Mass in anti-de Sitter spaces

PHYSICAL REVIEW D 72, 064021 (2005)

Mass in anti-de Sitter spaces

James T. Liu*Michigan Center for Theoretical Physics, Randall Laboratory of Physics, The University of Michigan,

Ann Arbor, Michigan 48109–1040, USA

W. A. Sabra†

Center for Advanced Mathematical Sciences (CAMS), Physics Department, American University of Beirut, Lebanon(Received 8 June 2005; published 29 September 2005)

*Electronic†Electronic

1550-7998=20

The boundary stress tensor approach has proven extremely useful in defining mass and angularmomentum in asymptotically anti-de Sitter spaces with CFT duals. An integral part of this method isthe use of boundary counterterms to regulate the gravitational action and stress tensor. In the presence ofmatter, however, ambiguities may arise that are related to the addition of possible finite counterterms. Wedemonstrate this explicitly for R-charged black holes in AdS5, where introduction of a finite countertermproportional to �2 is necessary to properly reproduce the expected mass/charge relation for the blackholes.

DOI: 10.1103/PhysRevD.72.064021 PACS numbers: 04.70.Bw

I. INTRODUCTION

While the notion of mass is perhaps intuitively obvious,much of this intuition is related to flat space, where massmay be used to label representations of the Poincare group.Once we consider curved space, some of this intuition fallsapart. Of course, the idea of mass as a source of curvature isan essential component of general relativity. Nevertheless,in the absence of Poincare symmetry, mass can no longerbe defined in a straightforward manner.

In fact, in a closed spacetime, there can be no intrinsicmeaning to the mass of the Universe in much the same wayas there cannot be any net charge in a closed space. On theother hand, there has been a long history of defining massfor spaces with an asymptotic region. Perhaps one of thebest known prescriptions is that of Arnowitt, Deser andMisner (ADM), which may be most straightforwardlyapplied in asymptotically flat spacetimes. This is essen-tially equivalent to reading off the mass from theNewtonian potential, ��r� � �M=r, where ��r� may beextracted from the time-time component of the metric,gtt ��1� 2��r��.

In general, the ADM prescription can also be applied tospacetimes with nonflat asymptotic regions, such asasymptotically anti-de Sitter (AdS) spaces [1]. In suchcases, the mass may be extracted by comparison to areference (e.g. vacuum AdS) background. However, caremust be taken to ensure that the deviation from the refer-ence background is sufficiently well controlled. This task isoften made difficult in practice because one must controlthe reparametrization invariance of both deformed andundeformed backgrounds to ensure a well defined result.A similar approach to mass has been taken by Brown and

address: [email protected]: [email protected]

05=72(6)=064021(9)$23.00 064021

York [2] in defining a quasilocal stress tensor through thevariation of the gravitational action

Tab �2��hp

�S�hab

; (1)

where hab is the boundary metric. In general, Tab divergesas the boundary is pushed off to infinity, and hence abackground subtraction is again necessary.

More recently, an alternative procedure has been dem-onstrated where the boundary stress tensor may be regu-lated by the introduction of appropriate boundarycounterterms [3–5]. The advantage of this method is thatthe regulated gravitational action and resulting boundarystress tensor may be obtained directly for the backgroundat hand, without having to introduce a somewhat artificialreference background. This counterterm method has be-come quite standard when applied to AdS/CFT, as theboundary counterterms have a natural interpretation asconventional field theory counterterms that show up inthe dual CFT.

In general, it is only necessary to introduce a handful ofboundary counterterms in order to cancel divergences inthe gravitational action. For example, in AdS5, only twocounterterms are necessary. However, one could equallywell add in an arbitrary amount of finite counterterms.While this would certainly change the values of the actionintegral and corresponding boundary stress tensor, this hasa natural interpretation in the dual CFT as simply the usualfreedom to change renormalization prescriptions.

Although one is in principle free to choose any desiredprescription, some are perhaps better motivated thanothers. For example, in a gauge theory, one tends to avoidnongauge invariant regulators, and in supersymmetrictheories, one generally chooses a ‘‘supersymmetric‘‘scheme. While the introduction of finite counterterms has

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.064021

JAMES T. LIU AND W. A. SABRA PHYSICAL REVIEW D 72, 064021 (2005)

often been overlooked, this can lead to somewhat surpris-ing results. In particular, it was shown in [6] that, in theabsence of finite counterterms, single R-charged blackholes in AdS5 obey a nonlinear mass/charge relation, M�32�� q�

13 �q=‘�

2, where� is the nonextremality parame-ter and ‘ is the AdS ‘‘radius.’’ While nothing prevents usfrom taking this as a definition of mass, it neverthelessappears to be in conflict with the Bogomol’nyi-Prasad-Sommerfield (BPS) expectation that M � jqj.

In this paper, we demonstrate that the inclusion of afinite counterterm related to the scalar fields will recoverthe expected linear relation M� 3

2�� q. Such mattercounterterms have been analyzed previously in [7–9],and are in general important for rendering the mattercoupled theory finite. In the present case, however, theappropriate �2 counterterm is finite for the black holesof [6]. We also show that for three-charge AdS5 black holesin the STU model, the mass/charge relation remains linear,namely M� 3

2�� q1 � q2 � q3, once the �2 counter-term is included. The boundary stress tensor method canalso be applied to the newly constructed Gutowski-Reallblack holes in AdS5 [10,11]. We compute the massesof these solutions and demonstrate equivalence with theresults obtained in [11] using the Ashtekar and Das ap-proach [12].

Recently, black hole masses in asymptotically AdSspacetimes were computed using a Hamiltonian formalismin [13] (see also [14]), where this extra finite scalar coun-terterm was also obtained for scalars saturating theBreitenlohner-Freedman bound. Under this formalism,the counterterm arose by demanding AdS-invariant bound-ary conditions, while in the present analysis it may beviewed as a requirement of preserving supersymmetry(i.e. the BPS condition). In both cases, these results suggestthat a proper choice of the finite counterterms correspondsto using a holographic renormalization prescription whichrespects the underlying symmetry.

We begin in Sec. II with a review of the boundarycounterterm procedure. While this is by now familiar, wefind it useful here to set the notation and prepare thegroundwork for the subsequent calculations. In Sec. IIIwe include matter fields (general scalars and vectors) andin Sec. IV we complete the regulation procedure by in-troducing a finite �2 counterterm. We verify in Sec. V thatthis counterterm results in the linear mass relation men-tioned above. Finally, we examine the Gutowski-Reallblack holes in Sec. VI and conclude in Sec. VII.

II. THE STRESS TENSOR FOR PURE GRAVITY

Before considering the matter coupled system, webriefly review the boundary counterterm method for apurely gravitational theory [3–5]. We work in five dimen-sions with a negative cosmological constant, so that theEinstein action may be written as

064021

S�g�� Sbulk � SGH

� �1

16�G5

ZMd5x

��gp

�R�

12

‘2

�

�1

8�G5

Z@M

d4x��hp

�; (2)

where ‘ is the radius of AdS5. The Gibbons-Hawkingsurface term is included to ensure a proper variationalprinciple for a spacetime M with boundary @M. Here,� is the trace of the extrinsic curvature �� of the bound-ary, defined by

�� 1

2�r�n� �r�n��; (3)

where n� is the outward-pointing normal on @M.In the holographic context, it is natural to single out a

radial coordinate r, and thus we decompose the bulk five-dimensional metric according to

ds25 � N2dr2 � hab�dxa � Vadr��dxb � Vbdr�: (4)

This is essentially an ADM decomposition, except thathere the radial coordinate r plays the role of time.Furthermore we will choose r so that the boundary @Mis reached as r! 1. The four-dimensional metric hab thenrepresents the induced metric on @M. Following [2], thequasilocal stress tensor on the surface @M is then definedthrough the variation of the gravitational action with re-spect to the boundary metric hab

Tab �2��hp

�S�hab

�1

8�G5��ab ��hab�: (5)

Given Tab, it is possible to extract the ADM mass andmomentum as appropriate conserved quantities. To do so,we foliate the boundary spacetime @M by spacelike sur-faces � with metric ��, so that

ds24 habdx

adxb

� �N2�dt

2 � ��dx� � V��dt��dx

� � V��dt�: (6)

The conserved charges are then obtained by integrating thetime component of the conserved stress tensor over thethree-dimensional surface �. More precisely, for an isome-try of the boundary geometry generated by a Killing vectora, the corresponding conserved charge is given by

Q �Z

�dx3

��p�uaTabb�; (7)

where ua is the timelike unit normal to the surface �. For atime-translationally invariant spacetime, we take theKilling vector to be a � N�ua, in which case the con-served charge Q corresponds to the total energy of thespacetime.

In general, it can be shown that the stress tensor definedin this matter (as well as the on-shell value of the action)diverges when the surface @M is pushed to infinity. While

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MASS IN ANTI-DE SITTER SPACES PHYSICAL REVIEW D 72, 064021 (2005)

Brown and York [2] remove this divergence through back-ground subtraction, the method of Refs. [3–5] is to insteadregulate the action (2) through the addition of boundarycounterterms, Sct�hab. This also yields a counterterm ad-dition to the stress tensor,

Tabreg �1

8�G5��ab ��hab� �

2��hp

�Sct�hab

: (8)

Only two counterterms, of the forms

S1 �1

8�G5

Zd4x

��hp

; S2 �1

8�G5

Zd4x

��hp

R;

(9)

are necessary for regulating the divergences of the gravi-tational action, (2). Here R is the scalar curvature of theboundary metric (6). The resulting action has the form

Sreg � Sbulk � SGH �3

‘S1 �

‘4S2: (10)

In addition, the counterterms contribute

Tab1 �1

8�G5hab; Tab2 �

1

8�G5�2Rab �Rhab�;

(11)

to the regulated stress tensor.

A. Mass of the Schwarzschild-AdS spacetimes

To illustrate the above general discussion, we review thecase of Schwarzschild-AdS5, which has attracted muchprevious attention as the spacetime corresponding to non-extremal D3-branes. In five dimensions, the metric may bewritten as

ds2 � �f�r�dt2 �dr2

f�r�� r2d�2

3; (12)

where f�r� � 1� �r0=r�2 � �r=‘�2. We implicitly definer� to be the location of the horizon, given by f�r�� 0.

When evaluated on shell, the bulk action may be reex-pressed in terms of a surface integral. We find

Ibulk ��!3

8�G5�r2�f� 1� � r2

��; (13)

where we use I to denote the value of the Euclidean actionintegral. Here, � � 2�=T is the periodicity along the timedirection and !3 � 2�2 is the volume of the unit 3-sphere.Note that, in the absence of matter, the action integral (13)is easily obtained through the substitution of the trace ofthe Einstein equation, R � �20=‘2, into Sbulk to obtain

Ibulk � �1

16�G5

Zd5x

��gp

��

8

‘2

�

��!3

8�G5‘2 �r

4 � r4��; (14)

064021

which is equivalent to (13) when the expression for f�r� istaken into account. However, as shown in the followingsection, the expression (13) is more general, and continuesto hold when matter is added to the system.

In addition to Ibulk, the Gibbons-Hawking boundary termgives the contribution

IGH � ��!3

8�G5

�1

2r3f0 � 3r2f

�: (15)

Thus the complete action is given by

IGH � Ibulk ��!3

8�G5

��3r2 � r2

� �3r4

‘2 � r20

�; (16)

where we have substituted in the explicit form of f.While the on-shell action diverges like r4 as we ap-

proach the boundary r! 1, this divergence is removedby the addition of the counterterms (9). The appropriatelyregulated action (10) is given by [4,5]

Ireg ��4G5

�r2� �

1

2r2

0 �3‘2

8

�; (17)

and remains finite. Likewise, the counterterms also lead toa finite stress tensor. Using (8) and (11), one finds

Ttt �1

8�G5r2

�3‘8�

3r20

2‘

�; (18)

resulting the familiar Schwarzschild-AdS5 energy [4]

E �3�8G5

�r2

0 �‘2

4

�; (19)

which naturally includes the CFT Casimir energy in addi-tion to the nonextremality parameter r0.

III. ADDITION OF THE MATTER SECTOR

In order to examine the mass of charged black holesolutions, we must first extend the standard countertermprocedure by introducing a matter sector to the bulk action.Since the action is no longer that of pure gravity, it maynow be necessary to include additional local countertermson @M constructed out of the boundary values of thematter fields in order to cancel all divergences.

Although we eventually turn to solutions of gaugedN � 2 supergravity in five dimensions, we first considerthe a general matter coupled gravity system with action

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S�g��;�i; AI� � �1

16�G5

ZMd5x

��gp

�R�

1

2g��ij@��i@��j �

1

4GIJ��FI��F��J � V��

�

�1

8�G5

Z@M

d4x��hp

�: (20)

To evaluate the on-shell value of the bulk action, we note that the Einstein equation, written in Ricci form, is given by

R�� 1

2g��ij@��i@��j �

1

2GIJ

�FI�F

J� �

1

6g��FI��F��J

��

1

3g��V: (21)

Taking the trace of this equation to obtain R, and substitut-ing it into the action integral gives

Ibulk � �1

16�G5

Zd5x

��gp

��

1

6GIJFI��F��J �

2

3V�:

(22)

While this is a simplification of the action integral, itappears to be as far as we may proceed without furtherinput. Thus we now focus on static electrically chargedblack hole solutions, and take an ansatz of the form

ds2 � �e�4B�r�f�r�dt2 � e2B�r��dr2

f�r�� r2d�2

3

�;

�i � �i�r�; AIt � AIt �r�;

(23)

where the 3-sphere may be parametrized as

d�23 � d 2 � sin2 d�2

2: (24)

In this case, the R component of the Einstein equation,(21), yields

2R � �1

6GIJFI��F��J �

2

3V; (25)

which has the same form as the integrand of (22). Thisgives a simple result for the action integral

Ibulk � �1

8�G5

Zd5x

��gp

R ; (26)

provided we follow the ansatz (23). Working out the R component explicitly, we obtain

Ibulk ��!3

8�G5

Zdr

ddr�r3fB0 � r2�f� 1�

��!3

8�G5�r3fB0 � r2�f� 1� � r2

��; (27)

where in the last line we have taken the range of r to befrom the horizon r� to the finite but large value rwhere wecut off the space.

To evaluate the Gibbons-Hawking surface term, we startwith the unit normal in the r direction, nr � e�Bf1=2.Evaluating its divergence yields

� � �r�n� � �e�Bf1=2

�B0 �

f0

2f�

3

r

�; (28)

064021

so that

IGH � ��!3

8�G5

�r3fB0 �

1

2r3f0 � 3r2f

�: (29)

Curiously, the B0 dependent terms in Ibulk and IGH cancelwhen added together. We find

IGH � Ibulk ��!3

8�G5

��2r2f�

1

2r3f0 � r2 � r2

�

�: (30)

This expression as it stands is divergent, and must beregulated by an appropriate counterterm subtraction.However, we emphasize that this expression includes alleffects of the scalars and gauge fields of (20), although theydo not show up explicitly here. It is remarkable that theunregulated action only depends explicitly on the ‘‘black-ening function‘‘ f�r� in (23). However, as a solution to theEinstein equation, f naturally includes residual informa-tion of all appropriate scalar and gauge charges.

For the metric ansatz, (23), the boundary countertermsS1 and S2 take the simple form

I1 ��!3

8�G5r3f1=2eB; I2 �

3�!3

4�G5rf1=2e�B; (31)

so that the regulated action integral, (10), is given by

Ireg ��!3

8�G5

��2r2f�

1

2r3f0 � r2 � r2

� � 3‘�1r3f1=2eB

�3

2‘rf1=2e�B

�: (32)

Although this is not manifestly finite, we demonstrateexplicitly that it is indeed so for R-charged black holes inAdS5.

A. R-charged black holes in AdS5

We now turn to the examination of R-charged blackholes. These electrically charged black holes are staticstationary solutions to gauged N � 2 supergravity, andhave a metric of the form [15]

ds2 � �H�2=3fdt2 �H 1=3

�dr2

f� r2d�2

3

�; (33)

where

-4


f � 1�r2

0

r2 �r2

‘2 H : (34)

The ‘‘harmonic function’’ H is related to e2B of the ansatz(23) by H � e6B. In the STU model, H is given by theproduct of three harmonic functions

H � H1H2H3 �

�1�

q1

r2

��1�

q2

r2

��1�

q3

r2

�: (35)

However, in general, we still expect H to have a large rexpansion of the form

H � 1�Q�1�

r2 �Q�2�

r4 �Q�3�

r6� � � � : (36)

It is now straightforward to substitute the metric func-tions f�r� and H �r� into the regulated action integral, (32).Up to terms that vanish in the limit r! 1, we obtain thefinite expression

Ireg ��4G5

�r2� �

1

2r2

0 �3‘2

8�

1

‘2

�Q�1�2

3�Q�2�

��: (37)

This expression is the generalization of (17) to the case ofR-charged black holes, where the charges are given byQ�1� �

Piqi and Q�2� �

Pi<jqiqj. Note that this expres-

sion is obtained directly from the metric (33), without evenspecifying the gauge fields and scalars associated with thesolution.

As can be seen from (37), the black hole charges enternonlinearly in the action integral. In particular, for thesingle charged black hole [Q�2� � 0] this expression re-duces to that derived previously in [6]. Although there isnothing inherently wrong with the nonlinear charge behav-ior, it is somewhat unexpected, especially considering thatit remains nonlinear in the BPS limit. Of course, theseblack holes actually become singular in the limit. Butnevertheless, the formal BPS expression could have beenexpected to hold. Indeed, it turns out that there is a simplemeans of removing the nonlinearity in (37) through theintroduction of finite boundary counterterms. This is whatwe now proceed to demonstrate.

IV. ADDITION OF FINITE COUNTERTERMS

In general, boundary counterterms have been introducedas a means of regulating divergences in the gravitationalaction. However, we wish to emphasize here that nothingprevents us from introducing finite counterterms as well.Such expressions yield a finite renormalization of thegravitational action, and are hence dual to finite shifts inthe renormalization of the CFT. As a result, they may beviewed as generating shifts between different renormaliza-tion prescriptions of the CFT.

Since we have introduced the matter fields �i and AI�into the action (20), it is natural to construct local boundarycounterterms such as

064021

S�2 �1

8�G5

Z@M

d4x��hp

gij�i�j;

S@�2 �1

8�G5

Z@M

d4x��hp

gij@a�i@a�j;

SF2 �1

8�G5

Z@M

d4x��hp

GIJFIabFabJ;

(38)

where the a, b indices correspond to the boundary surface@M. In particular, radial @r derivatives are absent, as theyare not local to the boundary.

For the spherically symmetric black holes, the fields areonly functions of the radial coordinate r. Hence the two-derivative counterterms in (38) will not contribute. As aresult, we consider only I�2 , which takes the form

I�2 ��!3

8�G5r3f1=2eB�gij�

i�j�: (39)

So far, the analysis has been completely general, at least forthis class of spherically symmetric and stationary solu-tions. However, at this stage, it is necessary to providethe explicit asymptotic form of the scalars correspondingto the black hole metric of (33).

To proceed, we consider the specific example of the STUmodel. Here, there are three U�1� gauge fields and twoscalars, with the scalars defined by

X1 � e��1=��6p��2=

��2p

� H�11 H 1=3;

X2 � e��1=��6p��2=

��2p

� H�12 H 1=3;

X3 � e2�1=��6p

� H�13 H 1=3;

(40)

where H � H1H2H3, so that X1X2X3 � 1. The two in-dependent scalars �1 and �2 may be reexpressed as

�1 �1��6p �logH1 � logH2 � 2 logH3�; (41)

�2 �1��2p �logH1 � logH2�: (42)

This gives in turn

~� 2 � �21 ��

22 �

1

r4

�2

3Q�1�2 � 2Q�2�

�� : (43)

Substituting this into (39) yields a finite contribution

I�2 ��

4G5‘

�2

3Q�1�2 � 2Q�2�

�; (44)

which has the exact same charge dependence as the finitepart of the action, (37). As a result, it may be used as a finitecounterterm to completely cancel the charge dependenceof the action. The regulated action integral, including finitecounterterm

Ireg �1

2‘I�2 �

��4G5

�r2� �

1

2r2

0 �3‘2

8

�; (45)

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is then identical to that of the Schwarzschild-AdS solution,(17). In fact, we are advocating the use of the full counter-term action

Icomplete � Ibulk � IGH �3

‘I1 �

‘4I2 �

1

2‘I�2 ; (46)

for black holes in AdS5 with or without R charge. For thelatter, of course, the I�2 counterterm vanishes. However,we may view this counterterm action as universal, with allcoefficients independent of charge.

V. THE REGULATED BOUNDARY STRESSTENSOR

In the previous section, we have shown that an appro-priate counterterm prescription exists for five-dimensionalR-charged black holes that preserves the standard expres-sion (17) for the action integral independent of charge. Wenow turn to the calculation of the boundary stress tensorand the extraction of the ADM energy.

We begin with the unregulated stress tensor, (5), givenby

Tab �1

8�G5��ab ��hab�: (47)

For the metric (23), the extrinsic curvature takes the form

�tt � �

��2B0 �

f0

2f

�htte�Bf1=2;

��

�B0 �

1

r

�h��e�Bf1=2;

(48)

so that

� � ��B0 �

3

r�f0

2f

�e�Bf1=2: (49)

Substituting these expressions into (47) gives

Ttt �1

8�G5‘

��

3r2

‘2 �Q�1�

‘2 �9

2

�1

r2

�9r2

0

2� 3Q�1� �

9‘2

8

��: (50)

At the same time, the local gravitational counterterms, S1

and S2, give rise to the contribution

~Ttt � �1

8�G5‘3gtt

�1�

‘2e�2B

2r2

�

�1

8�G5‘

�3r2

‘2 �Q�1�

‘2 �9

2�

1

r2

��2Q�1� � 3r2

0

�1

‘2

�Q�2� �

1

3Q�1�2

��

3‘2

2

��; (51)

so that the gravitationally regulated value of Ttt is

064021

Tregtt �

1

8�G5‘r2

�Q�1� �

3r20

2�

1

‘2

�Q�2� �

Q�1�2

3

��

3‘2

8

�:

(52)

While this expression yields a finite energy when insertedin (7), the term quadratic in charge gives rise to a nonlinearmass/charge relation, as first noted in [6]. In fact, settingQ�2� � 0 reproduces the single-charge black hole resultof [6].

Of course, the introduction of the finite counterterm S�2

also shifts the stress tensor according to

Tab�2 �

1

8�G5hab�gij�i�j�: (53)

For the STU model, the evaluation of �gij�i�j� followsfrom (43). Including Tab�2 results in cancellation of the

nonlinear charge term, so that the fully regulated value ofTtt takes on the simple form

Tcompletett �

1

8�G5‘r2

�Q�1� �

3r20

2�

3‘2

8

�: (54)

Therefore the energy is given by

E ��

4G5

�Q�1� �

3r20

2�

3‘2

8

�

�3�8G5

�r2

0 �2

3q1 �

2

3q2 �

2

3q3 �

‘2

4

�; (55)

where in the last line we have explicitly written out thethree charges of the STU model. This energy generalizesthe expression (19) to the case of charged black holes inAdS5.

By adding a finite counterterm, I�2 , we have been able toprovide a rigorous justification of the AdS5 black holemass originally given in [15]. We believe this expressionis natural from the viewpoint of thermodynamics, in thatthe three independent charges of the STU model (or equiv-alently the three commuting U�1�3 � SO�6�R charges ofthe four-dimensional N � 4 theory) contribute linearly tothe mass in (55). Note that the nonlinear term Q�2� �Q�1�2=3 vanishes identically for the three equal chargeblack hole. In this case, either mass expression yields thesame result. In fact, this must be true; since this black holemay be viewed as a solution of the pure five-dimensionalN � 2 supergravity, and there are no scalars in thistheory, the scalar counterterm cannot contribute to themass.

VI. GUTOWSKI-REALL SOLUTIONS

Another example where the importance of the S�2 coun-terterm shows up is in the case of the recently constructedGutowski-Reall supersymmetric black hole solutions[10,11]. Unlike the stationary R-charged black holes in-vestigated above which become singular in the BPS limit,

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the Gutowski-Reall solutions maintain a regular horizonthrough nonzero angular momentum.

The general rotating solution has a metric of the form[11]

ds2 � �f2dt2 � 2f2wdt�3L � f

�1g�1dr2

�r2

4�f�1��1

L�2 � ��2

L�2� � f2h��3

L�2; (56)

where the functions f, g, w and h are

f ��1�

�1

r2 ��2

r4 ��3

r6

��1=3

; g ��1�

�1

‘2 �r2

‘2

�;

w � ��r2

2‘

�1�

�1

r2 ��2

2r4

�; h � f�3g�

4

r2 w2:

(57)

Here, �iL are right-invariant 1-forms on SU(2) given by

�1L � sin�d � cos� sin d ;

�2L � cos�d � sin� sin d ;

�3L � d�� cos d :

(58)

In addition, the scalars and gauge fields are given generallyby

XI � f�

�XI �qIr2

�; AI � fXIdt� �UI � fwXI��3

L;

UI �9�4‘CIJK �XJ� �XKr2 � 2qK�: (59)

These expressions simplify for the STU model, in whichcase

f � �H1H2H3��1=3; XI �

1

3HIf; HI � 1�

�I

r2 ;

(60)

and

UI ��2‘�r2 � �1 ��I�: (61)

Note that

�1 � �1 ��2 ��3; �2 � �1�2 ��1�3 ��2�3;

�3 � �1�2�3; (62)

are analogous to the Q�i� of (36).Even in the STU model, where the scalar and gauge field

behavior is explicit, the analysis is somewhat complicatedby rotation. Foliating the boundary metric according to (6),we first rewrite (56) as

ds2 � �gfhdt2 �

dr2

fg�r2

4f

��1

L�2 � ��2

L�2

� f3h��3L �

4

r2

whdt�

2�: (63)

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This then allows us to introduce a natural vielbein basis

e0 � g1=2�fh��1=2dt; e4 � �fg��1=2dr;

e1 �r2f�1=2�1

L; e2 �r2f�1=2�2

L;

e3 �r2fh1=2

��3L �

4

r2

whdt�:

(64)

Given this solution, we may compute the regulated actionintegral (46) as well as the corresponding boundary stresstensor.

A. The action integral

While computation of the bulk action is in principlestraightforward, the simplification of (26) no longer fol-lows due to the rotation. In addition, the

RF ^ F ^ A term

of gauged supergravity contributes for these rotating solu-tions [19]. Starting from

Sbulk � �1

16�G5

ZMd5x

��gp

�R�

1

2GIJ@�X

I@�Xj

�1

4GIJF

I��F

��J � 2‘�2V

�

�1

16�G5

ZM

1

6CIJKFI ^ FJ ^ AK; (65)

we may use both the trace and the R11 component of theEinstein equation to rewrite the on-shell action as

Ibulk � �1

8�G5

Zd5x

��gp

�R11 �

1

2GIJFI12F

J12

�

�1

16�G5

Z 1

6CIJKF

I ^ FJ ^ AK; (66)

where we have also used the fact that the only nonvanish-ing vielbein components of the field strength are FI04, FI12and FI34.

For the STU model, the Ricci component R11 may bewritten as

R11 �f

r3

�ddr

�1

2r3g

f0

f� r2�2� g�

�� 2rhf3

�; (67)

while the gauge field term has the form

GIJFI12FJ12 �

f4

‘2r8 �3�22 � 8�1�3�: (68)

The Chern-Simons term is somewhat more lengthy. Wefind

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1

6CIJKF

I ^ FJ ^ AK � �1

‘2

��2

1��2 ��3�2 ��2

2�23

�r2 ��1��1 ��2��1 ��3��1��2

1��2 ��3�2 � 2�2

2�23 � �3��2 ��3�

3�r2 ��1�2��1 ��2��1 ��3�

� ��1 $ �2� � ��1 $ �3�

�dt ^ dr ^!3; (69)

where !3 �18�

1L ^ �

2L ^ �

3L is the volume form on the

unit 3-sphere.Combining these expressions and integrating from the

horizon at r � 0 to a large radial value r yields

Ibulk ��!3

8�G5

�r4

‘2 �2�1r2

3‘2 ��1

3��2

6‘2

�: (70)

In anticipation of the computation of the boundary stresstensor, we find that the nonvanishing components of theextrinsic curvature are

00 �1

2�fg�1=2

��f0

f�g0

g�h0

h

�;

03 �f2wr

�2

r�h0

h�w0

w

�;

11 � 22 �1

2�fg�1=2

��

2

r�f0

f

�;

33 �1

2�fg�1=2

��

2

r�

2f0

f�h0

h

�;

(71)

so that

�1

2�fg�1=2

��

6

r�f0

f�g0

g

�: (72)

This allows us to compute the Gibbons-Hawking term

IGH ��!3

8�G5

��

4r4

‘2 � 3r2 �8�1r

2

3‘2 ��1

3�

2�2

3‘2

�: (73)

For I2, we also need the intrinsic curvature on the bound-ary, R � 2

r2 f�4� hf3�. This allows us to obtain the coun-terterms

3

‘I1 �

�!3

8�G5

�3r4

‘2 �3

2r2 �

2�1r2

‘2 �3‘2

8��1

2��2

1

3‘2

��2

2‘2

�;

‘4I2 �

�!3

8�G5

�3

2r2 �

3‘2

4��1

2

�;

1

2‘I�2 �

�!3

8�G5

��2

1

3‘2 ��2

‘2

�:

(74)

Adding the above contributions according to (46) finallyyields the simple expression

Icomplete ��4G5

�3

8‘2

�: (75)

Note that this result matches that of (45) for the static black

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holes in the extremal limit (r0 � 0), provided we setr� � 0 in (45).

B. The boundary stress tensor

We now proceed to compute the boundary stress tensorand to extract the ADM energy and angular momentum ofthis solution. In fact, the result is rather simple, and we find

Tcomplete00 �

1

8�G5

‘

r4

�3

8‘2 � �1 �

3�2

2‘2 �2�3

‘4

�;

Tcomplete03 �

1

8�G5

�

‘r4

��2 �

2�3

‘2

�:

(76)

In addition, T11 � T22 and T33 are nonvanishing, but do notcontribute to conserved quantities.

Taking into account (7), the conserved ADM energy andangular momentum are

E ��

4G5

�3

8‘2 � �1 �

3�2

2‘2 �2�3

‘4

�;

J ��

8G5‘

��2 �

2�3

‘2

�:

(77)

These expressions agree with those obtained by Gutowskiand Reall [11] using the methods of Ashtekar and Das [12],provided one relates the ADM energy E and the Ashtekarand Das mass M through

E � M�3�‘2

32G5: (78)

The latter contribution is identified as the Casimir energy,and verifies the prediction of Gutowski and Reall.

VII. DISCUSSION

Computing black hole energies using the boundarystress tensor method is natural in the AdS/CFT context.What we have shown here is that, by incorporating a �2

counterterm, we are able to derive the expected ADMenergies for the nonrotating R-charged black holes, (55),and the rotating BPS solutions, (77). In the former case,this finite counterterm removes a nonlinear charge contri-bution to the energy, while in the latter case, it modifies butdoes not remove the nonlinearities.

For the case of the nonrotating black holes, the linearmass relation (55) verifies the result of [15]. As this was thebasis of the thermodynamic exploration of R-chargedblack holes in [16], we have shown that the standard resultsfollow naturally from the boundary stress tensor prescrip-tion, provided appropriate finite counterterms are incorpo-

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rated. The mass of rotating Einstein-Maxwell AdS5 blackholes was also examined using the boundary stress tensormethod in [17]. The result of [17] ought to be generalizableto the STU model after inclusion of the appropriate �2

counterterm.Of course, as we have indicated, the energy computed in

this manner is not unique, and depends on the nature offinite counterterms used in regulating the boundary stresstensor. This fact is understood in terms of having to specifya particular counterterm prescription with which to workwith; in a field theory language, this is simply the schemedependence of standard renormalization. Although theenergy, as so defined, is ambiguous up to finite counter-terms, physical quantities in the dual field theory mustalways be well defined. However, in practice, what is andis not scheme dependent is often a subtle issue, and sepa-rating the two may require care.

In order to deal with this ambiguity, it is natural toimpose some additional symmetry requirements on theregularization procedure. In the present case, our desireto expose a linear BPS-like relation between mass andR-charge in the dual CFT has led us to the addition of

064021

the finite �2 counterterm in (46). In fact, such a counter-term can be motivated by Hamilton-Jacobi theory, and canbe seen as a necessity for the preservation of supersymme-try in the boundary theory (or, equivalently, asymptoticsymmetries in the bulk theory [13]). Note, also, that for thecase of AdS4 with scalars, the �2 counterterm is no longeroptional, but necessary to render the action finite. Thisconnection between the counterterm action and the pres-ervation of symmetries is examined in [18] using theHamilton-Jacobi approach for matter coupled gravitysystems.

ACKNOWLEDGMENTS

This material is based upon work supported by theNational Science Foundation under Grant No. PHY-0313416 and by the US Department of Energy underGrant No. DE-FG02-95ER40899. J. T. L wishes to thankA. Batrachenko, A. Buchel, R. McNees, L. Pando Zayasand W. Y. Wen for discussions, and acknowledges thehospitality of Khuri lab at the Rockefeller University,where part of this work was completed.

[1] L. F. Abbott and S. Deser, Nucl. Phys. B195, 76(1982).

[2] J. D. Brown and J. W. York Jr., Phys. Rev. D 47, 1407(1993).

[3] M. Henningson and K. Skenderis, J. High Energy Phys. 07(1998) 023.

[4] V. Balasubramanian and P. Kraus, Commun. Math. Phys.208, 413 (1999).

[5] R. Emparan, C. V. Johnson, and R. C. Myers, Phys. Rev. D60, 104001 (1999).

[6] A. Buchel and L. A. Pando Zayas, Phys. Rev. D 68,066012 (2003).

[7] S. de Haro, S. N. Solodukhin, and K. Skenderis, Commun.Math. Phys. 217, 595 (2001).

[8] M. Bianchi, D. Z. Freedman, and K. Skenderis, J. HighEnergy Phys. 08 (2001) 041.

[9] M. Bianchi, D. Z. Freedman, and K. Skenderis, Nucl.Phys. B631, 159 (2002).

[10] J. B. Gutowski and H. S. Reall, J. High Energy Phys. 02(2004) 006.

[11] J. B. Gutowski and H. S. Reall, J. High Energy Phys. 04(2004) 048.

[12] A. Ashtekar and S. Das, Classical Quantum Gravity 17,L17 (2000).

[13] T. Hertog and K. Maeda, J. High Energy Phys. 07 (2004)051.

[14] T. Hertog, G. T. Horowitz, and K. Maeda, Phys. Rev. D 69,105001 (2004).

[15] K. Behrndt, M. Cvetic, and W. A. Sabra, Nucl. Phys.B553, 317 (1999).

[16] M. Cvetic and S. S. Gubser, J. High Energy Phys. 04(1999) 024.

[17] D. Klemm and W. A. Sabra, J. High Energy Phys. 02(2001) 031.

[18] A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra, andW. Y. Wen, J. High Energy Phys. 05 (2005) 034.

[19] We are grateful to H. Kunduri for pointing out the im-portance of this Chern-Simons term for the Gutowski-Reall solutions. This was overlooked in an earlier versionof the computation.

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Mass in anti-de Sitter spaces