Thermodynamics and stability of flat anti–de Sitter black strings

Thermodynamics and stability of flat anti–de Sitter black strings

Si Chen,1,* Kristin Schleich,1,2,† and Donald M. Witt1,2,‡

1Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada2Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

(Received 6 October 2008; published 4 December 2008)

We examine the thermodynamics and stability of 5-dimensional flat anti–de Sitter (AdS) black strings,

locally asymptotically anti–de Sitter spacetimes whose spatial sections are AdS black holes with Ricci flat

horizons. We find that there is a phase transition for the flat AdS black string when the AdS soliton string

is chosen as the thermal background. We find that this bulk phase transition corresponds to a 4-

dimensional flat AdS black hole to AdS soliton phase transition on the boundary Karch-Randall branes.

We compute the possibility of a phase transition from a flat AdS black string to a 5-dimensional AdS

soliton and show that, though possible for certain thin black strings, the transition to the AdS soliton string

is preferred. In contrast to the case of the Schwarzschild-AdS black string, we find that the specific heat of

the flat AdS black string is always positive; hence it is thermodynamically stable. We show numerically

that both the flat AdS black string and AdS soliton string are free of a Gregory-Laflamme instability for all

values of the mass parameter. Therefore thermodynamic stability implies perturbative stability for this

spacetime. This may indicate that a generalization of the Gubser-Mitra conjecture, in which the

assumption of a translational killing vector is weakened to that of a conformal killing vector of

translational form, holds under certain conditions.

DOI: 10.1103/PhysRevD.78.126001 PACS numbers: 11.25.Tq, 04.50.Gh, 04.70.Dy

I. INTRODUCTION

The thermodynamics of asymptotically anti–de Sitter(AdS) spacetimes is interesting both in its own right andin its connection to the AdS/CFT correspondence. Asshown by Hawking and Page, black holes in AdS space-times exhibit a first order phase transition to a thermal AdSbackground [1]. However, in marked contrast to theasymptotically flat case, the specific heat of sufficientlylarge black holes is positive in AdS spacetimes. Thereforethe canonical ensemble is well defined for these spacetimeswithout imposition of further conditions. In the context ofAdS/CFT, Witten showed that the Hawking-Page phasetransition corresponds to a deconfinement-confinementphase transition in the large N limit of supersymmetricYang-Mills gauge theory on the conformal boundary [2].Thus the thermodynamics of asymptotically AdS space-times also provides a useful probe in the study of the AdS/CFT correspondence.

The study of the thermodynamics of black strings andblack p-branes, higher dimensional spacetimes whose fo-liation yields a black hole on each spatial hypersurface, hasalso proven informative in the study of braneworlds. Inthese scenarios, an alternative to more traditional Kaluza-Klein compactification, the 4-dimensional universe corre-sponds to a brane embedded in a higher dimensional anti–de Sitter spacetime. The Randall-Sundrum model [3], aparticularly useful example of this scenario, was extended

by Karch and Randall to branes that are AdS slicings,corresponding to the case where the brane tension isslightly detuned from the critical Randall-Sundrum value[4]. This extension provides an advantageous forum inwhich to study the physics of braneworld black holes asthe thermodynamic stability of AdS black holes can beexploited. A first step, taken by Chamblin and Karch,utilized the Karch-Randall (KR) braneworld formulationto study the relation between bulk phase transitions of ablack AdS string and those of the braneworld black hole[5]. They found an exact correspondence between thethermodynamics of the AdS black string in the bulk andthat of the AdS black hole on the brane. The Hawking-Pagephase transition on the brane, corresponding to a CFTphase transition on AdSd�1 weakly coupled to gravity, isdescribed by a phase transition between the black stringand AdSd in the bulk. Furthermore, the temperature atwhich the specific heat of the black hole on the branebecomes negative, leading to black hole evaporation onthe brane to a thermal AdSd�1 state, qualitatively corre-sponds to the onset of the black AdS string instability forSchwarzschild-AdS slicings found by Hirayama andKang [6,7]. This analysis of the thermodynamics ofblack holes on asymptotically AdS branes provides a start-ing point for the further study of properties of blackholes on the brane and their connection to the AdS/CFTcorrespondence, in particular, the holographic conjecturefor black holes on the brane [8] (See [9] for a recentreview).Given the usefulness of this model, it is natural to

explore the connections between bulk and brane blackhole phase transitions for other asymptotically AdS space-

*[email protected]†[email protected]‡[email protected]

PHYSICAL REVIEW D 78, 126001 (2008)

1550-7998=2008=78(12)=126001(12) 126001-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.78.126001

times. A natural starting point is see if this connection canbe extended to AdS black hole solutions whose horizonshave nonspherical topology. The thermodynamics of suchspacetimes and various correspondences in a conformalfield theory on the conformal boundary have been wellstudied. The pioneering work of Vanzo on the thermody-namics of AdS black holes with nonspherical horizons [10]was extended and linked to the AdS/CFT correspondencein [11–13]. Surya, Schleich and Witt found that Ricci flatAdS black holes exhibit a phase transition when the AdSsoliton is chosen as the background solution [14]. Morerecent results include the study of the thermodynamics ofasymptotically AdS black holes of nonspherical topologyin higher derivative gravity [15], Gauss-Bonnet and dilatongravity [16], gravity dual to conformal field theory on aS2 � S1 � R boundary [17], charged Ricci flat AdS blackholes [18] and R-charged black holes [19]. Hence, thethermodynamics of black holes of nonspherical topologyon the brane is well understood. It is therefore natural tosee if the connection of brane and bulk thermodynamicspersists for such AdS black holes. We will do so in thispaper for a key case; we will examine the connectionbetween the thermodynamics of AdS black holes withRicci flat horizons on the brane and their bulk description.This case is especially interesting due to certain novelfeatures of the thermodynamics of flat AdS black holes.

Recall that the thermodynamics of Schwarzschild-AdSblack holes is determined by the black hole temperature Tand the AdS curvature radius l. Unlike the asymptoticallyflat case, the mass and horizon area of the Schwarzschild-AdS black hole are not uniquely specified by temperature;generically there is both a large and small black hole for agiven temperature. The large black hole has positive spe-cific heat and the small one has negative specific heat. Thefree energy of the 4-dimensional black hole relative to that

of thermal AdS is negative for T > 1�l and positive for

ffiffi3

p2�l <

T < 1�l . Hence the black hole is stable for T > 1

�l and is

locally stable with tunnelling possible between it and

thermal AdS forffiffi3

p2�l < T < 1

�l . For T <ffiffi3

p2�l , the black

hole has negative specific heat and the only stable state isthermal AdS spacetime.

Flat AdS black holes, locally asymptotically AdS space-times with Ricci flat horizons, exhibit quantitatively differ-ent behavior. Ricci flat AdS black holes also exhibit aphase transition when the AdS soliton is chosen as thebackground solution [14]. However, unlike theSchwarzschild-AdS case, the stability of these two phasesis determined by two variables, the temperature and hori-zon circumference; hot, very small AdS solitons are stableas are cold, very large flat AdS black holes. Furthermore,the curvature radius l does not characterize the temperatureof the phase transition. Additionally, the specific heat of theflat AdS black hole is always positive. Hence, in contrast tothe Schwarzschild-AdS case, flat AdS black holes arethermodynamically stable for all temperatures.

Given these important differences in the thermodynamicbehavior, it is natural to investigate whether or not theypersist in the thermodynamics of a braneworld scenario inwhich the KR branes contain flat AdS black holes. We doso in Section II. We find that the bulk thermodynamicsagain precisely parallels that of the braneworld. Because ofthe nature of the phase transition in flat AdS black hole andblack string spacetimes, a direct comparison of the phasetransition temperature of a 5-dimensional flat AdS blackhole to that of a flat AdS black string cannot be done.However, we can compute the possibility of a phase tran-sition from a flat AdS black string to a 5-dimensional AdSsoliton and show that, though it can occur for certain thinblack strings, the transition to the AdS soliton string ispreferred. In contrast to the Schwarzschild-AdS stringcase, the specific heat of the flat AdS black string is alwayspositive. Hence we find that that the bulk solution consist-ing of a flat AdS black string is thermodynamically stable.It is consequently interesting to examine the perturbativestability of the flat AdS black string as a generalization ofthe Gubser-Mitra conjecture [20] to the case of spacetimesconformal to those with a translational symmetry wouldimply perturbative stability for this case. We carry out anumerical analysis of this in Section III. We find that, incontrast to the Schwarzschild-AdS case, flat AdS blackstrings are always perturbatively stable. Hence, althoughthe flat AdS black string does not exhibit translationalinvariance, this generalization of the Gubser-Mitra conjec-ture holds.

II. THE THERMODYNAMICS OF FLATASYMPTOTICALLYADS STRING SPACETIMES

Locally asymptotically AdS spacetimes in five dimen-sions with negative cosmological constant foliated by hy-persurfaces of negative constant Ricci curvature can bewritten as

ds2 ¼ H�2ðzÞ�1

l24g��dx

�dx� þ dz2�; (1)

HðzÞ ¼ sinz

l; (2)

where l is the curvature radius of the 5-dimensional locallyasymptotically AdS spacetime and l4 that of the 4-dimensional one. These curvature radii are related to theirrespective cosmological constants by �5 ¼ �6=l2 and�4 ¼ �3=l24.The metric (1) is the bulk solution in a KR braneworld

model [4] in which the brane tension is slightly detunedfrom its critical Randall-Sundrum value. KR branes corre-spond to boundaries of this bulk solution at constant z. Themost general case is that of two branes, at z0 ¼ �=2þ �1and z1 ¼ �=2� �0, both with positive brane tension.These branes excise the boundary at infinity of the locallyasymptotically AdS spacetime; hence each provides a UV

SI CHEN, KRISTIN SCHLEICH, AND DONALD M. WITT PHYSICAL REVIEW D 78, 126001 (2008)

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cutoff CFT that communicate via transparent boundaryconditions. The physical curvature radius of a 4-dimensional KR brane is set by its location; for a braneat �=2þ �i, li ¼ l

cos�i. The special case �1 ¼ �0 ¼ � is of

particular interest. If the orbifold projection z ! �z is alsoimposed, it yields a single AdS brane with brane tension6 sin�8�G and curvature radius l

cos� whose holographic dual has

reflecting boundary conditions.1

Two well-known solutions whose conformal boundaryat infinity admits a Ricci flat metric are the flat AdS blackhole [10] and the AdS soliton [21]. The AdS soliton metriccan be written as

gss��dx�dx� ¼ �r2dt2 þ dr2

VssðrÞ þ VssðrÞd�2 þ r2d�2;

VssðrÞ ¼ r2

l24� k3ss

r; (3)

where �4 ¼ �3=l24. This spacetime is regular everywhere

if r � rssþ where VðrssþÞ ¼ 0 (hence rssþ ¼ kssl2=34 ) and �

is periodic with period

�ss ¼ 4�

�@VssðrÞ@r

��r¼rssþ

��1 ¼ 4�l243rssþ

:

The topology of its conformal boundary at infinity is S1 �R2 or T2 � R if � is identified with period 2�.

The flat AdS black hole metric can be written as2

gbs��dx�dx� ¼ �VbsðrÞdt2 þ dr2

VbsðrÞ þ r2d�2 þ r2d�2;

VbsðrÞ ¼ r2

l24� k3bs

r: (4)

The horizon, at VbsðrbsþÞ ¼ 0 (rbsþ ¼ kbsl2=34 ), has topol-

ogy S1 � R2 if � is identified with period �bs or T2 � R if

� is also periodically identified. The conformal boundaryat infinity has the same topology as the horizon and thus thesame topology as that of the AdS soliton.

The AdS soliton string, the 5-dimensional stringsolution (1) constructed with the AdS soliton (3), is aregular, asymptotically AdS spacetime with topologyD2 � R3 or D2 � S1 � R2 in the respective cases. Theflat AdS black string, that constructed with the flat AdSblack hole (4), has a spacetime topology exterior to thehorizon of S1 � R4 or T2 � R3 respectively. Both solutionsare foliated at large r by a family of timelike surfaces withRicci flat spatial slices of the same topology.

The Wick rotation t ! i� of the AdS soliton stringresults in the Euclidean metric

ds2ss ¼ l2

l24sin2z

�r2d�2 þ dr2

VssðrÞ þ VssðrÞd�2 þ r2d�2

þ l24dz2

�: (5)

Note that regularity places no constraint on � for thismetric; it can be chosen to have any (or no) periodicity.The Wick rotation t ! i�b of the flat AdS black stringresults in

ds2bs ¼l2

l24sin2z

�VbsðrÞd�2 þ dr2

VbsðrÞ þ r2d�2 þ r2d�2

þ l24dz2

�; (6)

where regularity now requires that �b be identified withperiod

bs ¼ 4�

�@VbsðrÞ@r

��r¼rbsþ

��1 ¼ 4�l243rbsþ

:

A second Wick rotation of � ! �it and a relabelling ofcoordinates �b ! �s results in the AdS soliton string. Thusthe same Euclidean instanton results fromWick rotation ofboth the AdS soliton string and the flat AdS black string.The free energy of the flat AdS black string relative to

that of the AdS soliton string determines the possibility of aphase transition between these two spacetimes. The differ-ence in the free energy is proportional to the difference oftheir Euclidean actions. The Euclidean action for Einsteingravity with cosmological constant � in n dimensions is

I¼� 1

16�G

ZMdnx

ffiffiffig

p ðR� 2�Þ� 1

8�G

Z@M

dn�1xffiffiffih

pK;

where h is the induced metric, K the extrinsic curvature ofthe boundary @M and G is the n-dimensional gravitationalconstant. This action diverges for asymptotically AdS so-lutions; therefore it must be computed using a regulariza-tion procedure. We will do so by calculating the action at afinite radius R, find the difference in the actions as afunction of R and then take the limit as R ! 1. Key tothis procedure is to match the induced geometry on thehypersurface at R of the two solutions. This matchingrequires that the periodicities of the coordinates for eachsolution are related by

bs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiVbsðRÞ

q¼ Rss; R�bs ¼ �ss

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVssðRÞ;

q

bs ¼ ss ¼ :(7)

For the case at hand, the difference of boundary terms atfixed R vanishes as R ! 1.3 Thus the action of the flatAdS black string relative to that of the AdS soliton string is

1Note that reflecting boundary conditions are those imposed inthe usual gravitational analysis of Schwarzschild-AdS blackholes.

2This is a special case of the more general metric of [10]. Wechoose this form for clarity; however our results are easilyextended to the general case. This extension does not changeour conclusions.

3The corresponding extrinsic curvature boundary term for theKR brane is cancelled by the contribution to the free energy fromthe brane tension.

THERMODYNAMICS AND STABILITY OF FLAT ANTI–DE . . . PHYSICAL REVIEW D 78, 126001 (2008)

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simply

Ibs � Iss ¼ l3

2�Gl44

Z z1

z0

dz

sin5z

�bs�bs

Z R

rbsþr2dr

� ss�ssZ R

rssþr2dr

�;

which, after substitution from (7) and expansion in R,yields in the limit that R ! 1,

�I ¼ 16�2l3l481G

��ss

2bs

��3

bs

�3ss

� 1

�Z z1

z0

dz

sin5z

¼ 4G4l3

3Gl24�I4ðCð�1Þ þ Cð�0ÞÞ;

CðuÞ ¼ 1

4

sinu

cos4uþ 3

8

sinu

cos2uþ 3

8ln

�1þ sinu

cosu

�; (8)

where�I4 is the action of the 4-dimensional flat AdS blackhole relative to the 4-dimensional AdS soliton with gravi-tational constant G4. Cð�1Þ þ Cð�0Þ is the contributionfrom the bulk integral over z; it is positive and divergesas �1 ! �=2 or �0 ! �=2. Thus, as in the spherical case,the physics of the bulk phase transition is completelydetermined by that on the brane. The sign of the relativeaction can be either positive or negative, depending on thevalue of bs and �ss. If bs >�ss the flat AdS black stringis unstable relative to the AdS soliton string. If bs <�ss,the opposite situation holds. At the Hawking-Page phasetransition point, bs ¼ �ss, or equivalently kbs ¼ kss.Hence, in contrast to the Schwarzschild-AdS case, thephase transition temperature is independent of the curva-ture scale.

To further clarify the properties of the phase transition, itis useful to recall the form of the area of a spatial crosssection of the black hole horizon of (4) with T2 topology;

A ¼ �bs

2bs

�4�l243

�2 ¼ �ss

2bs

�16�2l34

9

�:

Unlike Schwarzschild-AdS black holes, the area of thehorizon is not a function only of temperature; it alsodepends on two independent parameters �ss and , thecircumferences, characterizing the periodic identificationof the two flat coordinates. The instability condition bs >�ss corresponds to an inverse temperature bs larger thanthat of a preferred horizon circumference �bs ¼ �ss=l4.However, note that the two black hole circumferences arein fact physically equivalent. Either could be chosen tomatch with the AdS soliton by the physics of the phasetransition. This means that the phase transition occurswhen bs becomes larger than the smallest of the twocircumferences. Hence �bs is chosen to be the smallest ofthe two circumferences. Therefore black holes with onesmall horizon circumference become unstable to decay toan AdS soliton as their temperature is decreased. In con-trast, they become stable as their temperature is increased.

The thermodynamic stability of the flat AdS black stringand AdS soliton string exactly parallels that of this phasetransition on the brane. The bulk phase transition betweenthe flat AdS black string and AdS soliton string occursprecisely when the phase transition between the black holeand AdS soliton occurs on every AdS slice. The tempera-ture of the black hole on each slice varies; for fixed z, theflat AdS black hole metric (4) is scaled by a constant c ¼

ll4 sinz

. Under this constant scaling, the physical parameters

of the metric scale as �bs ¼ cbs, �rbsþ ¼ crbsþ and �l ¼cl4. The scale constant c has its minimum value at z ¼�=2; for this slice, the black hole has its maximum tem-perature, minimum cross-sectional area of the horizon andintrinsic curvature radius of l, that of the 5-dimensionalspacetime. As c increases, the temperature decreases andthe cross-sectional area of the horizon increases; in fact,the decrease in temperature for slices away from z ¼ �=2is exactly matched by the increase in black hole size.Therefore the flat AdS black hole on each constant z sliceis in equilibrium with its thermal bath. In summary, blackholes get big and cold as z approaches the conformalboundaries of the locally AdS spacetime at z ¼ 0 andz ¼ �.In the limit that both circumferences go to infinity, the

phase transition flat AdS black string to the AdS solitonstring disappears. This limit is equivalent to taking the AdSsoliton mass parameter kss to zero; in this limit, the AdSsoliton string metric becomes singular. Its geometry is alsothe same as that of the zero mass flat AdS black string. It isknown that no phase transition occurs when the zero massflat AdS black hole is taken as the background [10,12];hence this limit of the AdS soliton string reproduces theexpected result.The energy of the flat AdS black string can also be

computed; using the cutoff method [21], the energy isgiven by

E ¼ � 1

8�G

ZS

ffiffiffiffi

pd3xNðK � K0Þ;

where K is the extrinsic curvature of a surface S given bythe intersection of a constant radius and constant timesurface and K0 is that of the surface of the same geometryembedded in a static, asymptotically AdS spacetime. Thisyields, for an AdS soliton string reference spacetime,

Ebs � Ess ¼ 4�2l3l427G

�ss

�2

3bs

þ 1

�3ss

�Z z1

z0

dz

sin3ðzÞ

¼ G4l3

Gl24�E4ð �Cð�1Þ þ �Cð�0ÞÞ;

�CðuÞ ¼ 1

2

sinu

cos2uþ 1

2ln

�1þ sinu

cosu

�: (9)

As expected, the black string energy increases with in-creasing brane separation and the linear energy density isa minimum at z ¼ �=2. The energy can also be calculated


126001-4

using the respective zero mass backgrounds as a reference:

Ebs ¼ 8�2l3l427G

�ss

3bs

ð �Cð�1Þ þ �Cð�0ÞÞ;

Ess ¼ � 4�2l3l427G

�2ss

ð �Cð�1Þ þ �Cð�0ÞÞ:

Their difference yields (9). The entropy of the black stringis also easily found to be

S ¼ A

4G

Z z1

z0

dz

sin3ðzÞ ¼A

4Gð �Cð�1Þ þ �Cð�0ÞÞ; (10)

where A=4G is the 4-dimensional flat AdS black holeentropy.

It is easy to verify that on each constant z slice, theentropy, energy and free energy obey the thermodynamicrelation F4 ¼ �I4 ¼ �E4 � S4. However, this is notthe case for the 5-dimensional quantities; the integrands of(8)–(10) have different z dependence. This is because thesequantities, as well as the inverse temperature , vary in zfor the flat AdS black string; it is in local thermodynamicequilibrium. Thus the flat AdS black string behaves morelike a star than like a black hole. Hence (8)–(10) areproperly viewed as the total value of the correspondingthermodynamic quantities. Furthermore, there will be nonatural global definition of a nontrivial temperature for thisconfiguration.4 Note this effect is not due to the choice ofregularization method. Other regularization methods, suchas the counterterm method [22], may yield different nu-merical factors between terms in the computation of therelative action and energy. However, use of a differentregularization method will not change the differences inscaling of the z integrals seen here.

Given the proportionality of (9) to �E4, it is apparentthat specific heat of the black string, C ¼ dE

dT is positive and

proportional to the integrated cross-sectional area of theblack hole on each AdS slice. This result is, as one wouldanticipate, entirely due to the behavior of the specific heatfor the flat AdS black hole itself. This thermodynamicstability is associated with the nontrivial topology of thesespacetimes; it may indicate an obstruction to the evapora-tion of the flat AdS black string due to the nontrivialtopology of its spatial cross sections.

It is also interesting to consider whether or not there areother asymptotically AdS solutions with the same topologyof the boundary at infinity that contribute to the thermo-dynamics of the KR branes. Two natural candidates thatcan be chosen to have boundary topology T2 � R2 or S1 �R3 are the 5-dimensional flat AdS black hole

ds2b ¼ �fbðr0Þdt02 þ f�1b ðr0Þdr02 þ r02d�02

þ r02d�02 þ r02dz02;

fbðr0Þ ¼ r02

l2� k4b

r02;

(11)

and the 5-dimensional AdS soliton

ds2s ¼ �r02dt02 þ f�1s ðr0Þdr02 þ fsðr0Þd�02

þ r02d�02 þ r02dz02;

fsðr0Þ ¼ r02

l2� k4s

r02;

(12)

where, as for the 4-dimensional case, regularity requires �0be identified with period

�s ¼ �l2

r0sþ;

where r0sþ ¼ ksl1=2. Regularity of the Euclidean instanton

corresponding to (11) yields the inverse temperature

b ¼ �l2

r0bþ;

where r0bþ ¼ kbl1=2. The phase transition temperature of

the 5-dimensional flat AdS black hole to the AdS soliton atb ¼ �s or equivalently kb ¼ ks is again independent ofthe curvature scale. Hence a comparison of the phasetransition temperature of the black string to that of the 5-dimensional flat AdS black hole is not as straightforward asin the spherical case.One can, however, compute the likelihood of a phase

transition by comparing the action of the 5-dimensional flatAdS black hole relative to that of the zero mass AdS solitonto the action of the flat AdS black string relative to the zeromass AdS soliton string. This is possible as the zero massAdS soliton string (1) and the zero mass 5-dimensionalAdS soliton are locally the same spacetime; the coordinatetransformation

r0 ¼ lr

l4 sinzz0 ¼ l4 cosz

r�0 ¼ l

l4�

t0 ¼ t �0 ¼ �

brings the zero mass AdS soliton in the form (12) to themetric of a zero mass soliton string, (1) with (3) as the 4-dimensional metric.5 A KR brane boundary at z1 lies on thecurve z0r0 ¼ l cotz1. Hence the zero mass AdS solitonstring is a coordinatization of the 5-dimensional zeromass AdS soliton whose brane boundaries intersect theboundary at infinity of the 5-dimensional AdS soliton ata cusp. The zero mass solutions are singular; hence any

4Of course, this flat AdS black string spacetime has zerotemperature at infinity, in common with all other locally asymp-totically AdS spacetimes.

5A similar coordinate transformation relates the zero mass 5-dimensional flat AdS black hole to the zero mass flat AdS blackstring. In fact, the zero mass AdS soliton and zero mass blackhole are locally the same solution.


126001-5

periodic identification of the angular coordinate �0 is al-lowed. Therefore, in contrast to earlier calculations, theperiodicity of the angular coordinates are now fixed en-tirely in terms of flat AdS black string parameters;

00 ¼ 0 ¼ bs

l4; �0

0 ¼l

l4�0 ¼ l�bs;

00 ¼ 0 ¼ ;

(13)

that is, the periodicities 00, �

00,

00 of the zero mass 5-

dimensional AdS soliton are set by those of the zero massAdS soliton string, 0, �0, 0, which, in turn, are set bythose of the flat AdS black string.

The action of the flat AdS black string relative to thezero mass AdS soliton string is easily seen to be

�Ibs ¼ � 16�2l3l2481G

��bs

2bs

�ðCð�1Þ þ Cð�0ÞÞ: (14)

Similarly, the action of the AdS soliton string relative to thezero mass one is

�Iss ¼ � 16�2l3l2481G

�ss

�2ss

�ðCð�1Þ þ Cð�0ÞÞ: (15)

That of the 5-dimensional black hole relative to the zeromass AdS soliton is given by

I5b � I0 ¼ 1

2�Gl2

�b�b

Z R

r0bþr03dr0

� 00�

00

Z R

0r03dr0

�Z ðl cotz1Þ=r0

ðl cotz0Þ=r0dz0:

Note that the brane positions in the 5-dimensional flat AdSblack hole have been set by their positions in the zero massAdS soliton solution. After evaluation, using the matching

conditions 00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffifbðRÞ

pb=R, �

00 ¼ l�b and taking the

limit R ! 1, one finds

�I5b ¼ � 1

6�Gl

�sin�1cos�1

þ sin�0cos�0

�b�br

03bþ

¼ ��2l5

6G

�b

2b

�sin�1cos�1

þ sin�0cos�0

�: (16)

The action of the 5-dimensional AdS soliton relative tothe zero mass one is easily found by a similar computation:

�I5s ¼ ��2l5

6G

ss

�2s

�sin�1cos�1

þ sin�0cos�0

�: (17)

The differences in relative action can now be taken forall four possible cases. First, note that the matching (13)determines the 5-dimensional black hole temperature interms of the flat AdS black string; b ¼ l

l4bs, �b ¼ �bs.

The action for the flat AdS black string relative to the 5-dimensional black hole is then

�Ibs � �I5b ¼ ��2l3l24162G

�bs

2bs

�32Cð�1Þ � 27

sin�1cos�1

þ 32Cð�0Þ � 27sin�0cos�0

�: (18)

Although this difference depends on the brane positions, itis easy to see that the overall sign is independent of thischoice; Cð�iÞ is a monotonically increasing function thatgoes like sin�i for �i sufficiently near zero. Therefore, theflat AdS black string with horizon radius rbsþ is stablerelative to the 5-dimensional flat AdS black hole of radiusr0bþ ¼ l

4l4rbsþ.

The action of the flat AdS black string relative to the 5-dimensional AdS soliton is, noting that (13) and the usualmatching conditions give s ¼ bs=l4, �s ¼ l�bs,

�Ibs � �I5s ¼ ��2l3l24162G

�bs

2bs

�32Cð�1Þ � 27

3bs

l34�3bs

sin�1cos�1

þ 32Cð�0Þ � 273

bs

l34�3bs

sin�0cos�0

�: (19)

In contrast to 5-dimensional flat AdS black hole case, thedifference in actions can now have either sign as �bs is a

free parameter. At small �i, one sees that 273bs

l34�3bs

> 32; the

flat AdS black string will be unstable to the 5-dimensionalAdS soliton. As �i becomes larger, this will no longer bethe case; instead the transition depends explicitly on thebrane position:

3bs

l34�3bs

>32ðCð�1Þ þ Cð�0ÞÞ27ðsin�1cos�1

þ sin�0cos�0

Þ :

As C is increasing, the phase transition will occur onlyfor smaller and smaller �bs as the branes move outwardtoward the boundary. In addition, recall that the transitionbetween the flat AdS black string and AdS soliton stringoccurs when bs=�ss ¼ bs=ðl4�bsÞ ¼ 1. Hence a flat AdSblack string with a very small horizon circumference isalso unstable to a phase transition to an AdS soliton string.Therefore, a flat AdS black string of large horizon circum-ference will be stable and one with small horizon circum-ference will undergo a phase transition either to a 5-dimensional AdS soliton or an AdS soliton string.It is easy to see that the action for the AdS soliton string

relative to the 5-dimensional AdS soliton is

�Iss � �I5s ¼ ��2l3l24162G

ss

�2ss

�32Cð�1Þ � 27

sin�1cos�1

þ 32Cð�0Þ � 27sin�0cos�0

�; (20)

and that for the AdS soliton string relative to the 5-dimensional black hole is


126001-6

�Iss � �I5b ¼ ��2l3l24162G

ss

�2ss

�32Cð�1Þ � 27

�3ss

l343ss

sin�1cos�1

þ 32Cð�0Þ � 27�3ss

l343ss

sin�0cos�0

�: (21)

From the above results, it is clear that the 5-dimensionalAdS soliton is always unstable to the AdS soliton string.Thus if a flat AdS black string with a very small horizoncircumference undergoes a phase transition to the 5-dimensional AdS soliton, as this is only a local minimum,it in turn will undergo a phase transition to the AdS solitonstring. Similarly, there is a phase transition between the 5-dimensional flat AdS black hole and the AdS soliton stringwith stability determined by the black hole parameters.

At first glance, these results seem to contradict theconjecture that the AdS soliton is the ground state of anasymptotically AdS spacetime with Ricci flat boundary atinfinity. However, this is not the case; the above results arecomparing not a pure 5-dimensional AdS soliton, butrather one with brane boundaries. These branes in the 5-dimensional AdS soliton space are held in position by theirtension; hence, one is not comparing a bulk 5-dimensionalAdS soliton, but rather one constrained by these branes.Furthermore, the brane geometry induced by slicing the 5-dimensional AdS soliton

ds2 ¼ �r02dt02 þ�f�1s ðr0Þ þ l2

cot2z1r02

�dr02 þ fsðr0Þd�02

þ r02d�02 (22)

is not particularly nice. Although the spacetime is asymp-totically a 4-dimensional locally AdS spacetime withphysical curvature scale lp ¼ l

sin2z1as expected, the mass

of the spacetime is zero. Its Ricci tensor obeys the nullenergy condition,6 but exhibits radial dependence and isanisotropic. Hence the tension of this brane will alsoexhibit a corresponding unusual form.

Of course these calculations are somewhat unsatisfac-tory. Although the same space is used in the calculation ofthe relative actions, the cutoff at constant radius used in thecalculation of (14) and (15) and in (16) and (17) occurs ondifferent surfaces for each case. However, if the limit iswell defined, the interchange of the difference and limitshould not affect the result. Moreover, there is a certainphysical appeal to these results that leads one to expect thatthey hold at least qualitatively; although in certain circum-stances, a flat AdS black string can be unstable to a 5-dimensional AdS soliton with an unusual brane geometry,any such configuration is also unstable to an AdS solitonstring. Hence all paths lead to the same final outcome.

III. THE PERTURBATIVE STABILITY OFASYMPTOTICALLY RICCI FLATADS STRING

SPACETIMES

Unlike the Schwarzschild black hole which has beenproven to be perturbatively stable [23], it is well-knownthat black strings and black p-branes can be perturbativelyunstable [24,25] (See [26] for a recent review). This insta-bility is due to the existence of a metric perturbation thatdiverges exponentially in time associated with a transla-tional symmetry of the string spacetime. The Gubser-Mitraconjecture [20] connects this classical perturbative insta-bility of the spacetime to its thermodynamic instability,namely, a negative specific heat. This conjecture has beenproven for certain classes black branes with a noncompacttranslational symmetry [27–29]. Recently, the perturbativestability and its connection to the Gubser-Mitra conjecturehas been studied and verified in [30] for the translationallyinvariant, uniform AdS black string solutions of [31].However, the case in which there is a noncompact trans-

lational symmetry on a conformally related space is alsorelevant. The perturbative stability of black AdS stringswith Schwarzschild spacetime cross sections was first an-alyzed by Gregory in [32]. Hirayama and Kang [6] thenstudied the perturbative stability of the black AdS stringconstructed from the 4-dimensional Schwarzschild-AdSmetric. They found numerically that here is a thresholdvalue, proportional to the ratio of the 4-dimensional AdScurvature radius l4 to the string mass parameter kbs, abovewhich the black string is perturbatively stable. For valuesof the mass parameter smaller than this threshold value, thestring is unstable. It is this instability that was associatedwith the negative specific heat of the black string solutionin [5]. Hence, at least for the black AdS string constructedfrom Schwarzschild-AdS, a more general version of theGubser-Mitra conjecture holds, one with the weaker as-sumption of a conformal killing vector of translationalform.This generalization make sense for this case; if we

consider each AdS slicing as a spacetime in its own right,the Gubser-Mitra conjecture applies and identifies a per-turbative instability associated with the temperature atwhich the specific heat becomes negative for theSchwarzschild-AdS spacetime. Scaling of temperatureand horizon area indicates that this property persists foreach slice of the black string. Hence, though the spaceitself is not translationally invariant, its thermodynamicsbehaves in an invariant way with respect to the conformalkilling vector @z. In contrast, the flat AdS black hole andthe flat AdS black string have positive specific heat.Furthermore, the thermodynamics of the flat AdS blackstring is also invariant. Hence, we anticipate that the flatAdS black string will be perturbatively stable.In this section we examine the perturbative stability of 5-

dimensional AdS strings whose cross sections are AdSspacetimes with toroidal topology. Our analysis closely

6The null energy condition is that Rabkakb � 0 for all null

vectors ka.


126001-7

follows that of [6]. In the following we will only addressthe stability of the bulk AdS string solutions; we anticipatethat the presence of branes, though changing certain detailsin the calculation, will not change our conclusions.

It is convenient to write the general tensor perturbationof the metric (1) in a form compatible with that of [6] tofacilitate comparison with their results

ds2 ¼ �H�2ðzÞ½ðg�� þ h��ðx; zÞÞdx�dx� þ dz2�; (23)

�HðzÞ ¼ l4lsin

z

l4: (24)

We impose transverse traceless gauge7 on the brane

r�h�� ¼ 0 h�� ¼ 0: (25)

Let h��ðx; zÞ ¼ �H3=2ðzÞ�ðzÞh��ðxÞ; then to linear order,

the Einstein equations can be written as two decoupledequations

r�r�h��ðxÞ þ 2R��h��ðxÞ ¼ m2h��ðxÞ; (26)

½�@2z þWðzÞ��ðzÞ ¼ m2�ðzÞ

WðzÞ ¼ � 3

2

�H00�Hþ 15

4

� �H0�H

�2;

(27)

where the covariant derivative, the Riemann tensor, and theraising and lowering of indices are with respect to the 4-dimensional metric g��. Hence, as usual, the tensor per-

turbation (26) takes the form of a massive graviton and theminimum mass of this perturbation is set by the minimumeigenvalue of (27). Using (2), one finds that the explicitexpression of the effective potential is

WðzÞ ¼ 1

l24

�3

2þ 15

4cot2

z

l4

�;

which is bounded below by 32 and becomes infinite at z ¼

n�l4, n 2 Z. Thus (27) has the same spectrum as that ofthe Schwarzschild-AdS string [6]; solving, one findsmmin ¼ 2=l4. Therefore the difference between the spheri-cal case and that of toroidal AdS slicings is due entirely tothe different background geometry for the graviton in (26).For the toroidal AdS slicings, we shall seek a marginally

unstable tensor perturbation exhibiting the symmetry of thespacetime;

h�� ¼ e�t

BttðrÞ BtrðrÞ 0 0BtrðrÞ BrrðrÞ 0 00 0 B��ðrÞ 00 0 0 B��ðrÞ

0BBB@

1CCCA: (28)

At this point, the cases of the flat AdS black string andAdS soliton string must be analyzed separately.

A. The flat AdS black string

For the flat AdS black hole metric (4), the gauge con-ditions (25) yield the relations

� �

Vbs

Btt þ VbsB0tr þ

�V 0bs þ

2Vbs

r

�Btr ¼ 0

� �

Vbs

Btr þ VbsB0rr þ 2Vbs

rBrr þ V 0

bs

2V2bs

Btt þ 3V 0bs

2Brr � 1

r3ðB�� þ B��Þ ¼ 0

� 1

Vbs

Btt þ VbsBrr þ 1

r2ðB�� þ B��Þ ¼ 0:

(29)

Note 0 has been used to indicate differentiation by r. The graviton perturbation Eqs. (26) for the tr and rr components forthe flat AdS black hole slicing are equivalent to

r2V3bsB

00tr þ ðr2V 0

bsV2bs þ 2rV3

bsÞB0tr þ ð��2r2Vbs � 2V3

bs þ r2V 00bsV

2bs � r2V 02

bsVbs �m2r2V2bsÞBtr þ�r2V0

bsBtt

þ�r2V 0bsVbsBrr ¼ 0; (30)

�2r4V4bsB

00rr þ ð�6r4V3

bsV0bs � 4r3V4

bsÞB0rr þ ð2�2r4V2

bs � 2r4V3bsV

00bs � r4V2

bsV02bs � 4r3V3

bsV0bs þ 8r2V4

bs þ 2m2r4V3bsÞBrr

þ ðr4V2bs � 2r4VbsV

00bsÞBtt � 4�r4VbsV

0bsBtr þ ð�4V3

bs þ 2rV2bsV

0bsÞðB�� þ B��Þ ¼ 0: (31)

By eliminating Btt, Brr, B��, and B��, the above equations reduce to

C2B00tr þ C1B

0tr þ C0Btr ¼ 0; (32)

where

7This gauge, is also termed Randall-Sundrum gauge [3].


126001-8

C2 ¼ ð8þ 4�2Þr14 þ 4!2r12 þ ð�32� 12�2Þr3bsþr11 � 8!2r3bsþr9 þ ð39þ 12�2Þr6bsþr8 þ 4!2r6bsþr

6

þ ð�14� 4�2Þr9bsþr5 � r12bsþr2;

C1 ¼ ð48þ 24�2Þr13 þ 32!2r11 þ ð�168� 48�2Þr3bsþr10 � 28!2r3bsþr8 þ ð108þ 24�2Þr6bsþr7 � 4!2r6bsþr

5

þ 15r9bsþr4 � 3r12bsþr;

C0 ¼ ð32þ 8�2 � 4�4Þr12 þ ð24� 8�2Þ!2r10 þ ð�176� 12�2 þ 8�4Þr3bsþr9 � 4!4r8 þ ð24þ 8�2Þ!2r3bsþr7

þ ð12� 9�2 � 4�4Þr6bsþr6 � 3!2r6bsþr4 þ ð52þ 13�2Þr9bsþr3 � r12bsþ;

in which�2 ¼ m2l24 is the dimensionless KKmass, r3bsþ ¼k3bsl

24, the horizon radius and ! ¼ �l4. Note that, in con-

trast to the Schwarzschild-AdS case, rbsþ is interchange-able with the mass parameter kbs; this is due to the form ofVbs for the flat AdS black hole. Furthermore, also due to theform of Vbs, (32) is exactly invariant under the scaling r !�r, rbsþ ! �rbsþ, ! ! �!. Thus by rescaling we can fixthe value of either rbsþ or !; we will do so by takingrbsþ ¼ 1. Now (32) only has two arbitrary constants,� and!, in contrast to the Schwarzschild-AdS case [6]; thissimplifies the analysis of the stability. We will solve (32)as an eigenvalue problem, treating�2 as a parameter and!as the eigenvalue.

The asymptotic behavior of solutions of (32) approach-ing the horizon and spatial infinity is easily calculated;

r ! 1; Btr � ðr� 1Þ�1�!=3; (33)

r ! 1; Btr � rð�5=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ: (34)

As discussed by [24,25], the specification of the bound-ary conditions is key to a consistent perturbative solution.The perturbation must remain small exterior to the blackhole horizon. As the negative root solution in (39) divergesat the horizon, it should be excluded. Unfortunately, thepositive root solution will also diverge as r ! 1 if !< 3.However, this divergence is attributable to the failure of theSchwarzschild-like coordinates used here. A change ofcoordinates to a set regular on the horizon, such asKruskal coordinates, demonstrates that the perturbationwill be regular if it diverges more slowly than ðr� 1Þ�1.Therefore only the positive root of (39) exhibits the correctbehavior.

Furthermore, an additional condition, not present inasymptotically flat spacetimes, is necessary as AdS space-times are not globally hyperbolic. The usual condition formatter fields, and that imposed by us here, is that that theenergy in the perturbation be finite [33,34]. This requires afall-off of Btr � r with <�5=2 as r ! 1. This is trueof the behavior of the negative root of (40), but not the

positive one. Thus to summarize, the correct boundary

conditions for the perturbation are Btr � ðr� 1Þ�1þ!=3

as r ! 1 and Btr � rð�5=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ as r ! 1.

To numerically solve (32) we transform it into therelated nonlinear equation by dividing it by Btr. Then asB00tr

Btr¼ ðB0

tr

BtrÞ0 þ ðB0

tr

BtrÞ2, (32) becomes

C2ðY0 þ Y2Þ þ C1Y þ C0 ¼ 0; (35)

where Y ¼ B0tr=Btr. This form is solvable by the shooting

method. Y will be regular everywhere if Btr is nonvanish-ing on the interior. One anticipates that the marginallyunstable mode will exhibit such behavior. The boundaryconditions can now be imposed in a straightforward man-

ner; Y � ð�1þ!=3Þ=ðr� 1Þ at the horizon and Y �ð�5=2� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

9=4þ�2p Þ=r as r goes to infinity.8

We used the shooting method algorithm of [35] imple-mented in C. We tested this code on the equivalent equa-tions for the asymptotically flat string of [24] and thespherical AdS string of [6] and reproduced their results.We then applied this code to (41) and found no positiveeigenvalue ! that solves the equation for any chosen valueof the parameter �2. To be explicit, starting from eitherboundary, and with any value of �2 � 4, we find the valueof ! rapidly diverges to infinity. This means that higherfrequency modes solve (26) and (27) better, which indi-cates that an unstable mode with finite energy does notexist. Therefore, numerical evidence indicates that the flatAdS black string is perturbatively stable.

B. The AdS soliton string

It is useful to compare the results of the flat AdS blackstring case to that of the AdS soliton string. The steps aresimilar to those of the previous subsection.The gauge conditions for the AdS soliton metric (3) are

8The shooting method cannot be applied directly to (32) due tothe sensitivity of the asymptotic behavior to numerical errors.Numerical errors in the integration could drive the solution toexhibit the behavior of the larger asymptotic solution. Then wewould not be able to see the correct behavior by varying thevalue of !.


126001-9

��

r2Btt þ VssB

0tr þ

�V0ss þ 2Vss

r

�Btr ¼ 0;

��

r2Btr þ VssB

0rr þ 2Vss

rBrr þ 1

r3Btt þ 3V0

ss

2Brr � V 0

ss

2V2ss

B�� 1

r3B�� ¼ 0;

� 1

r2Btt þ VssBrr þ 1

Vss

B�� þ 1

r2B�� ¼ 0:

(36)

The graviton perturbation Eqs. (26) for the tr and rr components are equivalent to

2r2VssB00tr þ ð4rVss þ 4r2V 0

ssÞB0tr þ ð4rV0

ss � 2Vs þ r2V00ss � 2�2 � 2m2r2ÞBtr þ 4�rVssBrr ¼ 0; (37)

ð�4�V2ss ��r2ðV 0

ssÞ2 þ 2�rVssV0ss þ 2�r2VssV

00ssÞBtr þ ðr4V2

ssV0ss � 2r3V3

ssÞB00rr

þ ð�8r2V3ss � 2r3V 0

ssV2ss � 2r4V2

ssV00ss þ 4r4VssðV 0

ssÞ2ÞB0rr þ ð2�2rV2

ss ��2r2VssV0ss þ 2r4ðV 0

ssÞ3 � 12r2V2ssV

0ss

þ 8r3VssðV 0ssÞ2 � 8r3V2

ssV00ss � 2r4VssV

0ssV

00ss þ 2m2r3V2

ss �m2r4VssV0ssÞBrr ¼ 0: (38)

In contrast to the flat AdS black hole string, the AdS soliton equations are not invariant under the interchange of B�� andB��. Nonetheless, everything can be expressed in terms of Btr albeit at a price; the equation is now fourth order,

C4B0000tr þ C3B

000tr þ C2B

00tr þ C1B

0tr þ C0Btr ¼ 0; (39)

where

C4 ¼ r10 � 2r3ssþr7 þ r6ssþr4; C3 ¼ 22r9 � 26r3ssþr6 þ 4r6ssþr3;

C2 ¼ ð142� 2�2Þr8 � 2!2r6 þ ð�86þ 2�2Þr5 þ 2!2r3ssþr3 � 2r6ssþr2;

C1 ¼ ð288� 18�2Þr7 � 14!2r5 þ ð�72þ 6�2Þr3ssþr4 þ 2!2r3ssþr2;

C0 ¼ ð112� 32�2 þ�4Þr6 þ ð�10þ 2�2Þ!2r4 þ ð�8þ 2�2Þr3ssþr3 þ!4r2 � 2!2r3ssþrþ 4r6ssþ;

and the scaling freedom has again been used to takerssþ ¼ 1.

Again, the asymptotic behavior can be easily deter-mined; as r approaches infinity

Btr � rð�11=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ; (40)

Btr � rð�5=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ: (41)

For Btr to fall off faster than r�5=2 as r ! 1, the possible

asymptotic behavior is Btr � Crð�5=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ þ

Drð�11=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ for �2 > 27=4. The first term is

dominant. If �2 is smaller than this, the asymptotic

behavior is Btr � Crð�5=2�ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ þDrð�11�

ffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ þ

Erð�11=2þffiffiffiffiffiffiffiffiffiffiffiffiffi9=4þ�2

pÞ and the last term is dominant.

In contrast to the flat AdS black hole string case, thespacetime is regular as r ! 1. Therefore, the perturbation

must be regular at this point. Thus Btr � ðr� 1Þ�1þ ffiffi3

pas

r ! 1.As in the flat AdS black string case, (39) can also be

rewritten in terms of Y ¼ B0tr=Btr, resulting in the third

order equation

0 5 10 15 20 25 30−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

µ2

ω2

FIG. 1. Dependence of first eigenvalue of !2 on �2. Thisgraph shows that we find only negative eigenvalues of !2, whichcorrespond to stable oscillatory modes.


126001-10

C4ðY000 þ 4Y00Y þ 6Y0Y2 þ 3Y02 þ Y4Þþ C3ðY00 þ 3Y0Y þ Y3Þ þ C2ðY0 þ Y2Þ þ C1Y þ C0 ¼ 0:

(42)

Again, no solution is found numerically using the shootingmethod code, both for the case where �2 > 27=4 and for�2 < 27=4. However, in contrast to the flat AdS blackstring case, it is now possible to examine the behavior ofthis equation in more detail. The boundary conditions areindependent of !. Furthermore, (39) depends only on !2.These facts allow us to seek solutions with negative valuesof !2. Of course, any such solution will correspond to astable mode. In fact, we find such solutions; the numericalresults are shown in Fig. 1. This numerical result confirmsthat the AdS soliton string is stable.

IV. CONCLUSIONS

As in the Schwarzschild-AdS case [5], we find that thethermodynamics of flat AdS black holes on KR branes isexactly paralleled by that of the flat AdS black string in thebulk. The flat AdS black string is in local thermodynamicequilibrium on each constant z slice; thus its phase tran-sition occurs in parallel with that of the flat AdS black holetransition on the brane. Notably, both the brane and bulkthermodynamics retain the unique characteristic of the flatAdS black hole that the phase transition temperature isindependent of the curvature radius l. Thus this modelprovides a forum for the study of the properties of blackholes on asymptotically AdS branes and the AdS/CFTcorrespondence in which the phase transition temperatureis independent of the curvature radius.

In addition, we have shown, by a calculation of therelative action, that although very small circumferenceflat AdS black strings can be unstable to a 5-dimensionalAdS soliton with an unusual brane geometry, any suchconfiguration is also unstable to an AdS soliton string.Furthermore, the flat AdS black string is stable relative to

the 5-dimensional flat AdS black hole. In contrast to theSchwarzschild-AdS case, the specific heat of the flat AdSblack string is always positive. A numerical calculationindicates that the bulk black AdS string and AdS solitonstring are both perturbatively stable for all values of theirmass parameter. Thus thermodynamic stability impliesperturbative stability for the flat AdS black string. This,together with the similar relationship in the Schwarzschild-AdS case, may indicate that a generalization of the Gubser-Mitra conjecture, with the weaker assumption of a confor-mal killing vector of translational form, holds.Although our analysis is confined to the 5-dimensional

case, it is clear that the thermodynamics of higher dimen-sional flat AdS black strings will have qualitatively similarbehavior. However, it is not obvious that the more quanti-tative features in the analysis will carry through as dimen-sionality is known to play an important role inasymptotically flat string perturbative stability and phasetransitions [36,37].It is also natural to consider whether or not these results

will also hold true for AdS black strings whose spatialcross sections are hyperbolic AdS black holes. HyperbolicAdS black holes also have manifestly positive specific heat[10,11]. Hence one anticipates that hyperbolic AdS stringspacetimes will exhibit perturbative stability. However, thethermodynamics of these spacetimes will be quite differentin other aspects from that of the flat AdS black string asthere is no analog of the AdS soliton for hyperbolic case[38,39]. In particular, one expects that there will be no bulkphase transition for the hyperbolic AdS black string.

ACKNOWLEDGMENTS

This work was supported by the Natural Sciences andEngineering Council of Canada. In addition, K. S. andD.W. would like to thank the Perimeter Institute for itshospitality during the completion of this paper.

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