Frame-independent holographic conserved charges
aa r X i v : . [ h e p - t h ] F e b Frame-independent holographic conserved charges
Seungjoon Hyun ∗ , Jaehoon Jeong † , Sang-A Park ‡ , Sang-Heon Yi § Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea † Institute of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki 54124,Greece
ABSTRACT
We propose the modified form of the conventional holographic conserved charges which pro-vides us the frame-independent expressions for charges. This form is also shown to be indepen-dent of the holographic renormalization scheme. We show the frame and scheme independencethrough the matching of our holographic expression to the covariant bulk expression of conservedcharges. As an explicit example, we consider five-dimensional AdS Kerr black holes and showthat our form of holographic conserved charges gives us the identical expressions in the rotatingand non-rotating frames. ∗ e-mail : [email protected] † e-mail : [email protected] ‡ e-mail : [email protected] § e-mail : [email protected] Introduction
Holographic principle in modern physics has been introduced as the fundamental property ofquantum gravity, which was speculated on the basis of the area nature of the black hole entropy.After its concrete realization in the form of the AdS/CFT correspondence, it becomes oneof main research arena and has been studied in various contexts. Especially, the AdS/CFTcorrespondence has been used as a modern toolkit of strong coupling phenomena for the dualfield theory. In this context holography has many interesting applications and implications evenat the level of a classical theory of gravity, since the classical computation in gravity has thedual interpretation for quantum phenomena in the field theory side. Conversely, it also providesnew approaches to the classical theory of gravity through the perspective from the dual fieldtheory. One such application is the introduction of holographic approach to conserved chargesin the classical theory of gravity which have been explored in the huge number of literatures.Holographic conserved charges in the asymptotic AdS space [1] are introduced along with theconstruction of boundary stress tensor in gravity by using the Brown-York formalism [2], which isnow regarded as one of the AdS/CFT dictionary. Despite their successful applications to variouscases, holographic charges need to be compared and/or matched to traditional bulk chargessince their equivalence is not warranted a priori. In Einstein gravity with negative cosmologicalconstant, the equivalence between the holographic and traditional bulk conserved charges ofblack holes are shown in Ref.s [3, 4, 5]. Interestingly, it was observed that holographic conservedcharges of black holes might be different from those by the covariant phase space method whenthe conformal anomaly of the dual field theory does not vanish. In particular, it has been noticedthat the results from the conventional expression of holographic charges depends on the framesat the asymptotic AdS space in odd dimensions, while the charges in covariant phase spacemethod remain invariant. When the metric for the asymptotic AdS space in odd dimensionsis taken in the standard non-rotating form, the Casimir energy is given just by constant. Onthe other hand, the Casimir energy becomes dependent on the rotational parameters when themetric is taken in the rotating frame [6, 4, 7]. Furthermore, the conventional expression forholographic charges depends on the counter term subtraction scheme [8, 9].Since it was shown that conserved charges by the covariant phase space method shouldbe completely consistent with the first law of black hole thermodynamics [10], the differencebetween holographic and covariant phase space charges means that conserved charges by theholographic method require the modification of the first law of black hole thermodynamics,albeit the minimal modification of the first law is shown to be sufficient for harmless physicalinterpretation of holographic results [4]. Still it would be nice if there is a construction ofholographic charges in such a way that they are identical with the bulk ones and thus satisfythe standard form of the first law of black hole thermodynamics.1n this paper we would like to revisit the construction of the conventional holographic con-served charges and show how it can be modified to give identical results with the bulk construc-tions. Our approach is based on the recent works [11, 12, 13, 14, 15] which can be regarded asthe generalization of the traditional Abbott-Deser-Tekin(ADT) formalism [16, 17, 18, 19] to theholographic setup. It turns out that our construction is rather general and completely consistentwith the bulk covariant expression of conserved charges under a very mild assumption. As aresult, whenever the boundary stress tensor is well-defined and there is a continuous parameterin the black hole solution, our expression of holographic charges gives finite, frame and schemeindependent results and is completely consistent with the standard form of the first law of blackhole thermodynamics.
Let us start from the brief summary of holographic renormalization in this section. See [20] fora review. In terms of the boundary values ( γ, ψ ) of the bulk metric and matter fields Ψ ≡ ( g, ψ ),the on-shell renormalized action is given by (See the Ref. [15] for our convention) I onr [ γ, ψ ] = I [ g, ψ ] on − shell + I GH [ γ ] + I ct [ γ, ψ ] , where the Gibbons-Hawking and counter terms I GB , I ct are defined on a hypersurface. The on-shell condition renders the renormalized action I onr to be the functional of the boundary value( γ, ψ ) at the boundary B . The generic variation of the on-shell renormalized action is taken inthe form of δI onr [ γ, ψ ] = 116 πG Z B d d x √− γ h T ijB δγ ij + Π ψ δψ i . (1)In order to introduce the boundary ADT current in the renormalized boundary action, let usrecall that the boundary diffeomorphism results in the identity of the form: ∇ i (2 T ijB ζ Bj ) = T ijB £ ζ B γ ij + Π ψ £ ζ B ψ , (2)where £ ζ B denotes the Lie derivative on the boundary and T ijB does the modified boundarystress tensor defined by T ijB ≡ T ijB + 12 Z ijB , T ijB ≡ √− γ δI onr δγ ij . The above boundary identity can be regarded as the analog of the bulk Noether identity, of whichelementary derivation is given in [15]. Note that Z -tensor does not need to be a symmetric oneand is given in terms of Π ψ ’s.Let us introduce the boundary conserved current as J iB ( ξ B ) ≡ − δ T ijB ξ Bj − γ kl δγ kl T ijB ξ Bj − T ijB δγ jk ξ kB + 12 ξ iB (cid:0) T klB δγ kl + Π ψ δψ (cid:1) , (3)2here δ denotes a linearization with respect to the boundary fields, including the variations ofKilling vectors. This current can be written in the form of √− γ J iB ( ξ B ) = − δ (cid:16) √− γ T ijB ξ Bj (cid:17) + √− γ T iB j δξ jB + 12 √− γξ iB (cid:0) T klB δγ kl + Π ψ δψ (cid:1) . (4)One may note that the first term corresponds to the linearized form of the conserved currents inconventional holographic charges. For a boundary Killing vector ξ B , the conservation of the firstterm is the simple result of the identity given in Eq. (2). Interestingly, this identity also leadsto the conservation of the sum of the second and third terms as shown in the Appendix. Aftertaking the linearization of the boundary fields along the black hole parameters and integratingthe linearized form along the one-parameter path ds , the holographic charges are introduced by Q B ( ξ B ) ≡ πG Z ds Z d d − x i √− γ J iB . (5)We would like to emphasize that our choice of the conserved boundary currents is motivatedby the bulk off-shell extension of the conventional ADT formalism and its form in Eq. (3)is already written down in Ref. [15]. Our boundary current in Eq. (4) is a generalization inthe case of boundary Killing vectors varying under a generic variation. It turns out that thisgeneralization of conserved currents leads to the frame-independent expression of conservedcharges, which is also free from the ambiguity in the counter term subtraction. This advantagebecomes manifest by showing the equivalence of the boundary currents to the bulk ADT potentialexpressions for charges, which is given in the following section. In this section we argue that our boundary construction of currents leads to the scheme indepen-dent results by showing their equivalence with covariant bulk expression for the ADT potentialof conserved charges. To this purpose, we explain how to construct the off-shell ADT potentialeven when a bulk Killing vector is varied under a generic variation.In the bulk, there is an off-shell identity known as the Noether identity which can be writtenin the form of E Ψ £ ζ Ψ ≡ E µν £ ζ g µν + E ψ £ ζ ψ = − ∇ µ ( E µν ζ ν ) , E µν ≡ E µν + 12 Z µν , (6)where E Ψ denotes the Euler-Lagrange expression for the field Ψ and Z µν tensor is given in termsof matter Euler-Lagrange expressions, E ψ . For a Killing vector ξ which may be unpreservedunder a generic variation, one can introduce the off-shell ADT current, just like in the non-varying case [15] as J µADT ( ξ, δ Ψ) = δ E µν ξ ν + 12 g αβ δg αβ E µν ξ ν + E µν δg νρ ξ ρ + 12 ξ µ E Ψ δ Ψ , (7)3hich can be rewritten as √− g J µADT ( ξ, δ Ψ) = δ (cid:16) √− g E µν ξ ν (cid:17) − √− g E µν δξ ν + 12 √− g ξ µ E Ψ δ Ψ . (8)This expression may be regarded a slight generalization of the non-varying Killing vector case [11,15]. Note that this current takes the same structure as the boundary conserved current in theprevious section. The off-shell conservation of this current J µADT allows us to write this currentin terms of the potential as J µADT = ∇ ν Q µνADT at the off-shell level.For the bulk Killing vector ξ , one can see that the symplectic current [21, 10, 22] defined fora generic diffeomorphism parameter ζ by ω ( £ ζ Ψ , δ Ψ) ≡ £ ζ Θ µ ( δ Ψ ; Ψ) − δ Θ µ ( £ ζ Ψ ; Ψ), reducesto ω ( £ ξ Ψ , δ Ψ) = − Θ µ ( £ δξ Ψ ; Ψ) , (9)wherer Θ µ ( δ Ψ) is the surface term for a generic variation of the bulk Lagrangian L given by δ ( √− g L ) = √− g E Ψ δ Ψ + ∂ µ Θ µ ( δ Ψ). Through relations among the ADT current, symplecticcurrent and the off-shell Noether current for a diffeomorphism variation J µζ ≡ √− g E µν ζ ν + ζ µ √− g L − Θ µ , the final off-shell expression of the ADT potential, up to the irrelevant totalderivative term, turns out to be2 √− gQ µνADT ( ξ, δ Ψ ; Ψ) = δK µν ( ξ ; Ψ) − K µν ( δξ ; Ψ) − ξ [ µ Θ ν ] ( δ Ψ ; Ψ) . (10)This final expression can be regarded as a slight generalization of covariant phase space re-sults [10, 22], which has already been obtained in Einstein gravity in [23].The matching between the boundary current J iB and the bulk ADT potential Q µνADT goesin the same way just as in the case of δξ µ = 0 and δξ iB = 0, as follows. Let us take theFefferman-Graham coordinates for the asymptotic AdS space as ds = dη + γ ij dx i dx j . Addingthe Gibbons-Hawking and counter terms in holographic renormalization gives us the additionalsurface terms modifying the bulk surface term Θ µ as˜Θ η ( δ Ψ) = Θ η ( δ Ψ) + δ (2 √− γL GH ) + δ ( √− γL ct ) = √− γ (cid:16) T ijB δγ ij + Π ψ δψ (cid:17) , (11)where the second line equality comes from Eq. (1). Holographic renormalization condition and˜Θ-expression tells us that ˜Θ η ∼ O (1) in the radial expansion. Correspondingly, the modifiedon-shell Noether current ˜ J η for a diffeomorphism parameter ζ becomes˜ J η = ∂ i ˜ K ηi ( ζ ) = ζ η √− γ L onr − ˜Θ η ( £ ζ Ψ) , (12)where we have used the on-shell condition on the bulk background fields. Just as in the caseof δξ µ = 0 [15, 4], the asymptotic behavior of general diffeomorphism parameter ζ is givenby ζ η ∼ O ( e − dη ) and ζ i ∼ O (1), in order to preserve the asymptotic gauge choice and therenormalized action. This asymptotic behavior in the diffeomorphism parameter ζ allows us to4iscard the first term in the right hand side of Eq. (12) when we approach the boundary. Inthe following we keep only the relevant boundary values of parameters such that a bulk Killingvector ξ i is replaced by its boundary value ξ iB . For the diffeomorphism variation £ ζ Ψ, themodified surface term ˜Θ η becomes˜Θ η ( £ ζ Ψ) = √− γ (cid:16) T ijB ∇ i ζ j + Π ψ £ ζ ψ (cid:17) = ∂ i (cid:16) √− γ T ijB ζ j (cid:17) , where we have used the identity given in Eq. (2). By using this result, one can see thatthe Noether potential ˜ K ηi , up to the irrelevant total derivative term, is given by ˜ K ηi = − √− γ T ijB ζ j .As a result, the on-shell relation between the ADT and Noether potentials for a Killing vector ξ B is given by √− gQ ηiADT | η →∞ = √− γ J iB . (13)This shows us the scheme independence of the holographic charges since their currents areidentified with covariant bulk ADT potentials which are regardless of the counter terms. Wewould like to emphasize that the above potential-current relation holds up to the total derivativeterms which are irrelevant in the charge computation. Moreover this equality guarantees theSmarr relation since the relation was shown to hold in bulk formalisms [23, 13].Since we have presented formal arguments, it would be illuminating to show the frame andscheme independence of mass and angular momentum of five-dimensional AdS Kerr black holesas an explicit example, which is done in the following section. As a specific example, let us focus on the pure Einstein gravity on five dimensions. In thefollowing we will set the radius of the asymptotic AdS space as unity, L = 1. AdS Kerr blackhole solutions in Boyer-Lindquist coordinates [24] are given by ds = − ∆ r ρ (cid:16) dt − a ∆ φ dφ − b ∆ ψ dψ (cid:17) + ρ ∆ r dr + ρ ∆ θ dθ + ∆ θ sin θρ (cid:16) adt − r + a − a dφ (cid:17) + ∆ θ cos θρ (cid:16) bdt − r + b − b dψ (cid:17) (14)+ 1 + 1 /r ρ (cid:16) abdt − b ( r + a )∆ φ dφ − a ( r + b )∆ ψ dψ (cid:17) , where ρ ≡ r + a cos θ + b sin θ ,∆ r ≡ ( r + a )( r + b ) (cid:16) r (cid:17) − m , ∆ θ ≡ − a cos θ − b sin θ , ∆ φ ≡ sin θ − a , ∆ ψ ≡ cos θ − b .
5n order to use the holographic method, it is useful to take the radial expansion of the metricin Fefferman-Graham coordinates as ds = dη + γ ij dx i dx j , γ ij = X n =0 e − n − η γ ( n ) ij , (15)where the non-vanishing components of background metric γ (0) are given by γ (0) tt = − , γ (0) tφ = a ∆ φ , γ (0) tψ = b ∆ ψ , γ (0) θθ = 1∆ θ , γ (0) φφ = ∆ φ , γ (0) ψψ = ∆ ψ . In the computation of conserved charges, it turns out that the expansion up to the second orderis sufficient. The non-vanishing components of the first order γ (1) are given by γ (1) tt = −
12 ( a + b + ∆ θ ) , γ (1) tφ = a ∆ φ (cid:0) a − b − ∆ θ (cid:1) , γ (1) tψ = b ∆ ψ (cid:0) b − a − ∆ θ (cid:1) ,γ (1) θθ = (2 − a − b − θ )2∆ θ , γ (1) φφ = ∆ φ (cid:0) a − b − ∆ θ (cid:1) , γ (1) ψψ = ∆ ψ (cid:0) b − a − ∆ θ (cid:1) , and those of the second order γ (2) are γ (2) tt = 3 m −
18 ( a − b ) −
14 (2 − a − b )∆ θ + 38 ∆ θ ,γ (2) tφ = a ∆ φ h − m + 18 ( a − b ) −
14 ( a − b )∆ θ + 18 ∆ θ i ,γ (2) tψ = b ∆ ψ h − m + 18 ( a − b ) −
14 ( b − a )∆ θ + 18 ∆ θ i ,γ (2) θθ = 1∆ θ h m + (2 − a − b ) − θ − a − b ) + 9∆ θ i ,γ (2) φφ = ∆ φ h m (cid:0) a ∆ φ (cid:1) + ( a − b ) − ( a − b )∆ θ θ i ,γ (2) ψψ = ∆ ψ h m (cid:0) b ∆ ψ (cid:1) + ( a − b ) − ( b − a )∆ θ θ i ,γ (2) φψ = 4 abm ∆ φ ∆ ψ . Now, it is straightforward to obtain the expression of √− γ J iB ( ξ B ) by using Eq. (4). Sincethe first term in Eq. (4) was already given in [4], let us focus on the second and third terms.One may recall that the time-like Killing vector in this metric is given by ξ iT ∂ i = ∂ t − a∂ φ − b∂ ψ .After some computations[25] with 0 ≤ θ < π , 0 ≤ φ, ψ < π , it turns out that Z d x i √− γ h T iB j δξ jT + 12 ξ iT (cid:0) T klB δγ kl + Π ψ δψ (cid:1)i = − π ( a − b )(2 − a − b )6(1 − a )(1 − b ) (cid:20) aδa − a − bδb − b (cid:21) , (16)which results in the linearized mass expression of AdS Kerr black holes from the boundarycurrent as δM = δQ B ( ξ T )= π G (cid:20) maδa (5 − a − b − a b )(1 − a ) (1 − b ) + mbδb (5 − b − a − a b )(1 − a ) (1 − b ) + δm (3 − a − b − a b )2(1 − a ) (1 − b ) (cid:21) . δM and the conventional onein [4] resides only in absence of the rotational parameter dependence of Casimir energy part.The finite mass expression is given by M = 3 π G + πm (3 − a − b − a b )4 G (1 − a ) (1 − b ) , (17)where we have added the constant Casimir energy part as an integration constant. For rotationalKilling vectors ξ µR ∂ µ = − ∂ φ and ξ µR ∂ µ = − ∂ ψ , one can see that the additional terms, i.e. secondand third ones in Eq. (4), vanish and so the angular momentum expressions are identical withthose given in [4], which is also the case in the computation of Wald’s entropy of black holes.Now, let us check the frame independence for our expression by considering different coordi-nates. In asymptotically canonical AdS coordinates, the metric of AdS Kerr black holes can betaken in the form of [7] ds = − (1 + y ) dt + dy y − m ∆ θ y + y d ˆΩ (18)+ 2 m ∆ θ y ( dt − a sin ˆ θd ˆ φ − b cos ˆ θd ˆ ψ ) + · · · , where ∆ ˆ θ ≡ − a sin ˆ θ − b cos ˆ θ ,d ˆΩ ≡ d ˆ θ + sin ˆ θd ˆ φ + cos ˆ θd ˆ ψ . By using Fefferman-Graham coordinates, one can check explicitly that mass and angular mo-mentums in these non-rotating coordinates are given by the same expressions as in the rotatingones. (See also [7].)For comparison, let us turn to the bulk covariant expressions of ADT potentials. In Ein-stein gravity, the Noether potential K µν and the bulk surface term Θ µ can be taken respec-tively as K µν ( g ; ζ ) = 2 ∇ [ µ ζ ν ] and Θ µ ( g ; δg ) = 2 √− gg α [ µ ∇ β ] δg αβ . The ADT potential, Q µνADT ( ξ T ; δa, δb, δm ) for AdS Kerr black holes is composed of three terms which correspondto the variations of parameters a , b and m , respectively as Q µνADT ( ξ T ; δm ), Q µνADT ( ξ T ; δa ) and Q µνADT ( ξ T ; δb ).For the bulk Killing vector ξ T taken in the same form as the boundary time-like Killingvector, the relevant component of the Q µνADT ( ξ T ; δm ) term is given by2 √− gQ ηtADT ( ξ T ; δm ) = − δm sin 2 θ (1 − a ) (1 − b ) h ( a + b + a b −
3) + 2( a − b ) cos 2 θ i . Q µνADT ( ξ T ; δa ) term is given by2 √− gQ ηtADT ( ξ T ; δa ) = − aδa sin 2 θ (1 − a )(1 − b ) (cid:20) ( b − a )8 + 2 m ( − b + a + a b )(1 − a ) (1 − b )+ n
12 (2 − a − b − e η ) + 2 m (1 − b − a ) − a b )(1 − a ) (1 − b ) o cos 2 θ + 38 ( b − a ) cos 4 θ (cid:21) , where one may note that the potentially divergent term proportional to e η corresponds to theirrelevant total derivative one. Q µνADT ( ξ T ; δb ) is given just by exchanging ( a, δa ) by ( b, δb ) in theabove Q µνADT ( ξ T ; δa ) expression. One may note that the varying Killing vector contribution inEq. (10) does not vanish and is given by K ηt ( δξ T ) = 8 ma cos θ sin θ (1 − a ) (1 − b ) δa + 8 mb cos θ sin θ (1 − a )(1 − b ) δb . Now, it is straightforward to check the matching between the linearized mass expression of AdSKerr black holes as δM ADT = 116 πG Z dθdφdψ √− gQ ηtADT = δM , (19)It is also straightforward to obtain the ADT potentials for rotational Killing vectors and checkits equivalence with the results from the boundary currents. In this paper, we have proposed how to modify the conventional expression of holographic con-served charges in order to give the identical results with those from bulk formalisms. Ourconstruction of holographic charges is based on the conserved boundary current, of which formis motivated by the off-shell extension of the traditional ADT formalism for bulk charges. Thisboundary current is composed of two parts, one of which corresponds to the conventional expres-sion of holographic charges and the other of which does to the additional terms compensating theframe and scheme dependence of the first term. We would like to emphasize that our modifica-tion of holographic charge expression does not mean the change of the conventional AdS/CFTdictionary for boundary stress tensor. Rather, our modification corresponds to another pre-scription, in the gravity context, of holographic charge construction from boundary stress tensorin such a way that it does not depend on the frames for the asymptotic AdS space. In thebulk side, we have extended our previous covariant construction of quasi-local conserved chargeswhen Killing vectors are varied under a generic variation. By showing the equivalence of themodified holographic expression of conserved charges to the bulk covariant expression, we haveargued the consistency of our holographic expression with the standard form of the first law of8lack hole thermodynamics and the Smarr relation. Through the example, it is explicitly shownthat the boundary-bulk equivalence is satisfied up to the irrelevant total derivative term. It isalso shown that the additional terms in the boundary current vanish in the case of the angularmomentum and black hole entropy computation, while these remove the frame-dependence inthe mass computation.Since our boundary and bulk constructions of conserved charges are based on a single for-malism which depends only on the Euler-Lagrange expression of the given Lagrangian, ourconstruction can be presented in the unified manner and seems very natural. Furthermore,our bulk construction is completely consistent with the well-known formalisms. In all, variousconstructions are naturally connected and their relationships are revealed in a unified way. Itwould be very interesting to generalize our construction to the case of more general asymptoticboundary space.
Acknowledgments
SH was supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MOE) with the grant number 2012046278 and the grant number NRF-2013R1A1A2011548. S.-H.Yi was supported by the National Research Foundation of Ko-rea(NRF) grant funded by the Korea government(MOE) (No. 2012R1A1A2004410). J. Jeongwas supported by the research grant “ARISTEIA II”, 3337 “Aspects of three-dimensionalCFTs”, by the Greek General Secretariat of Research and Technology.
Appendix: Some formulae
In order to verify the conservation of boundary currents, let us start from the double variationof fields and actions. When the diffeomorphism parameter ζ is varied under a generic variation,the variation of any quantity F µν ··· containing ζ is taken as δF µν ··· ( ζ ; Ψ) ≡ F µν ··· ( ζ + δζ ; Ψ + δ Ψ) − F µν ··· ( ζ ; Ψ). For instance, the Killing conditions for the background field Ψ and thevaried field Ψ + δ Ψ are given respectively by £ ξ Ψ = 0 and £ ξ + δξ (Ψ + δ Ψ) = 0. When adiffeomorphism parameter is transformed under a variation such that δζ µ = 0, one needs tomodify the commutation of two generic variations as( δδ ζ − δ ζ δ )Ψ = δ δζ Ψ , ( δδ ζ − δ ζ δ ) I [Ψ] = δ δζ I [Ψ] . ξ B , one can see that( δ ξ B δ − δδ ξ B ) I onr [Ψ B ] = 116 πG Z d d x δ ξ B h √− γ (cid:16) T ijB δγ ij + Π ψ δψ (cid:17)i = 116 πG Z d d x ∂ i h ξ iB √− γ (cid:16) T klB δγ kl + Π ψ δψ (cid:17)i , (A.1)where we have used δ ξ B Ψ B = 0 and thus δ ξ B I onr [Ψ B ] = 0 in the first equality and δ ξ B = £ ξ B inthe second equality. The variation with respect to δξ iB can be written as δ δξ B I onr [Ψ B ] = 116 πG Z d d x √− γ (cid:16) − T Bij ∇ i δξ jB + Π ψ £ ξ B ψ (cid:17) = 116 πG Z d d x ∂ i (cid:16) − √− γ T iB j δξ jB (cid:17) , (A.2)where we have used the identity Eq. (2) in the second equality. By identifying Eq. (A.1) andEq. (A.2), one can finally see that ∇ i (cid:20) T iB j δξ jB + 12 ξ iB (cid:16) T klB δγ kl + Π ψ δψ (cid:17)(cid:21) = 0 . (A.3) References [1] V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun.Math. Phys. , 413 (1999) [hep-th/9902121].[2] J. D. Brown and J. W. York, Jr., “Quasilocal energy and conserved charges derived fromthe gravitational action,” Phys. Rev. D , 1407 (1993) [gr-qc/9209012].[3] S. Hollands, A. Ishibashi and D. Marolf, “Comparison between various notions of con-served charges in asymptotically AdS-spacetimes,” Class. Quant. Grav. , 2881 (2005)[hep-th/0503045].[4] I. Papadimitriou and K. Skenderis, “Thermodynamics of asymptotically locally AdS space-times,” JHEP , 004 (2005) [hep-th/0505190].[5] S. Hollands, A. Ishibashi and D. Marolf, “Counter-term charges generate bulk symmetries,”Phys. Rev. D , 104025 (2005) [hep-th/0503105].[6] A. M. Awad and C. V. Johnson, “Holographic stress tensors for Kerr - AdS black holes,”Phys. Rev. D , 084025 (2000) [hep-th/9910040].[7] G. W. Gibbons, M. J. Perry and C. N. Pope, “AdS/CFT Casimir energy for rotating blackholes,” Phys. Rev. Lett. , 231601 (2005) [hep-th/0507034].108] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic reconstruction of space-timeand renormalization in the AdS / CFT correspondence,” Commun. Math. Phys. , 595(2001) [hep-th/0002230].[9] M. Bianchi, D. Z. Freedman and K. Skenderis, “Holographic renormalization,” Nucl. Phys.B , 159 (2002) [hep-th/0112119].[10] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D , 3427 (1993)[gr-qc/9307038].[11] W. Kim, S. Kulkarni and S. H. Yi, “Quasilocal Conserved Charges in a Covariant Theoryof Gravity,” Phys. Rev. Lett. , no. 8, 081101 (2013) [arXiv:1306.2138 [hep-th]].[12] W. Kim, S. Kulkarni and S. H. Yi, “Quasilocal conserved charges in the presence of agravitational Chern-Simons term,” Phys. Rev. D , no. 12, 124004 (2013) [arXiv:1310.1739[hep-th]].[13] S. Hyun, S. A. Park and S. H. Yi, “Quasi-local charges and asymptotic symmetry genera-tors,” JHEP , 151 (2014) [arXiv:1403.2196 [hep-th]].[14] Y. Gim, W. Kim and S. H. Yi, “The first law of thermodynamics in Lifshitz black holesrevisited,” JHEP , 002 (2014) [arXiv:1403.4704 [hep-th]].[15] S. Hyun, J. Jeong, S. A. Park and S. H. Yi, “Quasi-local conserved charges and holography,”arXiv:1406.7101 [hep-th].[16] L. F. Abbott and S. Deser, “Stability of Gravity with a Cosmological Constant,” Nucl.Phys. B , 76 (1982).[17] L. F. Abbott and S. Deser, “Charge Definition in Nonabelian Gauge Theories,” Phys. Lett.B , 259 (1982).[18] S. Deser and B. Tekin, “Gravitational energy in quadratic curvature gravities,” Phys. Rev.Lett. , 101101 (2002) [hep-th/0205318].[19] S. Deser and B. Tekin, “Energy in generic higher curvature gravity theories,” Phys. Rev.D , 084009 (2003) [hep-th/0212292].[20] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. , 5849(2002) [hep-th/0209067].[21] J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. , 725 (1990).1122] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamicalblack hole entropy,” Phys. Rev. D , 846 (1994) [gr-qc/9403028].[23] G. Barnich and G. Compere, “Generalized Smarr relation for Kerr AdS black holes fromimproved surface integrals,” Phys. Rev. D , 044016 (2005) [Erratum-ibid. D , 029904(2006)] [gr-qc/0412029].[24] S. W. Hawking, C. J. Hunter and M. Taylor, “Rotation and the AdS / CFT correspon-dence,” Phys. Rev. D59