Thermodynamic and gravitational instability on hyperbolic spaces

0.37 MB·English

by  Ishwaree P. Neupane

Thermodynamic and gravitational instability on hyperbolic spaces
Ishwaree P. Neupane
∗
Department of Physics, National Taiwan University,
Taipei 106, Taiwan, R.O.C.
[email protected], [email protected]
Westudythepropertiesofanti–deSitterblackholeswithaGauss-Bonnettermforvarioushorizon
topologies(k=0, ±1)andforvariousdimensions,withemphasisonthelesswellunderstoodk=−1
solution. Wefindthatthezerotemperature(andzeroenergydensity)extremalstatesarethelocal
minimaoftheenergyforAdSblackholeswithhyperboliceventhorizons. ThehyperbolicAdSblack
holemaybestablethermodynamicallyifthebackgroundisdefinedbyanextremalsolutionandthe
extremalentropy is non-negative. Wealso investigate thegravitational stability of AdSspacetimes
of dimensions D > 4 against linear perturbations and find that the extremal states are still the
4 local minima of the energy. For a spherically symmetric AdS black hole solution, the gravitational
0 potentialispositive andbounded,with orwithout theGauss-Bonnet typecorrections, while, when
0 k = −1, a small Gauss-Bonnet coupling, namely, α ≪ l2 (where l is the curvature radius of AdS
2 space), is found useful to keep the potential bounded from below, as required for stability of the
n extremal background.
a
J PACSnumbers: 04.70.-s,04.50.+h,11.10.Kk,11.25.-w

2
1
I. INTRODUCTION TheAdSsolitonisanonsupersymmetricbackgroundbut

3 intheAdS/CFTcontextitisconjecturedtobeaground
v state for planar black holes; see also Ref. [12].
InparallelwiththedevelopmentofAdSconformalfield
2
theory (CFT) correspondence [1, 2]), black holes in AdS Another interesting issue with hyperbolic AdS black
3
1 space are known to play an important role in dual field hole spacetimesis the choice ofabackground. In agrav-
2 theory [3]. It has also been learned that the Einstein itational theory, it is necessary to make a Euclideanized
0 equations when supplemented by a negative cosmologi- action finite by assigning classically stable lowest energy
3 calconstantadmitblackholesasexactvacuumsolutions, configurations to AdS black hole spacetimes with a cur-
0 whoseeventhorizonsarehypersurfaces withzero,pos- vature k = 0, 1 of the horizons. For k = +1, the
h/ itive or negative constant curvature (kM= 0, +1, or 1). background is s±imply a global AdS space, which is the
t This mayberelatedtothe onwhichthedual−field solution at finite temperature [3, 13]. But, for k = 1,
p- theorMy is defined by a rescalinMg o′f the metric. the groundstate may be differentfrom a solutiontha−tis
e locally isometric to a pure (global) AdS space [4, 5].
Anti–de Sitter black holes with nonspherical event
h
horizons have been constructed in four and higher di- HyperbolicAdSblackholesareknowntoexhibitsome
:
v mensions [4, 5, 6]. Earlier work on closely related AdS new and interesting features, such as an increase in the
Xi thermodynamicscanbefoundin[7]. Theanalysisin[5]is entropy that is not accompanied by an increment in the
wellmotivatedfromthe AdS/CFT correspondence. Ref- energy [8, 10]. They are also relevant to studying CFTs
r
a erence [8] discusses AdS/CFT duals of k = 1 topolog- with less than maximal (or no) supersymmetry [14, 15].
ical black holes in the spirit of a holograph−ic countert- In supergravity theories, maximally symmetric hyper-
erm method developed in [9, 10]. Here we study the bolic spaces naturally arise as the near-horizon region
thermodynamic and gravitational stability of a class of of certain p-branes [16] and black hole geometries [17].
Gauss-Bonnet(GB)blackholesinAdSspace,whichalso Thusthe choiceofagroundstateforthehyperbolicAdS
havethefeaturethatthehorizon(hypersurface) isan black holes as well as their thermodynamic and gravita-
(n 1)-dimensional Einstein space with constanMt curva- tional (or dynamical) stability are the important issues.
−
ture (k = 1, 0, +1). Forasphericallysymmetricsolution(i.e.,k =+1),for
−
There is the issue of the positivity of the total en- instant,theAdSSchwarzschildsolution,thehypersurface
is usually a round sphere, while, for hyperbolic AdS
ergy when AdS black holes have non-spherical horizons.
M
Usually, the positive energy theorems show a stability black holes, is either a hyperbolic space Hn−1 or its
quotientHnM1/Γ. Therefore,fork = 1,onereasonably
ofspacetimeswhichbecomeasymptoticallyeitherlocally −
−
flatoranti–deSitter. Theknowntheoremsdonotextend assignsthe zero temperature (and zero energy)extremal
state as a background; see, for example, Refs. [4, 5]. In
to the Horowitz-Myers soliton [11], whose AdS asymp-
totic is a toroidal space with a zero constant curvature. this paper, we further show, with or without a Gauss-
Bonnet term, that the extremal states are local minima
of the energy for hyperbolic AdS black holes.
InRef.[18],GibbonandHartnollstudiedaclassicalin-
∗OnleavefromDept. ofPhysics,TribhuvanUniv.,Kathmandu stability ofspacetimes of dimensions D >4 againstmet-
2
ric perturbation, by considering generalized black hole A. Gauss-Bonnet black holes in AdS space
metrics in Einstein gravity. In this paper, we generalize
thoseresultsbyincludingaGauss-Bonnetterm. Wefind
The Einstein field equations modified by a Gauss-
that the black hole spacetime whose AdS asymptotic is
Bonnet term take the following form
a hypersurface of negative constant curvature could be
unstableundermetricperturbationsifthe backgroundis
1 1
a zero mass topological black hole. We show that, with Rab Rgab+Λgab =16πGn+1α′ GBgab 2Hab ,
− 2 2L −
orwithoutaGauss-Bonnetterm,the extremalstatesare (cid:20) (cid:21)
(2)
local minima of the energy for k = 1 AdS spacetimes
against linear perturbations. We fur−ther argue that the where Hab ≡RRab−2RacbdRcd+RacdeRbcde−2RacRbc
and = R2 4R Rab+R Rabcd. For α >0, we
extremal background, defined with a negative extremal LGB − ab abcd ′
have the well known Gauss-Bonnet black hole solution
mass, can be gravitationally (or dynamically) stable if
the groundstatemetric receiveshigher curvaturecorrec-
dr2
tions, like a GB term, with small couplings. ds2 = f(r)dt2+ +r2dΣ2 , (3)
− f(r) k,n−1
Thelayoutofthepaperisasfollows. InSec. IIwegive
black hole solutions inAdS space, compute extremalpa-
with
rameters, and define different reference backgrounds in
Einstein gravity modified with a Gauss-Bonnet term. In
r2 r2 8αΛ 4αµ
Sec. III we compute Euclideanized actions applicable to f(r)=k+ 1+ + , (4)
the curvature k = 0, 1 of the event horizons. In Sec. 2α ± 2αs n(n 1) rn
−
±
IV we relate the free energyandspecific heatcurves and
discuss the thermalphase transitions. In Sec. V we turn whereα=16πG (n 2)(n 3)α,andµisanintegra-
n+1 ′
− −
to stability analysis of the background metrics (vacuum tionconstant. Themetricofan(n 1)-dimensionalspace
−
solutions)inEinsteingravityundermetricperturbations, , whoseRicci scalarequals (n 1)(n 2)k, is denoted
by setting up a Sturm-Liouville problem. We extend Mas dΣ2 ; the latter is the unit−metric−onSn 1, IRn 1,
k,n 1 − −
this analysis in Sec. VI for the background metrics in or Hn 1−, respectively, for k =1, 0, or 1.
−
Einstein-Gauss-Bonnettheory. Section VII contains dis- When α = 0, the cosmological con−stant is fixed as
cussion and conclusion. Λ= n(n 1)/2ℓ2,while, forα>0,there isa rescaling,
name−ly, ℓ2− l2 =ℓ2/ 1 α/ℓ2 . For generality, hence-
→ −
forth,weusea commonscalel2,unless otherwisestated,
(cid:0) (cid:1)
but the convention that l2 ℓ2 in the limit α 0 is to
be understood. The dimens→ion of α is (length)2→.
II. BLACK HOLES IN ADS SPACE
Ourstartingpointis theLagrangianofgravityinclud-
B. Extremal Gauss-Bonnet black holes
ing a Gauss-Bonnet term
For α>0, the periodicity of the Euclidean time is
1
I = dn+1x√ g (R 2Λ)
16πG − −
n+1 Z β = 4πr+l2 r+2 +2αk . (5)
+α′ dn+1x√−g RabcdRabcd−4RcdRcd+R2 . nr+4 +k(n−2)r(cid:0)+2 l2+(n−(cid:1)4)αk2l2
Z
(cid:0) ((cid:1)1)
When k = 1, β starts from zero at the smallest radius
r = √2α,−except for the coupling 4α = l2. The space-
+
Usually, the action is supplemented with surface terms time region can be singular with no black hole interpre-
(oraHawking-Gibbonstypeboundaryaction),whichcan tation if 4α > l2 or/and r+ < √2α, and so one should
be found,forexample,inRef.[19],butthey havenorole be interested only in the regions 4α 6 l2 and r+2 > 2α.
in the AdS black hole calculations [3, 13]. The saturation limit 4α = l2 may be taken only if one
also approaches the critical limit r2 =2α.
The black hole solutions for the action (1) were first +
TheparameterµinEq.(4)maybe expressedinterms
given by Boulware and Deser [20], which were studied
of the horizon radius r , namely,
by MyersandSimon[21], within the contextof Lovelock +
gravity,by regularizingthe classicalaction; see [22] for a
r2 αk2 16πG M
dhiasscubseseionncoofnscihdaerrgaebdleGinatuesrse-sBtoinnnegtenbelraaclkizihnogletsh.oTsehesore- µ=r+n−2(cid:18)k+ l+2 + r+2 (cid:19)≡ (n−1n)+V1n−1 , (6)
lutions with Λ < 0 and k = 1 [23, 24, 25, 26], and also
6
within the context of dimensionally extended Lovelock where M is the Arnowitt-Deser-Misner (ADM) mass of
gravity [27]. a black hole, and V is the volume of dΣ2 . In the
n 1 k,n 1
− −
3
limit β , the k= 1 extremal parameters are gravity. In fact, the graviton propagators in AdS
n+1
→∞ −
spacetime, when n > 4, do not receive any corrections
µextr = 2nrenx−t4r2 n2 −s n−n 2 2− 4(nn−l24)α(,7) fαro=mlt2h/e4m(saeses,ivfoer(eKxaalmuzpal-eK, Rleienf.)[m29o]d),esanwdhesno kth=is0baacnkd-
− (cid:18) (cid:19) ground may be stable under linear perturbations.
  However, when k = 1, stability of the background
r2 = n−2 l2 1+ 1 4n(n−4) α . (8) requires α<l2/4. That−is, as in Einstein gravity [5, 10],
extr (cid:18) 2n (cid:19) s − (n−2)2 l2! the extremal mass µextr takes only a negative value. In
particular, when α = l2/4 and n = 4, the Ricci scalar
For α = l2/4, µextr = 0 for any n. It is somewhat of a and Kretschmann scalar K (=RabcdRabcd) read
misnomer to call the µ = 0 state an extremal state,
extr
because extremal black holes are defined to have zero 40 12 µl2
temperature, which requires α < l2/4. Moreover, the R = − l2 + r2l2 ,
solution with µ = 0 saturates the bound r2 > 2α = p
extr + 160 48µ 96 µl2
l2/2, so the proper extremal black holes are those that K = + . (13)
l4 r4l2 ∓ r2l4
satisfy α<l2/4 and have zero Hawking temperature. p
The metric spacetime is only asymptotically AdS when
µ = 0, and µ < 0 is not allowed in this case. Next,
C. Choice of backgrounds con6 sider, for example, the coupling α = l2/12. When r
is large, the curvature scalars approximate to
For α =0, the harmonic function f(r) is defined by
′
40 10(15 4√6)µ 1
R = 3 √6 ± + ,
r2 µ −l2 ∓ − r4 O r6
f(r)=k+ l2 − rn−2 . (9) 480(5(cid:16) 2√6)(cid:17) 1 (cid:18) (cid:19)
K = ∓ + , (14)
For k = 0, a zero mass ground state is still legitimate l4 O r8
(cid:18) (cid:19)
and is an acceptable background [4, 5]. However, within
the AdS/CFT context, Horowitz and Myers [11] have in satisfying µ > 2r4/l2, r2 > l2/2√3. K diverges
proposedan AdS solitonas a candidate ground state for when µ= 2r4/l2,−so µextr should be greater than this
−
planarblackholes. TheAdSsolitonmetric,forexample, value. The extremal solution is obtained after replacing
in five dimensions, has the form µ in Eq. (12) by µextr. This is a physically motivated
choice of background for topological black holes [4, 5].
r2 r4 l2 r4 −1
ds2 = 1 0 dφ2+ 1 0 dr2
l2 − r4 r2 − r4
(cid:18) (cid:19) (cid:18) (cid:19) III. BACKGROUND SUBTRACTION AND
2
+r2 dt2+ dθi 2 , (10) THERMODYNAMIC QUANTITIES
− !
i=1
X(cid:0) (cid:1) For a solution ofthe equations ofmotion, the classical
where r is a constant related to the AdS soliton mass. action (1) becomes
0
We are interested here in the k = 1 case. When α=0,
kµ = −=1, al2n/d4,nan=d s4o,tthheeeexxttrreemma−allmmetarsisc hpaasratmheetfeorrmis I = 16πG1 dn+1x√−g −n 2 3R+ n8Λ3 .
extr − n+1 Z (cid:18) − − (cid:19)
(15)
r2 l2 2 l2 l2 −2 TheGauss-Bonnettermisatopologicalforn=3,andso
ds2 = 1 dt2+ 1 dr2
− l2 − 2r2 r2 − 2r2 wemustconcernourselvesherewithspacetimesforwhich
(cid:18) (cid:19) (cid:18) (cid:19) n > 4. The action (15) diverges for a classical solution
+r2dH2 . (11)
3,k= 1 whenintegratedfromzerotoinfinity. Tomaketheaction
−
(or energy) finite, there should be a regularized setting
For this to be a candidate ground state, the spacetime
– a cutoff in the radial integration and subtraction of a
region should be restricted to r >r =l/√2.
e
suitablychosenbackground. Fork=+1,thebackground
Next,considertheAdSblackholesolutionwithα >0:
′
is simply an AdS space with µ = 0, while, for k = 1,
−
the subtraction of a non zero mass extremal background
r2 r2 4α 4αµ appears more physical [5, 26].
f(r)=k+ 1 + . (12) For a pure anti–de Sitter space (X ) any value of the
2α − 2α − l2 rn 1
r periodicity β is possible, while the black hole spacetime
′
For k =+1, µ=0 itself defines a reference background, (X ) has a fixed periodicity β. One may thus adjust
2
and it is obvious that the constraint α 6 l2/4 must β such that the geometry of the hypersurface at r =
′
hold. Interestingly, a gravitational action defined with R isthesameforAdSspaceandAdS-Schwarzschild
α=l2/4,insomecases[28],isequivalenttothe Einstein spa→ce∞if k = +1 [3], and for the extremal solution and
4
hyperbolic AdS black hole spacetime if k = 1 [5]. The OnereadsM fromEq.(6),andE isanintegrationcon-
0
−
surface terms have no role in the large r limit because stant. The energy E, obtained directly using the Eu-
the black hole correctionsto the AdS metric or extremal clideanized action, after a lengthy but straightforward
state vanish too rapidly at r =R [3, 13]. Thus, as calculation, takes a remarkably simple form
→∞
in Refs. [3, 5], we may fix β by demanding
′
∂I
E = =M M δ . (22)
µextr r2 µ r2 ∂β − extr k,−1
β k + β k + . (16)
′ − r2 l2 ≃ − r2 l2 b
r r Comparing Eqs. (21) and (22), we easily identify that
In doing this, we find the (Euclidean) action difference E0 = Mextrδk, 1 > 0. For k = +1, the thermody-
namic−energy is gi−venby the black hole mass M and the
I =I(X ) I(X ) to be
2 1
− AdSsolutionwithµ=0istheonewithlowestactionand
b I = −V1n6−π1Gr+nn−+41((nn−−13))β"2kr+2 −µr+4−n+ 4lr2+4 aeEnned=rgE−y.>MTe0hxitosrthi>seru0wsuiwsaehl.leynImnµopto=rtth0ae,ntcElays,=ethf0oerwtkoht=eanl−eµn1e=,rrgaµytehixsetrra,,
b 8πr3 positive,concavefunctionoftheblackhole’stemperature
− (n 1+)β#−βMextrδk,−1, (17) for all values of k (=0, ±1) when Mextr <0.
−
where M =(n 1)V µ /16πG . [This canbe
extr n 1 extr n+1 A. Extremal state as the ground state
easilywritten, usin−gEq.−(6), inthe formreportedearlier
inRef.[26];cf.,Eq.(10)]. Thefreeenergyofablackhole
As we mentioned in the Introduction, the hyperbolic
is given by F =I/β, namely,
AdS black holes present some special features, such as
5
F = (1n6π−G1b)Vn(−n1r+n−3)4 kr+2−αk2 + 3rl+24 that the3tVotarl2(thermodry2namicα) energ3yl2V 4α
n+1 − (cid:20) (cid:21) E = 3 + 1+ + + + 3 1
+2(Vnn−−13r)+nG−n1+1β1 −(cid:0) Mextrδk,−1(cid:1) = 136Vπ3G (cid:18)−r2 + lr2+4 +rl+22 (cid:19)>064πG (cid:18) − l2 ((cid:19)23)
= Vn−1r+n−2 k r+2 + n−1 αk2 16πG(cid:18)− + l2 4(cid:19)
16πGn+1 "(cid:18) − l2 (cid:19) n−3 r+2 is independent of the coupling α. This result may have
somenewandinterestingconsequencesinthefieldtheory
2n αk 2r2 +kl2
+ M δ . (18) dual, if the latter can exist with a GB term. We can
−n−3 l2 (cid:0)r+2 +2αk(cid:1)#− extr k,−1 perhaps use the relation
This is modified from(cid:0)the result(cid:1)in Ref. [25] only by the α′ = 2g2 N −1/2
last term, which is non zero when k = 1 and α=l2/4. l2 YM
− 6
But, since Mextr is r+ or β independent, the black hole (in units 16πG5 = 1, α′(cid:0)= α/2).(cid:1)Then a small α′ corre-
entropy turned out to be the same as was given in [25]: spondstothestrongcouplinglimit(i.e.,g2 N islarge.)
YM
The black hole entropy is given by
=β2 ∂F = Vn−1r+n−3 r2 + 2(n−1)αk . (19)
S ∂β 4G + (n 3) = β (E F).
n+1 (cid:18) − (cid:19) S | | −
For k = 0 or +1, we find lim βF = 0. This is usu-
(See also Ref. [23] for an alternative derivation.) The β
ally not the case for a hyperbo→lic∞AdS black hole, since
entropy flow is given by 5
the free energyandthe entropybothdepend onthe cou-
dS = (n−14)GVn−1r+n−4 r+2 +2αk pnloitn.gSαo,biuntptahreticeunlearrg,yfoEra(osrmeanlelr(goyrvdaennissihtyinEg)/Vco3u)pdloinegs
n+1
(cid:16) (cid:17) α′, we find limβ βF = finite. This result obviously
= β (n16−π1G)nV+n1−1"nrl+n2−1 +(n−2)kr+n−3 egxenteenradlslyt,oththeevarer→liaa∞tteiodndoifscfruesesieonnesrgmya(daenidneRnterfo.p[8y]);wmhoerne
going from strong to weak coupling limits.
+(n−4)αk2r+n−5#. (20) theThcoeutpoltianlgeαnewrghyenofnhy>p4er,baonldicibtlahcaks haolloecsadlempeinnidmsuomn
atthe extremalhorizonposition, whichis seenalso from
In the second line above, we have used Eq. (5). Using theplotsinFigs.1and2. Evenforthecouplingα=l2/4,
the thermodynamic relation d = βdE, and after inte- the total energy, for example, when n=6,
S
grating with respect to r , we arrive at
+ 5V r6 r2l2
E = 5 r4 + + + + (24)
E =M +E0. (21) 16πG7 (cid:18)− + l2 4 (cid:19)
5
M, Mextr E
6
10
4 8
6
2
4
0.2 0.4 0.6 0.8 1 1.2 1.4 r+ 2
0.2 0.4 0.6 0.8 1 1.2 1.4 r+
-2
-2
-4 -4
M, Mextr E
1 8
0.5 6
4
0.2 0.4 0.6 0.8 1 1.2 1.4 r+
2
-0.5
0.2 0.4 0.6 0.8 1 1.2 1.4 r+
-1
-2
-1.5 -4
FIG.1: TheADMmassM (curvedlines)andextremalmass FIG. 2: The energy E as a function of horizon (r+). The
Mextr (horizontal lines) as functionsof thehorizon. The val- values are fixed at l =1, 16πG=1, and n=4 (upper plot),
ues are fixed at l = 1, 16πG = 1, and (upper plot) n = 4, V3=2π2,k=+1(curvesthatgrowwithr+)atα=1/4,1/12
V3 = 2π2, k = −1, and α = 1/4, 1/12, 0 (top to bottom); (upper to lower), and k = −1 (curve that has minimum at
(lower plot) n=6, V3 =π3, k=−1, and α=1/4, 17/100, 0 r+ = rextr); n = 6 (lower plot), V3 = π3, k = −1, and
(top to bottom). α=1/4,17/100,0 (top to bottom along the E-axis).
is vanishing at the extremal horizon r = l/√2. But, along the path parameterized by x is
+
since E is negative in the range 0<r <l/√2, a mass-
+
less extremal state can violate the positive energy theo- ∂2Iˆ 1 dβ 2 dE
rem, but a negative mass extremal state with α < l2/4 = , (25)
∂2x β2 dx dT
always respects the energy condition E >0. !T (cid:18) (cid:19)
where x=x(T) is the parameter that labels the path in
the Euclidean path integral formulation. Thus, as dis-
B. Thermodynamic instability for k=−1 cussedin[30],the blackholeisnotthe localminimumof
the action when the specific heat dE/dT is negative.
Here we adopt a Euclidean path integral formulation
along with a consideration that the (extremal) entropy
and specific heat are non-negative at the background.
IV. SPECIFIC HEAT AND FREE ENERGY
Onealsonotesthattheblackholeentropyisalwayspos-
CURVES AND THERMAL PHASE TRANSITION
itive for the k = 0,1 family. However, for k = 1,
{ } −
the inequality r2 > 2α, which must hold in order to
+
haveablackholeinterpretation,doesnotguaranteethat InEinsteingravity(α =0),smallsphericalblackholes
′
the (extremal) entropy and specific heat are always pos- havenegativespecificheatbutlargesizeblackholeshave
itive, rather one must satisfy r2 > 2(n 1)α/(n 3). positive specific heat [31]. There exists a discontinu-
+ − −
Thereisindeedanupper boundinthe couplingconstant ity of the specific heat as a function of temperature at
α which ensures a thermodynamicalstability; e.g., when r =l/√2, and so small and large black holes are some-
+
n + 1 = 7, the positivity of extremal entropy requires whatdisjointobjects. However,thisisnotessentiallythe
r2 > 10α/3 and hence α/l2 < 17/100. In particular, case when α is nontrivial, and specially, in the spheri-
+ ′
when one approaches the µextr = 0 limit at 4α = l2, so cal AdS5 case, the small black holes also have positive
Mextr = 0, the specific heat could be negative, which specific heat [25] (see also the discussion in Ref. [23]).
mimics thermodynamic instability of the solution.
Here we study the AdS and AdS black holes, which
5 7
In the canonical ensemble, which should be the case are of some particular interest in string or M theory as
here as we are considering uncharged black holes, the they may provide duals for CFTs describing the world-
second derivative of the Euclidean action I (= βE ) volume theory for N parallel D3- or M5-branes.
−S
b
6
A. Specific heat for hyperbolic black holes beta, C
40
30
In the AdS5 case, the specific heat curve has a single 20
branch for α=l2/4 and two branches for α<l2/4. The 10
first cusp on the left, which will almost coincide with 0.2 0.4 0.6 0.8 1 r+
the r axis for α l2, has negative specific heat, so is -10
+
unstable. Moreove≪r,the entropy formula -20
-30
V r3 6α -40
= 3,k=−1 + 1 (26)
S 4G − r2
5 (cid:18) +(cid:19) beta, C
20
shows that the extremal entropy is non-negative only if
15
α6l2/12. It is then relevant to ask what would happen 10
in the limit l2/4>α>l2/12. As a specific example, for 5
αis=nelg2a/t8i,vtehoenplyeriiondtβhe=rπanl2ge4r0+2.5−<l2r/[2<r+0.27r0+27.−Bl2ut] -5 0.2 0.4 0.6 0.8 1 r+
+
whenα=l2/8,theentropyin(cid:0)Eq.(26)(cid:1)becom(cid:0)esnegativ(cid:1)e -10
-15
in the range l2/2 < r2 < 3l2/4, although β is positive
+ -20
there. Thus, hyperbolic AdS black holes are thermody-
5
namically stable only if α 6 l2/12, i.e., when > 0
extr beta, C
S
(see Fig. 3). 15
10
beta, C 5
100 0.2 0.4 0.6 0.8 1 r+
50 -5
-10
0.2 0.4 0.6 0.8r+
-15
-50
-100
FIG. 4: The Euclidean period (curves that asymptote to the
C or/andr+ axes)andspecificheat(curveswithtwoormore
beta, C cusps) vs horizon position r+. The values are fixed at l =1,
40 n= 6, V5/16πG=π3, k =−1, and α=0.25, α=0.23, and
α=17/100 (top to bottom plots).
20
0.2 0.4 0.6 0.8 1 r+
-20 The free energy of a black hole is background depen-
-40 dent, while the Hawking temperature is not. But the
formulas C = dE/dT and C = dM/dT give the same
-60
answer since M is r or T independent. In the AdS
extr + 5
case,thespecificheatcanbenegativeforasmallr ,spe-
+
beta, C
cially, if α > l2/12. This particular feature is, although
40
not totally new, different from the k = +1 case. For
30
k =+1, a thermodynamic instability may arise due to a
20
finitesizeeffect,viz.,ablackholeofsizer 6l/√2isun-
10 +
stable,r >lisstable,andl >r >l/√2isonlylocally
0.2 0.4 0.6 0.8 1 r+ preferred+butgloballyunstable. H+owever,fork= 1,in-
-10 −
stability may arise from the both a large coupling effect
-20
(α l2) and a small size effect (r <l).
-30 ∼ +
For the AdS case, the extremal entropy is zero when
-40 7
α=17l2/100,for which the Euclidean period is
FIG. 3: The Euclidean period β (curves that asymptote to β = 4πr+l2(50r+2 −17l2) . (27)
the C or/and r+ axes) and specific heat (curves with one or (10r2 l2)(30r2 17l2)
more cusps) vs horizon position r+. The values are fixed at +− +−
l=1, n=4, V3/16πG=2π2, k=−1, and α=1/4, α=1/8 This is negative when 0 < r < 0.3162 and 0.5831 <
+
and α=1/12 (top tobottom plots).
r <0.7528. Thefreeenergyisalsonegativeinthelatter
+
range. That is, the first and third cusps in the bottom
7
plotofFig.4shouldbemirrorreflected. Thereisnoblack beta,F
holeinterpretationfor the firstcusp. Forα=l2/4,there 20
are only two cusps, because in this case two unstable
15
branches(thesecondandthirdcuspsthatappearforα<
l2/4) merge to a single cusp, which has negative specific 10
heat. We easily see that the thermodynamic stability of 5
hyperbolic black holes in n = 6 requires α/l2 6 17/100. 0.25 0.5 0.75 1 1.25 1.5 r+
As a result, the specific heat and extremal entropy are
-5
positive at the background.
Moreover,fork = 1,the entropyofablackhole may -10
−
appear to be negative, typically in the strong coupling
limit α l2; see, for example, [32]. But this parameter beta,F
∼ 20
space must be excluded as an unphysical region because
15
such a state is not stable classically nor do the super-
10
gravity approximations allow one to take α in the same
order as l2. As a result, for α l2, the hyperbolic black 5
hole is still stable and has pos≪itive entropy. 0.2 0.4 0.6 0.8 1 1.2 1.4 r+
-5
-10
-15
B. Free energy for hyperbolic black holes
beta,F
In Ref. [14], it was argued that the stringy corrections 20
of order α3R4 do not give to rise a thermal phase tran- 15
′
sition for flat and hyperbolic horizons, as a function of 10
temperature. But using an AdS soliton as the thermal 5
backgroundofAdSblackholeswithRicciflathorizons,a 0.2 0.4 0.6 0.8 1 1.2 1.4r+
phasetransitionthatisdependentnotonlyontheHawk- -5
-10
ingtemperaturebutalsoontheblackholeareawasfound
in[33]. ThismayimplythatfortheflatspaceIRn 1com- -15
−
pactified down to a torus, and possibly also for the quo-
tient of hyperbolic space Hn 1/Γ, the choice of ground
− FIG. 5: The Euclidean period (curves that asymptote to the
state is crucial to see a possible phase transition. Here
F orr+ axis) and free energy (curvesthatare boundedfrom
we want to use the obtained thermodynamic quantities
aboveortakefinitevaluesat r+=0) vshorizon position r+.
to determine a thermal phase structure in the Einstein- The values are fixed at k = −1, n = 4, l = 1, 16πG = 1,
Gauss-Bonnet gravity. V3=2π2, and, α=1/6, 0.1305, 1/8 (top to bottom plots).
Inthelimit0<r2 62(n 1)α/(n 3),thebehaviorof
+ − −
extremal entropy is somewhat exotic, and so the second
law of thermodynamics may not hold. In addition, for a
singularity at r = √2α, where β = 0, is hidden inside
hyperbolic AdS black hole, the free energy scales with +
5
the extremal horizon, so is harmless.
the coupling α, but the energy (or energy density) does
not (Fig. 5 and Eq. (23)). The total energy E is vanish- When n = 4 and 0.25l2 > α > 0.1305l2, the free en-
ing at the extremal horizon, where β = , but the free ergy appears to be positive in a certain range r >r >
2 +
∞
energycanbepositive,zero,ornegativetheredepending rextr,wheretheHawking-Pagetemperatureisfinite,but,
uponthecouplingα. Asaresult,theentropyisnonzero for a small coupling α l2, the free energy takes only
≪
at the extremal state, in particular, for a zero or small the negative value for positive β. It takes a maximum
coupling α. This result is consistent with the earlier ob- value at the extremal state if = 0 (for example,
extr
servationsmade in [8, 10] with α =0; a commonthread at α = l2/12 when n = 4). TShere is no thermal AdS
′
intheseresultsisthatlim βF =finite. Themeaning phaseforthecouplingα<0.1305l2,andhence nophase
β
ofanonzeroextremalentro→p∞ymaybetracedbacktothe transition appears to occur in this case.
observationof“precursor”statesfirstmadein[34],when
studying a short distance behavior of AdS/CFT duality.
The first plot in Fig. 6 corresponds to the µ = 0
extr
statewithα=l2/4. Inthe lowertwoplots,theextremal
C. Free energy for spherical black holes
state is shown by an asymptotic on the right, and the
critical state, which is absent in AdS (cf. Fig. 5), is
5
shown by another asymptotic on the left. The middle The spherical Gauss-Bonnet black holes were studied
vertical line, where the free energy diverges and β = 0, in [23, 25] in some detail, so we will be brief here. For
correspondstothehorizonatr2 =2α. Forα<l2/4,the thiscase,sinceE =M,thefreeenergyisvanishingwhen
+
8
beta, F beta,F
50
40 10
30
20 5
10
0.25 0.5 0.75 1 1.25 1.5 r+ 0.2 0.4 0.6 0.8 1 r+
-10 -5
-20
beta, F beta, F
6 5
4 4
3
2
2
0.2 0.4 0.6 0.8 1 1.2 1.4r+ 1
-2 0.5 1 1.5 2 r+
-4 -1
beta, F
6 FIG.7: Inversetemperatureβ (curvesthatasymptotetothe
4 F and r+ axes) and free energy (curves that cross r+ axis)
vs horizon position r+ for spherical black holes. The values
2 are fixed at k = +1, n = 4, l = 1, and, from top to bottom,
α=0.0278,0.01. Anewbranchofastableblackholeappears
0.2 0.4 0.6 0.8 1 1.2 1.4r+ in the AdS5 case.
-2
-4
FIG. 6: The Euclidean period (curves that asymptote to the
small r region, F nearly approaches zero but it never
F or/and r+ axis) and free energy (curves that have finite touches+the F = 0 axis, e.g., when α/l2 = 3 10 7,
vvaalluueess aarterfi+xe=d a0tokr=bo−u1n,dned=fr6o,ml=ab1o,v1e)6πvGs h=or1i,zoVn5.=Tπh3e, F ∼ 4 × 10−6. That is, the free energy curve×cros−ses
and α=0.25,0.2162,0.2072 (top to bottom plots). the F = 0 axis only once, namely, when r+ . l, which
correspondstotheHawking-Pagephasetransitionpoint.
For α/l2 & 0.0278, the free energy is a monotonically
decreasing function of r and the second branch (peak)
+
α=α˜ =r2 4 n(n−3)−(n−3) . (28) on the right disappears. Using the relation α′/l2 =
+" p 6(n−1) # 1/ 2g2N, we see that the Hawking-Pagetransition dis-
appears at sufficiently small ’t Hooft coupling g2N.
p
This separates two other F =0 trajectories [23]:
As for the charged AdS black holes [35], the inverse
α = r+2 9r+4 + (15−n)r+2 (n+1)(7n−25) temperature and free energy curves in Fig. 7 remind us
± ± 2 s l4 (n 1)l2 − 4(n 1)2 of the behavior of a van der Waals gas in the Clapeyron
− −
(V,P)phase,wherethepointofinflectioninβ(r)signals
3r2 (n 7)
+r2 + − . (29) a critical point. This behavior may be seen for spherical
+ 2l2 − 4(n 1)
(cid:18) − (cid:19) AdS black holes that receive stringy corrections of the
form α3R4 [14] or R Rabcd [36].
Obviously, n > 4 is needed to ensure that α > 0. For ′ abcd
example, when n = 4, we find r2 = 18α/7 > 2α. The
+
α branch of the solutions gives a stable AdS solution Onealsonotesthat,inalldimensionsn>4,thebehav-
w−hen α l2, while the α branch may be unstable. ior of a Hawking-Page transition is qualitatively similar
+
≪
As in the α = 0 case, a spherical Gauss-Bonnet black forsmallandlargecouplings,exceptthatthefreeenergy
hole presents an interesting thermal feature, namely, the becomes more positive as α is decreased. But still, as in
Hawking-Pagephase transition at finite temperature. In the AdS case, a low temperature phase corresponds to
5
the AdS case, as α is decreased, the free energy lowers athermalAdSspacewithF >0andahightemperature
5
toward zero at low temperature. For α l2, in the phase corresponds to an AdS black hole with F <0.
≪
9
beta, F with
3
1 8αΛ 1 4α
2 C = 1 1+ 1 1 .
2α ∓s n(n−1)!≡ 2α ∓r − l2 !
1 (32)
Fortransversetrace-free(deDonder)perturbations,hc =
0.2 0.4 0.6 0.8 1 1.2 1.4r+ ¯ah =0, we obtain c
ab
∇
-1
2δR = (∆ h)
ab L ab
beta, F 2nC Cα(3n2 3n+2)
4 = 1 − hab.
−1 2Cα − n(n 3)
− (cid:20) − (cid:21)
3
(33)
2
If the base is hyperbolic, then it is not a priori suffi-
1 cientto proMvestabilityforthe tensormodes only. Inthis
0.25 0.5 0.75 1 1.25 1.5 1.75 r+ case, the scalar and vector modes can also be unstable
but we will not consider such modes here.
-1
We wish to study the stability ofthe backgroundmet-
rics under the conditions
FIG.8: Inversetemperatureβ (curvesthatasymptotetothe
h =h =0. (34)
r+ axis) and free energy (curves that cross the r+-axis) vs 0a 1a
horizon position r+ for spherical black holes. The values are
Withtheseconstraints,alongwitharequirementthatthe
fixed at k = +1, n = 6, l = 1, and, from top to bottom,
α=0.1, α=0.01. base is itself an Einstein space Rij(h)=(n 2)kg˜ij,
M −
the transverse trace-free property of h is characterized
ab
by hij, where i, j, run from 2 toen ( D 1),
··· ≡ −
V. STABILITY ANALYSIS: METRIC
PERTURBATIONS 2δRij =(∆Lh)ij =−2nAhij, (35)
with
The stability of background metrics under gravita-
tional perturbations can normally be checked from an 1 d2 4
(∆ h) = ∆˜ h f + h
analysis of whether bound states with a negative eigen- L ij r2 L ij − dr2 r2 ij
state exist. If the corresponding Schr¨odinger equations (cid:18) (cid:19)
rf +(n 5)f d 1 d2
allowastatewithE <0and/ortherearegrowing(quasi) ′ − h + h ,
normalmodes atthe eventhorizon,and/orthe potential − r dr ij f dt2 ij
is negative and unbounded from below, then the space- C Cα(3n2 3n+2)
A = 1 − ,(36)
time metrics can be unstable. 1 2Cα − n(n 3)
− (cid:20) − (cid:21)
where ∆˜ h is the Lichnerowicz operator acting on .
L ij
A. Linearized field equations We look for unstable tensor modes of the form M
Consider a linear perturbation of the metric hij(x)=r2φ(r)eωthij(x˜), (37)
g¯ab →gab =g¯ab+hab, gab =g¯ab−hab where (x˜) are coordinates on M.e It is convenient to
assume that
with ha 1.Forα =0,the linearizedequationsare
b ′
| |≪
∆ h =λh , (38)
L ij
δR = nch , (30) ij
ab ab
− (cid:16) (cid:17)
where c hasthe dimensionofℓ2. For α >0 andk =+1, where λ is the eigenvealuee of theeLichnerowicz operator.
′
Equation (35) then takes the form of a Sturm-Liouville
as evaluatedin the Appendix, the perturbationsδg (=
ab
type problem [18]
h ) must satisfy
ab
(1 2Cα)δRab = nC 1 (3n2−3n+2)Cα hab −fddr frn−1 dφd(rr)
− − − n(n 3) (cid:18) (cid:19)
(cid:18) − (cid:19)
2f λ
2Cα g¯ ¯c¯dh ¯2h+C(n+1)h . (n 2)f +rf′ nAr2 φ(r)
− n 3 ab ∇ ∇ cd−∇ − r2 (cid:18) − − 2 − (cid:19)
−
(cid:2) (cid:3) (31) =ω2φ(r)rn 1. (39)
−
10
In terms of Regge-Wheeler type coordinates dr fdr positive potential, while, for n > 10, the eigenvalue λ
and φ(r) Φr (n 1)/2, this takes the form ≡ ∗ may take a negative value, so ν >1, but this solution is
− −
≡
always non-normalizable.
d2Φ For Λ<0, the potential (42) is modified to
V (r(r ))Φ=ω2Φ, (40)
dr2 − ∗
∗ 4λ+(n 2)(n 10) 1 (n 1)(n+1)
V(r) = − − − + −
where 4r2 4l2
(cid:18) (cid:19)
λf [(n 2)(n 10) 1]f2 (n 5)ff r2
V(r) = + − − − + − ′ 1+
r2 4r2 2r × l2
(cid:18) (cid:19)
+2nAf. (41)
1 4(λ n+1)+(n 3)(n 5)
V(x) = − − −
l2 4x
A requirement of finite energy is equivalent to the nor-
malization condition that Φ2dr = f(r) 1Φ2dr is
− (n 1)(n+1)
finite. Usually, the stability of a∗potential depends on + − (1+x), (45)
the eigenvalues of the LichRnerowicz opRerator, ensuring 4 !
that the potential is positive and bounded from below.
where x r2/l2. This is always positive and bounded
≡
frombelowinsatisfyingEq.(43). TheSchr¨odingerequa-
B. Stability of massless state in general relativity tion may be expressed as a hypergeometric equation
(x+1)(3x+1)
The Schwarzschild black hole is stable in four dimen- x(x+1)2Φ′′(x)+ Φ′(x)
2
sions [37]. It is also learnt that a spherically symmetric
AdS black hole is stable against small electromagnetic x+1 4λ+(n 2)(n 10) 1
4 − − −
and gravitational perturbations [38]. The issue of insta- − 4 × 4x
bility of tensor modes, however,arisesonly for n+1>4
since there are no tensor harmonics on S2 or H2. (n 1)(n+1) ω2l2
+ − + Φ(x)=0, (46)
Foravanishingcosmologicalconstant(Λ=0),sothat 4 (x+1)
!
A=0,thespacetimemetricisflatifthebaseisasphere,
and if the base is not a sphere then the full spacetime is with no singularities in the range 0 < x < . To sim-
a cone which is singular at the origin. Thus, for Λ = 0, plify (46) one can make the following substit∞utions:
wemusttakek =1,sothatf(r) 1asymptoticallyand
→
r =r . Then the asymptotic potential is iωl
∗ 2ν = 4λ+(n 2)(n 10), σ = ,
− − − 2
4λ+(n 2)(n 10) 1 (2ν 1)(2ν+1)
V (r)= − − − = − . p 2ν (n 2) 2ν+(n+2)
∞ 4r2 4r2 a = σ+ − 4 − , b=σ+ 4 ,
(42)
c=ν+1. (47)
This potential is non-negative only if
1+(n 2)(10 n) Underthe decompositionΦ(x)=x(1+2ν)/4(1+x)σϕ(x),
λ > − − , two independent solutions of Eq. (46) are found to be
4
1 1
or ν = (n 2)(n 10)+4λ> , (43) F (a, b; c; x),
2 − − 2 ϕ(x)= 2 1 −
x1 c F (a c+1, b c+1; 2 c; x),
p (cid:26) − 2 1 − − − −
where V (r) = 0 if equality holds. Stability of a po- (48)
tential re∞quires that there are no bound negative energy where 2F1(a,b;c;x) is the hypergeometric function, and
states or no growingmodes with ω >0 at the horizon,if theparameterc=0, 1, 2, 3, forthefirstsolution,
6 − − − ···
the latter exists. A requirement that λ > 0 indeed con- and c = 2,3,4, for the second solution. A general
6 ···
straintsthespacetimedimensionsD =(n+1)ton610. solution of Eq. (46) is given by a linear combination of
The asymptotic solution for Φ(r) that decays as r goes
to + is 2ν (n 2)
∞ Φ (x) = x(1±2ν)/4 (1+x)−iωl/22F1 ± − −
± 4
Φ = C √rJ ω2r +C √r Y ω2r
1 ν 2 ν
∞ − − iωl 2ν+(n+2) iωl
= Re √rK(cid:16)(pωr) . (cid:17) (cid:16)p ((cid:17)44) , ± ; ν+1; x .
ν − 2 4 − 2 ± − !
For small r a(cid:2)nd a real p(cid:3)ositive ν, Φ (r) behaves as (49)
r ν+1/2. This solution is divergent b∞ut normalizable∼if
−
1>ν >1/2, and divergent and non-normalizable if ν > A stability check may be needed in both limits x 0
→
1. For n 6 10, an imaginary ν does not exist with a and x . This is the case studied by Gibbons and
→ ∞