A note on the mass of Kerr-AdS black holes in the off-shell generalized ADT formalism
aa r X i v : . [ g r- q c ] J u l A note on the mass of Kerr-AdS black holes in the off-shell generalizedADT formalism
Yi-De Jing and Jun-Jin Peng ∗ School of Physics and Electronic Science, Guizhou Normal University,
Guiyang, Guizhou 550001, People’s Republic of China
Abstract
In this note, the off-shell generalized Abbott-Deser-Tekin (ADT) formalism is ap-plied to explore the mass of Kerr-anti-de Sitter (Kerr-AdS) black holes in various di-mensions within asymptotically rotating frames. The cases in four and five dimensionsare explicitly investigated. It is demonstrated that the asymptotically rotating effectmay make the charge be non-integrable or unphysical when the asymptotic non-rotatingtimelike Killing vector associated with the charge is allowed to vary and the fluctuationof the metric is determined by the variation of all the mass and rotation parameters.To avoid such a dilemma, we can let the non-rotating timelike Killing vector be fixedor perform calculations in the asymptotically static frame. Our results further supportthat the ADT formalism is background-dependent. ∗ Corresponding author: [email protected] Introduction
Till now, exact rotating black hole solutions with cosmological constant have been foundin various dimensions within the context of Einstein gravity. In 1968, Carter first founda generalization of the four-dimensional (4D) rotating Kerr black hole with a cosmologicalconstant [1]. Since this black hole has asymptotically de Sitter (dS) or anti-de Sitter (AdS)boundary conditions, it is usually called as Kerr-dS or Kerr-AdS black hole in the literature.Many years later, Hawking, Hunter and Taylor-Robinson found the five-dimensional (5D)generalization of the 4D Kerr-(A)dS black hole, as well as the solutions with just onenonzero angular momentum parameter in all dimensions [2]. In fact, the 5D Kerr-(A)dSblack hole can also be regarded as a generalization of the 5D Ricci-flat rotating Myers-Perry black hole [3] including a cosmological constant. Subsequently, Gibbons, L¨u, Pageand Pope further constructed the general Kerr-(A)dS black holes with arbitrary angularmomenta in all higher dimensions [4, 5], which exactly satisfy the vacuum Einstein fieldequation with a cosmological constant. For the Kerr-AdS black holes, inspired by stringtheory, especially by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence,as well as black hole thermodynamics, much work has been done on their diverse aspects[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].Recently, in Ref. [27], relieving the restriction that the background spacetime satisfiesthe field equations, Kim, Kulkarni and Yi proposed a quasi-local formalism of conservedcharges within the framework of generic covariant pure gravity theories by constructing anoff-shell ADT current to generalize the conventional on-shell Noether potential in the usualADT formulation [28, 29, 30, 31] to the off-shell level, as well as following the Barnich-Brandt-Compere (BBC) method [32, 33, 22] to incorporate a single parameter path in thespace of solutions into their definition. Since the current and potential in the modifiedapproach is off-shell, one may refer to it as the off-shell generalized ADT formalism. Incontrast with the usual one, the off-shell generalization makes it more operable to derive theNoether potential from the corresponding current and the procedure of computation becomemore convenient to manipulate. Owing to these, it provides another fruitful way to evaluatethe ADT charges for various theories of gravity. For several developments and applicationsof the off-shell generalized ADT formalism see the works [34, 35, 36, 37, 38, 39, 40, 41, 42].As usual, after obtaining the Kerr-AdS black hole solutions, it is of great necessity toidentify their mass and angular momenta. Because of the asymptotically AdS structure,2he usual Arnowitt-Deser-Misner (ADM) formalism, as well as the Komar integral, failsto produce their mass. So it is desired to seek for other feasible approaches. Fortunately,some works [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have succeeded toyield the conserved charges of the Kerr-AdS black holes through a series of methods, suchas the Ashtekar-Magnon-Das (AMD) formalism [25, 26], the (off-shell) ADT formulation,the BBC method and so on. For example, in the work [21], the usual ADT formulationhas been applied to obtain the mass of Kerr-AdS black holes in various dimensions. Torealise this, the perturbation of the metric is set as the divergence between the metric anda fixed reference background, which is the spacetime got through letting the parameterassociated with the mass in the original metric be zero. In [22], the same background wasalso adopted to calculate the conserved charges of the Kerr-AdS black holes via the BBCmethod. There the fluctuation of the metric only depends on the parameter related tothe mass rather than all the solution parameters. Besides, the potential used to define theconserved charges coincides with that in the usual ADT formulation. Due to these, theBBC method adopted in [22] is essentially in accordance with the usual ADT one.Notably, the usual ADT formulation and the BBC approach are background-dependent.Actually, in the works [21, 22], the reference spacetimes adopted to calculate the mass ofKerr-AdS black holes in all dimensions are non-rotating at infinity. Otherwise, both thetwo methods may fail to yield physical results if the timelike Killing vector associated withthe mass is chosen as the usual one ξ µ = − δ µt . For the off-shell generalized ADT formalism,we also wonder what will happen if it is used to deal with the mass of the Kerr-AdS blackholes in an asymptotic rotating frame. On the other hand, in the light of the fact that themass together with the angular momenta enters into the first law of thermodynamics forblack holes as thermodynamical variables, the rotation parameters should be the membersto determine the fluctuation of the metric for the Kerr-AdS black holes. However, inRef. [22], only the mass parameter was regarded as a variable to fluctuate the metric,while the rotation parameters were fixed. In view of above-mentioned issues, unlike theworks [21, 22], we shall utilize the off-shell generalized ADT formalism to explore the massof the Kerr-AdS black holes in a more general manner. Namely, we do this under theconditions that the background spacetime is asymptotically rotating and the fluctuation ofthe metric is determined by the variation of all the mass and rotation parameters. Theresults demonstrate that the way that the Kerr-AdS black holes behave at infinity, rotatingor not, plays a key role in determining whether the off-shell generalized ADT formalism can3uccessfully produce their physically meaningful mass.The remainder of this work goes as follows. Sections 2 and 3 are devoted to investigatingthe off-shell generalized ADT mass of the Kerr-AdS black holes in four and five dimensionsrespectively. In Section 4, the mass of the Kerr-AdS black holes in arbitrary dimensionsis calculated by generalizing the results in the 4D and 5D cases. The last section is ourconclusions. As is well-known, the theory of Einstein gravity in D dimensions is described by the Einstein-Hilbert Lagrangian L EH = √− g (cid:0) R − (cid:1) (2.1)with Λ = − ( D − D − ℓ /
2, where ℓ − is the radiu of AdS spaces. The field equationfor the gravitational field is R µν = − ( D − D − ℓ g µν . (2.2)The 4D Kerr-AdS black hole [1] is an exact rotating solution with asymptotic AdS behaviorof Eq. (2.2) in the case D = 4. In Boyer-Lindquist coordinates, the metric for the 4DKerr-AdS black hole takes the form ds = − ∆ (4) Σ (4) h dt − a sin θ (cid:16) dφ Ξ − ω φ dt Ξ (cid:17)i + Σ (4) ∆ (4) dr + Σ (4) F (4) dθ + F (4) sin θ Σ (4) h adt − ( r + a ) (cid:16) dφ Ξ − ω φ dt Ξ (cid:17)i , (2.3)in which ∆ (4) = ( r + a )(1 + ℓ r ) − mr , Σ (4) = r + a cos θ ,F (4) = 1 − a ℓ cos θ , Ξ = 1 − ℓ a , (2.4)and the constant ω φ = ω φ ( m, a, ℓ ), only depending on the integral parameters m and a , aswell as the constant ℓ . Particularly, when ω φ = 0, the metric (2.3) becomes the usual formof the 4D Kerr-AdS black hole, which is asymptotic to AdS in a rotating frame with the4ngular velocity Ω ∞ φ = − aℓ . To guarantee that the black hole is static at infinity, observedrelative to a frame that is non-rotating, one only needs to set ω φ = aℓ .We now go on to compute the mass of the 4D Kerr-AdS black hole via the off-shellgeneralized ADT method proposed in [27]. With the choice of a Killing vector ξ µ , thedefinition of the ADT conserved charge related to the Einstein-Hilbert Lagrangian (2.1) isread off as δQ c = 116 π ( D − Z ∂ Σ √− g Q µνADT ǫ µνµ µ ··· µ ( D − dx µ ∧ · · · ∧ dx µ ( D − , (2.5)where the quantity ǫ µνµ µ ··· µ ( D − is the totally antisymmetric Levi-Civita tensor, which isdefined through the equation ǫ µ µ ··· µ D = D ! δ µ δ µ · · · δ D − µ D ] , and the off-shell ADT potentialis defined by Q µνADT = Q µνADT + ∇ [ µ δξ ν ] ,Q µνADT = ξ σ ∇ [ µ h ν ] σ − h σ [ µ ∇ σ ξ ν ] + 12 h ∇ [ µ ξ ν ] − ξ [ µ ∇ σ h ν ] σ + ξ [ µ ∇ ν ] h , (2.6)in which h µν = δg µν , h = g µν h µν , and Q µνADT is the conventional (off-shell) ADT potential.The term with δξ µ comes from the variation of the off-shell Noether potential in accordancewith the generalized off-shell ADT potential in [36], whose contribution to the off-shell ADTcurrent is ∇ ν ∇ [ µ δξ ν ] = R µν δξ ν + L ξ Θ µ , where the surface term Θ µ = 2 g µ [ ρ ∇ ν h ν ] ρ . Let uspay attention to the behaviour of the charge in the case where the Killing vector ξ µ isassumed to be fixed, that is δξ µ = 0. In such a case, it has been shown that the charge Q c associated with the linear combination of two Killing vectors preserves the linearity propertyin Appendix A . However, for the other cases where δξ µ = 0, the linear property of thecharge may break down because of the appearance of terms proportional to the Komarintegral.For the 4D Kerr-AdS black hole described by the metric (2.3), the fluctuation of thespacetime h µν is determined by the infinitesimal change of both the parameters ( m, a ) ratherthan the single mass parameter m like in [22], that is m → m + dm , a → a + da . (2.7)Under such conditions, we calculate the charge associated with the 4D timelike Killing ξ µ ( t ) = ( − , , , t, r ) component of the off-shell5DT potential is computed as √− g Q trADT [ ξ µ ( t ) ] = Υ (4) r + O (cid:16) r (cid:17) + sin θ (2Ξ + 3 aω φ sin θ ) dm Ξ +3 m sin θ [2 aℓ Ξ + (4 − ω φ sin θ ] da Ξ , Υ (4) = − aℓ sin θ (1 − θ ) da Ξ . (2.8)Substituting the above equation into the formula (2.5) for the conserved charge, togetherwith the condition that the Υ (4) term makes no contribution to the mass since R π Υ (4) dθ = 0,we have dQ c [ ξ µ ( t ) ] = 1Ξ n Ξ dm + 4 maℓ da + (cid:0) ω φ − aℓ (cid:1)(cid:2) a Ξ dm + m (4 − da (cid:3)o . (2.9)A straightforward computation then shows that dQ c [ ξ µ ( t ) ] = dM (4) + (cid:0) ω φ + Ω ∞ φ (cid:1) dJ (4) , (2.10)in which, the quantities M (4) and J (4) are read off as M (4) = m Ξ , J (4) = ma Ξ , (2.11)respectively. They are the usual mass and angular momentum of the 4D Kerr-AdS blackhole presented in the literature [15, 16, 17, 18, 19, 20, 21, 22]. On the other hand, the charge Q c [ ξ µ ( φ ) ] associated with the spacelike Killing vector ξ µ ( φ ) = (0 , , ,
1) is computed as dQ c [ ξ µ ( φ ) ] = dJ (4) , (2.12)from which one can obtain the angular momentum J (4) . In fact, it essentially results fromthe Komar integral in Eq. (A.2).Since the 4D Kerr black hole is rotating at infinity, the Killing vector correspondingto its mass should be chosen as the asymptotic non-rotating timelike Killing vector ˆ ξ µ ( t ) = ξ µ ( t ) − (cid:0) ω φ + Ω ∞ φ (cid:1) ξ µ ( φ ) , which is just the linear combination of the Killing vector ξ µ ( t ) and ξ µ ( φ ) .With the help of Eq. (A.1) in Appendix A , we have d f M (4) = dM (4) − J (4) (cid:16) ∂ (cid:0) ω φ + Ω ∞ φ (cid:1) ∂m dm + ∂ (cid:0) ω φ + Ω ∞ φ (cid:1) ∂a da (cid:17) , (2.13)where Q c [ ˆ ξ µ ( t ) ] = f M (4) . Obviously, for a general ω φ , the second term in the right side of Eq.(2.13), arising from the variation of the Killing vector ˆ ξ µ ( t ) , may make f M (4) be non-integrable6n the case where both the parameters m and a are variables when the angular velocity( ω φ + Ω ∞ φ ) = 0. In contrast to this, in the case where the metric perturbation h µν merelydepends on the change of the mass parameter m , like in [22], f M (4) is integrable. Nevertheless,it is not “physically meaningful” unless ( ω φ + Ω ∞ φ ) vanishes or ω φ is independent on themass parameter. Therefore, in order to guarantee that the mass f M (4) = M (4) , a ratherefficient way is to let the angular velocity at infinity vanish, namely, ω φ = − Ω ∞ φ = aℓ , orto fix the Killing vector ˆ ξ µ ( t ) , although such conditions are not required when the off-shellgeneralized ADT method is applied to compute the angular momentum of the 4D Kerr-AdSblack hole. The 5D Kerr-AdS black hole, which behaves like an asymptotic
AdS space and possessestwo independent rotations, is an exact solution of the field equation (2.2) in five dimensions.This black hole was first constructed in [2], and it can be seen as a special case of the generalsolutions in arbitrary higher dimensions, found in [4, 5]. For the sake of convenience on ouranalysis, the metric for the 5D Kerr-AdS black hole takes the form ds = − ∆ (5) Σ (5) h dt − a (1 − x ) d ˆ φ − bx d ˆ ψ i + Σ (5) ∆ (5) dr + Σ (5) F (5) dx − x + 1 + ℓ r r Σ (5) h abdt − b (1 − x )( r + a ) d ˆ φ − ax ( r + b ) d ˆ ψ i + F (5) Σ (5) (cid:0) − x (cid:1)h adt − ( r + a ) d ˆ φ i + F (5) Σ (5) x h bdt − ( r + b ) d ˆ ψ i , (3.1)where ∆ (5) = ( r + a )( r + b )(1 + ℓ r ) r − m , Σ (5) = r + a x + b (1 − x ) ,F (5) = 1 − a ℓ x − b ℓ (1 − x ) , Ξ a = 1 − a ℓ , Ξ b = 1 − b ℓ ,d ˆ φ = dφ Ξ a − ω φ dt Ξ a , d ˆ ψ = dψ Ξ b − ω ψ dt Ξ b . (3.2)In Eq. (3.1), both the coordinates φ and ψ range from [0 , π ], while x takes the value [0 , ω φ and ω ψ , which depend on the four parameters ( m, a, b, ℓ ), disappearand the coordinate x is substituted by a new one θ through the relation x = cos θ , themetric (3.1) returns to the usual form of the 5D Kerr-AdS black hole in the literature,7hose angular velocities along the φ and ψ directions are Ω ∞ φ = − aℓ and Ω ∞ ψ = − bℓ respectively at infinity. Unlike this, in the more general cases ω φ , ω ψ = 0, the angularvelocities of the metric (3.1) at infinity become ω φ + Ω ∞ φ and ω ψ + Ω ∞ ψ .In order to calculate the off-shell generalized ADT mass of the 5D Kerr-AdS blackhole (3.1), the fluctuation of the metric is determined by the infinitesimal variation of theparameters ( m, a, b ) as m → m + dm , a → a + da , b → b + db , (3.3)respectively. As before, we first consider the conserved charge associated with the Killingvector ξ µ (5 t ) = ( − , , , , Q µνADT in Eq. (2.6), acomplex calculation gives the following expression for the ( t, r ) component of the potential: √− g Q trADT [ ξ µ (5 t ) ] = 1Ξ a Ξ b (cid:2) xdm − b (2 + Ξ b − Ξ a ) (cid:0) x − x (cid:1) db (cid:3) +2 X (5) + 4 ω φ (cid:0) x − x (cid:1) Y (5) + 2 X (5) ( a ↔ b )+4 ω ψ x Y (5) ( a ↔ b ) + Υ (5) r + O (cid:16) r (cid:17) , (3.4)in which X (5) = axda Ξ a Ξ b (cid:2) a (Ξ a − Ξ b ) x + 2Ξ a (1 + Ξ b − Ξ a ) x − Ξ a + 4 mℓ (cid:3) ,Y (5) = 1Ξ a Ξ b (cid:2) a Ξ a Ξ b dm + m Ξ b (4 − a ) da + 2 abm Ξ a ℓ db (cid:3) , Υ (5) = 2 ℓ Ξ a Ξ b (cid:0) x − x (cid:1)(cid:0) ada − bdb (cid:1) . (3.5)In the above equation, the quantity Υ (5) has the property R Υ (5) dx = 0, which implies thatthe contribution to the mass from the first term in Eq. (3.4) can be neglected. Anotheravenue to cancel the contribution from the r term is to let the perturbation of the metric beindependent of the rotation parameters a and b , like in [22]. This holds for the cancellationof the contribution from the r term in Eq. (2.8) as well. Further making use of the definition(2.5) for the off-shell ADT conserved charges, one gets dQ c [ ξ µ (5 t ) ] = dM (5) + (cid:0) ω φ + Ω ∞ φ (cid:1) dJ (5 φ ) + (cid:0) ω ψ + Ω ∞ ψ (cid:1) dJ (5 ψ ) , (3.6)where M (5) , J (5 φ ) and J (5 ψ ) are the mass and angular momenta along the φ and ψ directions8ot through other methods in the literature [18, 19, 20, 21, 22], which are read off as M (5) = π m Ξ a Ξ b (2Ξ a + 2Ξ b − Ξ a Ξ b ) ,J (5 φ ) = π ma Ξ a Ξ b , J (5 ψ ) = π mb Ξ a Ξ b . (3.7)By using the off-shell generalized ADT formula (2.5), the angular momenta J (5 φ ) and J (5 ψ ) ,associated with the Killing vectors ξ µ (5 φ ) = δ µφ and ξ µ (5 ψ ) = δ µψ respectively, can also beobtained through dQ c [ ξ µ (5 φ ) ] = dJ (5 φ ) , dQ c [ ξ µ (5 ψ ) ] = dJ (5 ψ ) . (3.8)It was explicitly proved in [19] that the conserved charges M (5) , J (5 φ ) and J (5 ψ ) strictlysatisfy the first law of thermodynamics. What is more, by making use of Eqs. (3.6), (3.8)and (A.1), the off-shell ADT charge of the 5D Kerr-AdS black hole Q c [ ˆ ξ µ (5 t ) ], where theasymptotic non-rotating timelike Killing vector ˆ ξ µ (5 t ) = ξ µ ( t ) − (cid:0) ω φ +Ω ∞ φ (cid:1) ξ µ (5 φ ) − (cid:0) ω ψ +Ω ∞ ψ (cid:1) ξ µ (5 ψ ) ,is computed as d f M (5) = dM (5) − J (5 φ ) (cid:16) ∂ (cid:0) ω φ + Ω ∞ φ (cid:1) ∂m dm + ∂ (cid:0) ω φ + Ω ∞ φ (cid:1) ∂a da + ∂ (cid:0) ω φ + Ω ∞ φ (cid:1) ∂b db (cid:17) − J (5 ψ ) (cid:16) ∂ (cid:0) ω ψ + Ω ∞ ψ (cid:1) ∂m dm + ∂ (cid:0) ω ψ + Ω ∞ ψ (cid:1) ∂a da + ∂ (cid:0) ω ψ + Ω ∞ ψ (cid:1) ∂b db (cid:17) , (3.9)in which, f M (5) = Q c [ ˆ ξ µ (5 t ) ]. Note that Eq. (3.9) is similar to Eq. (2.13) for the 4D Kerr-AdSblack hole. As before, in order to make the quantity f M (5) coincide with the physicallymeaningful mass M (5) , a rather effective way is to set ω φ = − Ω ∞ φ and ω ψ = − Ω ∞ ψ , that is,the angular velocities of the 5D Kerr-AdS black hole vanish at infinity.A remark is in order here. In the above, we have demonstrated that it is a better choice toevaluate the mass of the 4D and 5D Kerr-AdS black holes in an asymptotically non-rotatingframe. However, Eq. (A.3) allows the possibility to compute their mass in a rotating frameat infinity when the asymptotic angular velocity is independent of the mass parameter. Forinstance, to get the mass M (4) for the 4D Kerr-AdS black hole (2.3) with ω φ = 0, one canset that the variation of the metric only depends on that of the single parameter m and theKilling vector corresponding to the mass is chosen as the asymptotic non-rotating timelikeKilling vector ˆ ξ µ ( t ) = ξ µ ( t ) + aℓ ξ µ ( φ ) = ( − , , , aℓ ). The former makes the charge Q c [ ξ µ ( t ) ] beintegrable. Its integral yields Q c [ ξ µ ( t ) ] = M (4) − aℓ J (4) , while Q c [ ξ µ ( φ ) ] = J (4) . Thus the linearcombination of Q c [ ξ µ ( t ) ] and Q c [ ξ µ ( φ ) ] gives rise to Q c [ ˆ ξ µ ( t ) ] = Q c [ ξ µ ( t ) ] + aℓ Q c [ ξ µ ( φ ) ] = M (4) .9n the other hand, in the case where the parameters ω φ , ω ψ = 0 within the metric form(3.1) for the 5D Kerr-AdS black hole, if the fluctuation of the metric is still determined bythe parameter m rather than the ones ( m, a, b ), and the timelike Killing vector is set asˆ ξ µ (5 t ) = ξ µ (5 t ) + aℓ ξ µ (5 φ ) + bℓ ξ µ (5 ψ ) , one obtains M (5) . Besides, in the more general cases, ifthe Killing vectors are fixed, the terms proportional to the angular momenta in Eqs. (2.13)and (3.9) vanish, yielding the physical mass of the 4D and 5D Kerr-AdS black holes in thegeneral asymptotically rotating frames. In the present section, we deal with the off-shell generalized ADT mass for the general Kerr-AdS black holes in D ( D ≥
4) dimensions by generalizing the analysis for the 4D and 5Dones. The general Kerr-AdS black holes in D = (2 N + 1) + ǫ dimensions, where ǫ = 1 when D is even and ǫ = 0 when D is odd, were constructed in [4, 5]. They possess N independentrotations in N orthogonal 2-planes, characterized by the parameters a i ’s (1 ≤ i ≤ N ) andthe azimuthal angles φ i ’s. The metric for the D -dimensional Kerr-AdS black hole is readoff as ds D ) = d ¯ s D ) + 2 mUV ( V − m ) dr + 2 mU (cid:16) W dt − N X i =1 a i µ i d ˆ φ i Ξ i (cid:17) ,d ¯ s D ) = − W (1 + ℓ r ) dt + N X i =1 µ i ( r + a i ) d ˆ φ i Ξ i + N + ǫ X i =1 ( r + a i ) dµ i Ξ i − ℓ W (1 + ℓ r ) (cid:16) N + ǫ X i =1 r + a i Ξ i µ i dµ i (cid:17) + UV dr , (4.1)in which U = r V ℓ r N + ǫ X i =1 µ i r + a i , V = ( r ǫ − + ℓ r ǫ ) N Y i =1 ( r + a i ) , W = N + ǫ X i =1 µ i Ξ i ,d ˆ φ i = dφ i − ω ( i ) ( m, a j , ℓ ) dt , Ξ i = 1 − a i ℓ , (4.2)and the coordinates µ i ’s obey the constraint P N + ǫi =1 µ i = 1. The metric (4.1) describesrotating black holes in an asymptotically rotating frame with angular velocities Ω ∞ ( i ) = ω ( i ) .It becomes the one in Eq. (4.2) of the paper [19] when the parameters ω ( i ) ’s vanish, thatis, the Kerr-AdS black holes are non-rotating at infinity.10e turn our attention to evaluating the off-shell generalized ADT mass of the Kerr-AdS black holes in D dimensions. As is shown in both the 4D and 5D cases, we choosethe asymptotic non-rotating timelike Killing vector associated with the mass as the oneˆ ξ µ ( Dt ) = ξ µ ( Dt ) − P i Ω ∞ ( i ) ξ µ ( i ) , where the timelike Killing vector ξ µ ( Dt ) = − δ µt and the spacelikeKilling vectors ξ µ ( i ) = δ µφ i , while the perturbations of the metric rely on the infinitesimalchanges of all the parameters ( m, a , · · · , a N ) through m → m + dm , a i → a i + da i ( i = 1 , · · · , N ) . (4.3)Under these conditions, it is proposed that the variation of the off-shell generalized ADTcharge Q c [ ˆ ξ µ ( Dt ) ] for the D -dimensional Kerr-AdS black hole described by Eq. (4.1) takesthe following form dQ c [ ˆ ξ µ ( Dt ) ] = dM ( D ) − N X i =1 (cid:16) ∂ Ω ∞ ( i ) ∂m dm + N X j =1 ∂ Ω ∞ ( i ) ∂a j da j (cid:17) J ( i ) , (4.4)where M ( D ) = A D − π m Q j Ξ j (cid:16) N X i =1 i − − ǫ (cid:17) , J ( i ) = A D − π ma i Ξ i Q j Ξ j . (4.5)In the above equation, A D − is the volume of the unit ( D −
2) sphere. M ( D ) and J ( i ) arethe physical mass and angular momenta in the literature [18, 19, 20, 21, 22]. J ( i ) ’s, whichcan result from the Komar integrals, coincide with the off-shell generalized ADT charges Q c [ ξ µ ( i ) ]. It should be emphasized that the following equation dQ c [ ξ µ ( Dt ) ] = dM ( D ) + N X i =1 Ω ∞ ( i ) dJ ( i ) (4.6)is utilized in order to obtain Eq. (4.4). In fact, due to Eq. (A.1) in Appendix A , bothEqs. (4.4) and (4.6) are equivalent.Equation (4.4) can be regarded as the generalization of Eqs. (2.13) and (3.9) in D dimensions. By adopting Eq. (B.2) in Appendix B to calculate the ( t, r ) componentof the off-shell ADT potential, this equation has been checked to hold in D ≤ Q c [ ˆ ξ µ ( Dt ) ] = M ( D ) , one has to cancel the contribution arising fromthe variation of the asymptotic velocities Ω ∞ ( i ) ’s. To realize this, a rather simple methodis to perform the coordinate transformations φ i → φ i + ω ( i ) t to keep the Kerr-AdS blackholes to be static at infinity. This is to say, as long as the symmetry related to the mass11s generated by the usual timelike Killing vector ξ µ ( Dt ) and the fluctuations of the metricdepend on the variation of all the mass and rotation parameters, only under the conditionthat these black holes are asymptotically non-rotating, can the off-shell generalized ADTformalism yield their physical mass.For the Kerr-AdS black holes in the asymptotic rotating frame with the angular velocitiesΩ ∞ ( i ) ’s irrelevant to the mass parameter m , when the perturbations of the metric are onlydependent on the variation of this parameter, together with the relevant Killing vectorgiven by the asymptotic non-rotating timelike one ˆ ξ µ ( Dt ) , one is able to obtain the mass Q c [ ˆ ξ µ ( Dt ) ] = M ( D ) according to Eq. (4.4).Besides, if the Killing vector ξ µ in Eq. (2.6) is supposed to be fixed, that is δξ µ = 0,giving rise to that the conventional ADT potential Q µνADT rather than Q µνADT enters into thedefinition of the conserved charge Q c , one observes that the terms with angular momentain Eq. (4.4) disappear, yielding dQ c [ ˆ ξ µ ( Dt ) ] = dM ( D ) . Its integral gives the mass of the D -dimensional Kerr-AdS black holes with general asymptotic angular velocities, which isconsistent with the AMD mass [19, 25, 26]. In this note, we have made use of the off-shell generalized ADT formalism to compute themass of the asymptotically-rotating Kerr-AdS black holes in all dimensions. Without therequirement to fix the Killing vector, for 4D and 5D Kerr-AdS black holes in a generalasymptotic rotating frame, when the timelike Killing vectors associated with the mass arechosen as the asymptotic non-rotating timelike one and the perturbations of the metric aredetermined by the variation of all the mass and angular momentum parameters, the con-served charges take the forms in Eqs. (2.13) and (3.9) respectively. A similar equation (4.4)holds for the Kerr-AdS black holes in diverse dimensions. All the equations show that thecharges are non-integrable or unphysical because of the appearance of the general asymp-totic angular velocities. Thus, in order to obtain charges coinciding with the physicallymeaningful mass in the literature, the spacetime has to be non-rotating at infinity. In thissense, all the results support that the (off-shell) ADT formalism depends on the referencebackground.By comparison, in an asymptotic rotating frame with angular velocities that are onlydependent of the rotation parameters, one can obtain the physical mass of the Kerr-AdS12lack holes, however, it is required that the fluctuations of the metric merely rely on themass parameter and the related Killing vector is set as the asymptotic non-rotating timelikeone. Besides, in the more general asymptotic rotating frame, one can still get the physicalmass if the asymptotic non-rotating timelike Killing vector is assumed to be fixed. In thelight of the above, we suggest that one had better perform calculations in an asymptoticnon-rotating frame when the off-shell generalized ADT formalism is applied to compute themass of asymptotically AdS black holes. This also holds for the usual ADT formulationand the BBC method since they are essentially equivalent to the off-shell generalized ADTformalism for the Einstein gravity theory.It should be emphasized that the cosmological constant is fixed in our calculations so thatit is not involved in the perturbation of the metric. Otherwise, the charges of the Kerr-AdSblack holes are non-integrable. To overcome this, the off-shell generalized ADT formalismhas to be modified by reevaluating the contribution from the cosmological constant [37].Accordingly, the mass of the Kerr-AdS black holes should be reconsidered with the varyingcosmological constant.
Acknowledgments
This work was supported by the Natural Science Foundation of China under Grant No.11505036 and No. 11275157. It was also partially supported by the Technology Departmentof Guizhou Province Fund under Grant No. (2016)1104.
A The linear combination of the off-shell ADT charges
Suppose that there exist two Killing vectors ξ µ (1) and ξ µ (2) , which correspond to the conservedcharges Q c [ ξ µ (1) ] and Q c [ ξ µ (2) ] respectively. In terms of the formula (2.5), one sees that thevariation of the conserved charge Q c [ c ξ µ (1) + c ξ µ (2) ] associated with the linear combinationof the two Killing vectors c ξ µ (1) + c ξ µ (2) takes the form δQ c [ c ξ µ (1) + c ξ µ (2) ] = c δQ c [ ξ µ (1) ] + c δQ c [ ξ µ (2) ] + ( δc ) Q K [ ξ µ (1) ] + ( δc ) Q K [ ξ µ (2) ] , (A.1)where the charge Q K is defined through the Komar integral Q K = 116 π ( D − Z ∂ Σ √− g ∇ [ µ ξ ν ] ǫ µνµ µ ··· µ ( D − dx µ ∧ · · · ∧ dx µ ( D − . (A.2)13or another equivalent form of the Komar integral that can be conveniently extended tothe higher-derivative gravity theories see the work [43]. If the two constants ( c , c ) satisfythe restrictions δc = 0 and δc = 0, or more generally, the Killing vector ξ µ in Eq. (2.6) isfixed, the charge Q c [ c ξ µ (1) + c ξ µ (2) ] preserves the property of the linearity, namely, δQ c [ c ξ µ (1) + c ξ µ (2) ] = c δQ c [ ξ µ (1) ] + c δQ c [ ξ µ (2) ] . (A.3) B The ( t, r ) component of the off-shell ADT potential asso-ciated with the Killing vector ξ µ = − δ µt In this appendix, we shall present the explicit form of the ( t, r ) component of the off-shellADT potential for a general metric ansatz that covers all known stationary and axisymmet-ric black hole solutions in D dimensions when the timelike Killing vector is set as ξ µ = − δ µt .For a black hole in D dimensions, there can exist N = D − − ǫ independent rotations,where ǫ = 1 when D is even and ǫ = 0 when D is odd. They correspond to N azimuthaldirections φ i ’s. In the coordinate system ( t, r, θ a , φ i ), where θ a ’s denote the ( D − N −
2) lat-itudinal angles, the general metric ansatz describing all known stationary and axisymmetricblack hole solutions can be expressed as the following form ds = − B (cid:0) dt + Y i dφ i (cid:1) + F dr + ˜ g ab dθ a dθ b + ˆ g ij dφ i dφ j , (B.1)in which all the functions merely depend on the coordinate r and θ a ’s. In our notation, theindices a, b, c range from 1 to ( D − N −
2) while the indices i, j, k = 1 , · · · , N .By letting the Killing vector ξ µ = − δ µt , in terms of the off-shell ADT potential in Eq.(2.6), its ( t, r ) component for the metric (B.1), defined by Q = Q trADT [ − δ µt ], is given by Q = 14 BF (cid:2) F δ ( αB ) − αδ ( BF ) + 2 F B ˆ g ij ˙ Y i δY j + ˜ P + ˆ P (cid:3) , ˜ P = F (cid:0) ˙ B + αB (cid:1) ˜ g ab ˜ h ab − BF ˜ g ab ˙˜ h ab − B ˙˜ g ab δ (cid:0) F ˜ g ab (cid:1) , ˆ P = F (cid:0) ˙ B + αB (cid:1) ˆ g ij ˆ h ij − BF ˆ g ij ˙ˆ h ij − B ˙ˆ g ij δ (cid:0) F ˆ g ij (cid:1) . (B.2)In the above equation, ˜ h ab = δ ˜ g ab , ˆ h ij = δ ˆ g ij , α = B ˆ g ij Y i ˙ Y j , (B.3)and the dot “ ˙ ” denotes the partial derivative with respect to the coordinate r , for instance,˙ Y i = ∂ r Y i , ˙ˆ h ij = ∂ r ˆ h ij , ˙ˆ g ij = ∂ r ˆ g ij and so on. Equation (B.2) simplifies the calculations14f the ADT potential quite drastically and it can be widely applied to the Einstein gravitytheory coupled with matter fields or not. References [1] Carter B 1968
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