ( r − ) O ( r − ) O ( r − )ˆ r O ( r − ) O ( r − ) O ( r − ) x − O ( r − ) O ( r − ) θ O ( r − ) . (6.11)Then, the following asymptotic symmetry group satisfies this boundary condition: ξ = (cid:16) ǫ ξ ( x + ) + ǫ ′′ ξ ( x + )2ˆ r (cid:17) ∂ + + (cid:16) − ˆ rǫ ′ ξ ( x + ) + ǫ ′′′ ξ ( x + )2ˆ r (cid:17) ∂ ˆ r + (cid:16) C − ǫ ′′ ξ ( x + )ˆ r (cid:17) ∂ − + O (ˆ r − ) . (6.12)Since the period of φ is 2 π , the coordinates x + and x − have the following periodicity: x + ∼ x + + 2 πnǫ , x − ∼ x − + 2 πn . (6.13)The central extension is given by an integration of two-from k ξ [ £ ξ ¯ g, ¯ g ] on a timeslice. Aswe have discussed in the previous section, the charge should be defined by the integrationon the timeslice of the original coordinates. As in the case of the five-dimensional Myers-Perry black hole, the near horizon geometry is usually taken such that the time directionof the near horizon coordinates is equivalent to that of the original one, up to a constantfactor. In this case, only ˜ k trξ component contributes to the charge and the central charge18ecomes zero [8]. However, in the new near horizon coordinate (6.8), the coordinate x + is not the time direction of original coordinates. Then, ˜ k − rξ also contributes to thecentral extension and we obtain18 πG N Z ∂ Σ k − rξ m [ £ ξ n ¯ g, ¯ g ] = δ n + m, n a ǫ . (6.14)Then, the scale-covariant central charge is c ( cov )+ = 12 a G N ǫ . (6.15)This result agrees with that obtained in [8]. It should be noted that this is the expectedvalue of c R ≡ c + = 12 J if we use the scale-invariant definition.The Frolov-Thorne temperatures associated to ∂ + and ∂ − are given by T + = ǫ − r + − r − πa → ˆ r H π , (6.16) T − = r + + r − πa → π . (6.17)Then the Cardy formula gives the entropy S = 2 πa G N ( a + ǫ ˆ r H ) , (6.18)which agrees with the Bekenstein-Hawking entropy up to O ( ǫ ), S = 2 πM r + G N = 2 πa G N (cid:0) a + ǫ ˆ r H + O ( ǫ ) (cid:1) . (6.19) In this paper, we introduced a new near horizon limit. In this limit, structure ofthe near horizon geometry is the same as that introduced in [4], while the originaltime direction is embedded in a different way. Then, conserved charges are slightlymodified since the timeslice is different in our limit and that in [4]. This limit is usefulto describe the right mover in the Kerr/CFT correspondence, and the central chargecan be calculated explicitly.By using our new definition of the near horizon limit, the charge density depends onthe angular coordinates and hence we can define generators simply integrating on thetimeslice. The central charge can be calculated by using the definition of [15, 16], anddoes not have ambiguities of the cut-off. It turns out that the Virasoro algebra does nothave the standard form and depends on the definition of the coordinate. By redefining19he generators to have the standard algebraic relation, the central charge becomes finiteand satisfies the expected relation of c L = c R = 12 J .In our new near horizon coordinates, the combinations of coordinates are equivalentto those in analyses of the hidden conformal symmetry. The presence of the hiddenconformal symmetry implies the decoupling of the right and left movers. In order toapply the Cardy formula separately, the right and left movers should be decoupled.Hence it is natural that the appropriate choice of the coordinates is equivalent to thosefor the hidden conformal symmetry.Even though the scale-invariant central charge is finite, it is natural to use the covari-ant definition for the temperature. Then, the (covariant) central charge for right movertakes an infinitesimally small value, and hence, the right mover gives subleading contri-butions in the near-extremal limit. We took the near-extremal limit and considered itsleading corrections. However, we did not include all the next-to-leading contributions inthis limit. Hence, we cannot exclude the possibility that these contributions affect thenear-extremal corrections. In order to see this, we have to study subleading corrections,or consider more general non-extremal cases.We computed the central charges for each sector separately by giving each boundarycondition. Although we reproduce the Bekenstein-Hawking entropy by summing upthe entropy of each sector, it is desirable to find a boundary condition that admit twoVirasoro algebras as asymptotic symmetry groups. One example is given by [19], butit cannot fix higher order corrections of the asymptotic symmetries for right moverwhich contribute to the central charges. Investigation in this direction will give furtherevidence for the Kerr/CFT correspondence.
Acknowledgements
We are grateful to T. Hartman and N. Matsumiya for valuable discussions, and N.Matsumiya for collaboration on a new near horizon limit at an earlier stage. TN wouldlike to thank all members of the High Energy Physics Theory Group of the Universityof Tokyo for hospitality during his stay. The work of TN was supported in part by theUS NSF under Grants No. PHY-0844827 and PHY-0756966. If we use the scale-invariant definition, the Frolov-Thorne temperature becomes infinitesimallysmall. Therefore, the right mover contributes to the subleading corrections, independent to the defini-tion of the central charge. ppendix A: Counter terms for the quasi-local charge In this appendix, we consider counter terms for the quasi-local charges. The quasi-local charge is defined by (3.6), and we have defined the regularized surface energy-momentum tensor τ µν by (3.5). Instead of taking difference from the background in(3.5), a counter term can be introduced to regularize the surface energy-momentumtensor. In this case, any covariant counter terms cannot terminate all of divergentterms in the surface energy-momentum tensor. Here, we take the counter term suchthat Q QL ξ becomes finite. Then, the surface energy-momentum tensor is given by τ µν = T µν + λg µν , (A.1)where g µν is the induced metric on the boundary and λ is a constant. Using thisdefinition of τ µν , the quasi-local charge for (2.14) has the following divergent terms: Q QL ξ = − πG N Z dφ dψ dθ Λ k i k j γ ij ( θ ) p γ ( θ ) f θ ( θ )2 f ( θ )+ λ πG N Z dφ dψ dθ Λ p f ( θ ) γ ( θ ) f θ ( θ ) + O (Λ ) . (A.2)The constant λ is chosen such that these two terms cancel each other. For the nearhorizon geometry of Myers-Perry black hole (2.10), it turns out that λ = ( a / − b / ) √ a + b a − b ) . (A.3)By using this condition, the central extension becomes δ ξ Q QL ∂ ˆ t = λ πG N Z dφ dψ dθ p f ( θ ) γ ( θ ) f θ ( θ )Λ ǫ ′′′ ξ (ˆ t )= 18 πG N Z dφ dψ dθ k i k j γ ij ( θ ) p γ ( θ ) f θ ( θ )2Λ f ( θ ) ǫ ′′′ ξ (ˆ t ) , (A.4)and then, we obtain the same result to (3.9). If we allow the coefficient λ to have θ -dependence, we can make the charge density to be finite. In this case, the coefficient ofthe counter term becomes λ ( θ ) = k i k j γ ij ( θ )2 (cid:18) f ( θ ) (cid:19) / , (A.5)where f ( θ ), γ ij ( θ ) and k i just specify the θ -dependence but do not respond to thevariation with respect to the metric. In the limit of b →
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