Thermodynamic instability of topological black holes in Gauss-Bonnet gravity with a generalized electrodynamics

0.36 MB·English

by  S. H. Hendi

Thermodynamic instability of topological black holes
in Gauss-Bonnet gravity with a generalized electrodynamics
S. H. Hendi1,2∗ and S. Panahiyan1
1Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
Motivated by the string corrections on the gravity and electrodynamics sides, we consider a
quadraticMaxwellinvarianttermasacorrectionoftheMaxwellLagrangiantoobtainexactsolutions
of higher dimensional topological black holes in Gauss-Bonnet gravity. We first investigate the
asymptotically flat solutions and obtain conserved and thermodynamic quantities which satisfy
the first law of thermodynamics. We also analyze thermodynamic stability of the solutions by
calculatingtheheatcapacityandtheHessianmatrix. Then,wefocusonhorizon-flatsolutionswith
adS asymptote and produce a rotating spacetime with a suitable transformation. In addition, we
5
calculate the conserved and thermodynamic quantities for asymptotically adS black branes which
1
satisfy thefirstlawofthermodynamics. Finally, weperform thermodynamicinstability criterion to
0
investigate theeffects of nonlinear electrodynamics in canonical and grand canonical ensembles.
2

n
a
I. INTRODUCTION
J

2
2 Nonlinear theories have been studied extensively in the context of various physical phenomena. In other words,
becauseofthenonlinearnatureofmostphysicalsystems,lineartheoriescouldnotpreciselydescribetheexperimental

] consequences and we must inevitably consider the nonlinear models.
c
Classicalelectrodynamics,whosefieldequationsoriginatefromthe MaxwellLagrangian,containsvariousproblems
q
- which motivate one to consider nonlinear electrodynamics (NLED). One of the main problematic items of Maxwell
r theory is infinite self-energy of a point-like charge. The first successful Lorentz invariance theory for solving this
g
[ problem was introduced by Born and Infeld [1]. After Born-Infeld (BI) theory, although some different NLED have
been introduced with various motivations, their weak field limits lead to Maxwell theory [2–5]. Amongst the NLED

1
theories, the so-called BI-types, whose first nonlinear correction is a quadratic function of Maxwell invariant, are
v
completely special[4–6]. Inadditiontothe interestingpropertiesofBI-type NLEDtheories[7],itmay be shownthat
1
these theories may arise as a low energy limit of heterotic string theory [8], which leads to increased interest. Taking
8
4 into account the first leading correction term of BI-type theories in which originated from the string theory, one can
5 investigate the effect of NLED versus Maxwell theory [9, 10].
0 Inthispaper,weconsiderGauss-Bonnet(GB)gravityasanaturalgeneralizationofEinsteingravitywithaquadratic
1. power of Maxwell invariant in addition to the Maxwell Lagrangian as a source. On the gravity point of view, string
0 theories in their low energy limit give rise to effective models of gravity in higher dimensions which involve higher
5 curvature terms, while from the electrodynamics viewpoint, a quadratic power of Maxwell invariant supplements to
1 the Maxwell Lagrangian with the development of superstring theory. In the weak field limit, GB gravity and the
:
v Lagrangianof the mentioned NLED reduce to Einstein gravity and Maxwell Lagrangian,respectively.
i On the other hand, it has been shown that there is a close relationship between black hole thermodynamics and
X
the nature of quantum gravity. For black holes to be physical objects, it is necessary for them to be stable under
r externalperturbation. Therearetwokindsofstability: physical(dynamical)stabilityandthermodynamicalstability,
a
which is of interest in this paper. The most interesting parts of black hole thermodynamics are thermal instability
and Hawking–Page phase transition [11]. It has been shown that there is no Hawking–Page phase transition for the
Schwarzschild–adSblack hole whose horizons have vanishing or negative constant curvature [12].
Thermodynamic properties of black holes in GB gravity with NLED have been studied before [13]. In this paper,
wewillobtaintheblackholesolutionsofGB-Maxwellcorrected(GB-MC)gravityinarbitrarydimensionsanddiscuss
thermal instability conditions.
One may ask the motivation for considering the GB-MC gravity. As we know, the Maxwell theory has acceptable
consequences, to a large extent, in various physical areas. So, in transition from the Maxwell theory to NLED, it is
allowableto considerthe effects ofsmallnonlinearityvariations,notstrongeffects. Inother words,inorderto obtain
physical results with experimental agreements, one should regard the nonlinearity as a correction to the Maxwell
field. Eventually, we should note that, although various theories of NLED have been created with different primitive
motivations, only their weak nonlinearities have physical and experimental importance. So the effects of nonlinearity
∗ emailaddress: [email protected]
2
should be considered as a perturbation to Maxwell theory. In addition, in a gravitational framework the GB gravity
is a natural generalization of Einstein gravity (not a perturbation in general) in higher dimensions. One may regard
GB and MC as the corrections of an Einstein-Maxwell black hole.
The layout of the paper will be in this order: In Sec. II, we will introduce the structure of action which contains
GB gravity coupled with MC. Then, we will solve the field equations and discuss the geometric properties of the
topological black holes. We will study thermodynamic properties and stability of the asymptotically flat spacetime
in Sec. III. In the next section, we will generalize our solutions to rotating black branes with adS asymptote and
calculate thermodynamic and conserved quantities of rotating solutions. Last part is devoted to investigating the
stability of the solutions in canonical and grand canonical ensembles. Conclusions are drawn in the last section.
II. TOPOLOGICAL BLACK HOLES
The Lagrangianof Einstein-GB gravity coupled to NLED can be written as
L =L 2Λ+αL +L(F), (1)
tot E GB
−
wheretheLagrangianofEinsteingravityisthe Ricciscalar,L =R,andΛisthe cosmologicalconstant. Inthethird
E
term of Eq. (1), α is the GB coefficient with dimension (length)2 and L is the Lagrangianof GB gravity,
GB
L =R Rabcd 4R Rab+R2. (2)
GB abcd ab
−
The last term in Eq. (1) is the Lagrangian of NLED, which we choose a quadratic correction in addition to the
Maxwell Lagrangian
L( )= +β 2+O(β2), (3)
F −F F
where β is the nonlinearity parameter and the Maxwell invariant = F Fab, where F = ∂ A ∂ A is the
ab ab a b b a
F −
electromagnetic field tensor and A is the gauge potential. For the vanishing the nonlinearity parameter, this La-
b
grangianyields the standardMaxwelltheory,asit should. Regardingthe gauge-gravityLagrangian(1)andusing the
variational method, one can obtain the following field equations:
1
GEab+Λgab+αGGabB = 2gabL(F)−2LFFacFbc, (4)
∂a √ gLFFab =0, (5)
−
where GEab is the Einstein tensor, GGabB =2 RacdeR(cid:0)bcde−2Racbd(cid:1)Rcd−2RacRbc+RRab − 12LGBgab and LF = dLd(FF).
Now,weareinterestedintopologicalstaticblackholesolutions;therefore,wetakeinto accountthe followingstatic
(cid:0) (cid:1)
metric:
dr2
ds2 = f(r)dt2+ +r2dΩ2, (6)
− f(r) k
wheredΩ2 representsthelineelementofan(n 1)-dimensionalhypersurfacewiththeconstantcurvature(n 1)(n 2)k
k − − −
and volume Vn−1 with the following explicit form:
n−1i−1
dθ2+ sin2θ dθ2 k =1
1 j i
 i=2j=1
dΩ2k = dθ12+sinh2θ1dθ22P+n−si1Qnh2θ1niP=−31jiQ−=12sin2θjdθi2 k =−1 . (7)
dφ2 k =0
Since we are looking for the black hole solution withiP=a1radiial electric field, we know that the nonzero components
of the electromagnetic field are
F = F . (8)
tr rt
−
3
One can use Eq. (5) to obtain the explicit form of F with the following form:
tr
q 4q3β
F = +O(β2), (9)
tr rn−3 − r3n−3
which for small values of β reduces to Maxwell electromagnetic field tensor.
We find that the nonzero independent components of Eq. (4) can be written by applying electromagnetic field
tensor (9):
2(1 f)α′
e = r3 1+ − f′ α′(n 4)(1 f)2+
tt r2 − − −
(cid:18) (cid:19)
2Λr4 2q2 4q4β
+ +O(β2). (10)
(n 1) (n 1)r2n−6 − (n 1)r4n−8
− − −
2(1 f)α′
e = r4 1+ − f′′ 2α′r2f′2
θθ r2 −
(cid:20) (cid:21)
(n 4)
+2r(n 2) r2+2α′ − (1 f) f′
− (n 2) −
(cid:20) − (cid:21)
α′(n 4)(n 5)(1 f)2 (n 2)(n 3)r2(1 f)
− − − − − − − −
2q2 12q4β
+2Λr4 + +O(β2). (11)
− r2n−6 r4n−8
where α′ =(n 2)(n 3)α
− −
It is straightforwardto show that the following metric function satisfies all of the field equations simultaneously:
r2
f(r)=k+ 1 Ψ(r) , (12)
2α′ −
(cid:16) p (cid:17)
with
8α′ n(n 1)m nq2 2nq4β
Ψ(r)=1+ Λ+ − + +O β2 , (13)
n(n 1) 2rn − (n 2)r2n−2 r4n−4(3n 4)
− (cid:18) − − (cid:19)
(cid:0) (cid:1)
wheremisanintegrationconstantthatisrelatedto mass. Onecanseethatforsmallvaluesofβ, themetric function
reducestousualGB-Maxwellgravity. Expansionofthemetric functionforsmallvaluesofthe GB parameterleadsto
4q4 (k f )2 8q4(k f )
f(r)=f β+ − EM α′+ − EM α′β+O α′2,β2 , (14)
EM − (n 1)(3n 4)r4n−6 r2 (n 1)(3n 4)r4n−4
− − − −
(cid:0) (cid:1)
where the metric function of Einstein–Maxwell gravity is
2Λr2 m 2q2
f =k + . (15)
EM − n(n 1) − rn−2 (n 1)(n 2)r2n−4
− − −
In order to consider the asymptotic behavior of the solution, we put m=q =0 where the metric function reduces
to
r2 8α′Λ
f(r)=k+ 1 1+ . (16)
2α′ −s n(n 1)!
−
Thisresultputsarestrictiononα′whichstatesthatforα′ −n(n−1),themetricfunctionwillberealasymptotically.
≤ 8Λ
On the other hand, for Λ=0, asymptotically flat solutions are available only for k =1.
The next step is to look for the singularities. It is a matter of calculation to show that the Kretschmann scalar of
metric (6) is
f′2 f2
R Rαβγδ =f′′2+2(n 1) +2(n 1)(n 2) , (17)
αβγδ − r2 − − r4
4
where its series expansion for small and large values of r will be
lim R Rαβγδ r−4(n−1) (18)
αβγδ
r−→0 ∝
4(n+2)Λ ς (n 1)(n+2)
lim R Rαβγδ = 1+ − , (19)
r−→∞ αβγδ nα′ − α′ α′
(cid:18) (cid:19)
n 2 8α′Λ
ς = − + 1, (20)
s n n(n 1) −
−
which, for small values of the GB parameter, it will lead to
8(n+2) 4
lim R Rαβγδ = Λ2 Λ+O(α′) (21)
r−→∞ αβγδ (n+2)n2 − n(n 1)
−
Eq. (18)-(21) show that there is an essential singularity located at r = 0 and the asymptotical behavior of the
solutions are adS with an effective cosmological constant. If this solution contains the horizon then our metric
functionis interpretedasablackhole. Inordertoinvestigatethe existence ofthe horizon,one shouldfindthe rootof
grr =f(r)=0. If we suppose that the metric function is positive for large and small r, by considering the possibility
of the existence of only one extreme root r = r , we know that f(r) has a minimum at r = r . So, we can
+ ext ext
investigate the roots of f′,
α′(n 4)
k(n 1)(n 2)r4n−6 − +1 +4q4β 2r2n−2q2 2r4n−4Λ+O β2 =0, (22)
− − ext (n 2)r2 − ext − ext
(cid:20) − ext (cid:21)
(cid:0) (cid:1)
where, for example, in the Ricci-flat case (k =0), we obtain
q2Λ2n−3( √1+8Λβ 1) 2n1−2
r = ± − +O β2 . (23)
ext
Λ
(cid:2) (cid:3)
(cid:0) (cid:1)
It is matter of calculation to show that the extremal mass for k =0 is
2q2 2rn Λ 4q4β
m = ext +O β2 . (24)
ext rn−2(n 1)(n 2) − n(n 1) − r3n−4(3n 4)(n 1)
ext − − − ext − −
(cid:0) (cid:1)
We should note that the obtained solutions may be interpreted as black holes with inner and outer horizons
provided m > m , an extreme black hole for m = m , and naked singularity otherwise. One can obtain the
ext ext
Hawking temperature by surface gravity definition with the following explicit form:
k(n 1)(n 2)r4n−6 1+ (n−4)α′ 2r4n−4Λ 2r2n−2q2+4q4β
T = − − + (n−2)r+2 − + − + +O β2 . (25)
4π(cid:16)(n 1)r4n−5(cid:17) 1+ 2kα′
− + r+2 (cid:0) (cid:1)
(cid:16) (cid:17)
It is worthwhile to mention that extremal black holes have zero temperature.
III. THERMODYNAMICS OF ASYMPTOTICALLY FLAT BLACK HOLES (k=1 & Λ=0)
Inthis sectionwe areinterestedinthe thermodynamicsofasymptoticallyflatsolutions. Using the seriesexpansion
of the metric function (12) with k=1 for large values of distance, we obtain
m 2q2 1
f(r)=1 + +O , (26)
− rn−2 (n 1)(n 2)r2n−4 r2n−2
− − (cid:18) (cid:19)
which shows that the solutions are asymptotically flat.
5
According to area law, the entropy of black holes is one-quarter of the horizon area. This relation is acceptable for
Einstein gravity, whereas we are not allowed to use it for higher derivative gravity. For our GB case we can use the
Wald formula for calculating the entropy of asymptotically flat black hole solutions
1
S = dn−1x√γ 1+2αR (27)
4
Z (cid:16) (cid:17)
e
where R is the Ricci scalar for the induced metric γ on the (n 1) dimensional boundary. Calculations show that
ab
−
one can obtain
e
S = Vn−1 1+ 2(n−1)α′ rn−1, (28)
4 (n 3)r2 +
(cid:18) − + (cid:19)
which confirms that asymptotically flat black holes violate the area law.
Consideringthe flux ofthe electricfieldatinfinity, onecanfindthe electric chargeofblackholes withthe following
form:
q
Q= . (29)
4π
Next, for the electric potential, Φ, we use the following definition
q 4(n 2)q2β
Φ= A χµ A χµ = 1 − +O β2 , (30)
µ |r−→∞− µ |r−→r+ (n−2)r+n−2 − (3n−4)r+2n−2!
(cid:0) (cid:1)
where χµ is the null generator of the horizon. It is notable that although the electric potential depends on the
nonlinearity of electrodynamics, the electric charge does not.
In order to calculate the finite mass of the black hole we use the ADM (Arnowitt-Deser-Misner)approachfor large
values of r, which will result in [14]
Vn−1
M = m(n 1). (31)
16π −
We shouldnote that one canobtain m fromf(r =r )=0, sothe total massdepends onGB andNLED parameters.
+
Next, byusing Eqs. (12), (28), (29) andconsideringM asa functionofthe extensiveparametersS andQ,wehave
rn−2 α′ 32π2Q2 1024π4Q4β
M(S,Q)= + (n 1) 1+ + +O β2 . (32)
16π " − (cid:18) r+2 (cid:19) (n−2)r+2n−4 − (3n−4)r+4n−6#
(cid:0) (cid:1)
Now, we calculate the temperature and electric potential as the intensive parameter with the following relation
∂M ∂M ∂S ∂M
T = = , Φ= , (33)
∂S ∂r ∂r ∂Q
(cid:18) (cid:19)Q (cid:18) +(cid:19)Q,(cid:18) +(cid:19)Q (cid:18) (cid:19)S
which are the same as the ones that were calculated in Eqs. (25) and (30). Thus, the conservedand thermodynamic
quantities satisfy the first law of thermodynamics,
dM =TdS+ΦdQ. (34)
A. Stability of the solutions
Here, we investigate the thermodynamic stability of charged black hole solutions of GB gravity with NLED. In
the grandcanonicalensemble, the thermodynamic stability may be carriedout by calculating the determinant of the
HessianmatrixofM(S,Q)withrespecttoitsextensivevariablesX , HM = ∂2M . Inthe canonicalensemble,
i XiXj ∂Xi∂Xj
theelectricchargeisafixedparameter,andthereforethepositivityoftheheatc(cid:16)apacityC(cid:17) =T / ∂2M issufficient
Q + ∂S2
(cid:16) (cid:17)
6
to ensure the localstability. Since the physicalblackhole solutions havepositive temperature, itis sufficient to check
the positivity of ∂2M = ∂T / ∂S ,
∂S2 Q ∂r+ Q ∂r+ Q
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
∂2M 2(n 4)α′2 (n 8)α′r2
= r4n−8(n 1)(n 2) − + − + +r4
∂S2 − + − − (n 2) (n 2) +
(cid:18) (cid:19)Q (cid:26) (cid:20) − − (cid:21)
2r2n−2q2 2α′(2n 5)+(2n 3)r2 +2βq4[4α′
− + − − +
(4n 7)+(cid:2)(8n 10)r2 / π(n 1)(cid:3)2r5n−10 2α′+r2 3 +O β2 . (35)
− − + − + +
(cid:3)(cid:9) h (cid:0) (cid:1) i (cid:0) (cid:1)
One may study the behavior of ∂2M for small values of the GB parameter with the following form:
∂S2
Q
(cid:16) (cid:17)
∂2M (n 1)(n 2)r4n−4 2(2n 3)r2nq2 4(4n 5)q4
= − − + − − + − β
∂S2 − (n 1)2r5n−4π − (n 1)2r5n−6π
(cid:18) (cid:19)Q − + − +
(5n 4)r4n 16r2n+4q2 64q4
+ − + − + α′+ βα′
(n 1)r5n+2π (n 1)r5n−4π
(cid:2) − + (cid:3) − +
+O α′2,β2 . (36)
In addition, it is worthwhile to mentio(cid:0)n that(cid:1)for small and large values of r , we obtain
+
(4n−7)βq4 <0, α′ =0,β =0
−π(n−1)2α′2r5n−10 6 6
+
(cid:18)∂∂2SM2 (cid:19)Q(cid:12)(cid:12)(cid:12)(cid:12)Smallr+ =  2−π(πn24((−(n2(241n−nn)−2−1−α)5325)′)2rq)qβr+522n+3qn−4−68><>00,0,, ααα′′′ 6===000,,,βββ =6==000 , (37)
∂2M (cid:12)  nπ(n−21)2r+3n−4
= − <0, (38)
(cid:18) ∂S2 (cid:19)Q(cid:12)(cid:12)Larger+ −π(n−1)r+n
(cid:12)
(cid:12)
which indicate that for nonzero α′ an(cid:12)d β, asymptotically flat black holes with a large or small horizonradius are not
stable. In other words, there is an upper limit, r , as well as a lower limit, r , for the asymptotically flat
+max +min
stable black holes (r < r < r ). Although this result does not depend on the value of the GB parameter,
+min + +max
asymptotically flat black holes with a small horizon radius may be stable in the presence of a pure Maxwell field
(absence of nonlinearity of electrodynamic correction). In other words, taking into account thermal stability in the
canonical ensemble, the nonlinearity of electrodynamics has a reasonable influence on a small horizon radius.
In order to investigate the effects of GB gravity and NLED on the local stability for arbitrary r , we plot Figs. 1
+
– 6. Here, we consider left figures which are appropriate for the canonical ensemble analysis.
In Figs. 1 and 2 we investigate the effects of considering GB gravity. One can find that when the GB parameter is
less than a criticalvalue (α′ <α′), T has a realpositive root at r . In this case, ∂2M >0 for r <r <r .
c + 0 ∂S2 0 + +max
Q
In other words, small black holes are not physical and large black holes are no(cid:16)t stab(cid:17)le. Moreover, we find that
r (r ) decrease (increase) when we increase α′. In Fig. 2, we consider α′ >α′, in which T is a positive definite
0 +max c +
function of r . In this case one can obtain stable solutions for r < r < r , in which confirms that small
+ +min + +max
and large black holes are unstable. We should note that increasing α′ leads to increasing (decreasing) the value of
r (r ) and, therefore, decreases the range of stability.
+min +max
Besides,weplotFigs. 3and4toanalyzetheeffects ofNLED.Wefindthatforthefixedvaluesofq,nandα′,there
is a critical value for the nonlinearity parameter, β , in which for β < β , the temperature is positive for r > r .
c c + 0
This case is the same as that in Fig. 1 with α′ < α′ and we may obtain asymptotically flat stable black holes for
c
r < r < r . In Fig. 4 we set β > β to investigate the stability condition for positive definite temperature.
0 + +max c
One finds there are lower and upper limits for the horizon radius of stable black holes. It means that for β > β ,
c
asymptotically flat black holes are stable when r <r <r .
+min + +max
WestudytheeffectsofelectricchargeinFigs. 5and6. Thesefiguresshowthatforfixedvaluesofα′ andβ,thereis
a criticalvalue for the electric charge,q , in which the temperature is positive definite for q <q . When q >q , there
c c c
is a real positive root (r ) for the temperature which is an increasing function of q. We should note that the general
0
behaviors of Figs. 5 and 6 are, respectively, the same as those in Figs. 1 and 2. In other words, asymptotically flat
7
FIG. 1: Asymptotically flat solutions: (cid:16)∂∂2SM2(cid:17)Q (left figure) and (cid:12)(cid:12)HMS,Q(cid:12)(cid:12) (right figure) versus r+ for n = 5, q = 1, β = 0.001
and α=0.1 (solid line), α=0.3 (dotted line) and α=0.5 (dashed line). ”bold lines represent the temperature”
FIG.2: Asymptotically flat solutions: (cid:16)∂∂2SM2(cid:17) (leftfigure)and |H1MS0,Q| (rightfigure)versusr+ forn=5,q=1,β =0.001and
Q
α=0.6 (solid line), α=0.8 (dotted line) and α=0.1 (dashed line). ”bold lines represent the temperature”
black holes are stable for r < r < r and one should replace r with r for q > q . It is worthwhile to
+min + +max +min 0 c
mention that both r and r are increasing functions of q.
+min +max
Finally, considering Figs. 1 – 6, we should note that, in canonical ensemble, asymptotically flat black holes are
stable for r <r <r , in which one must replace r with r when T has a real positive root (q >q or
+min + +max +min 0 + c
β <β or α′ <α′).
c c
Next, in the grand-canonical ensemble, we calculate the determinant of the Hessian matrix of the asymptotically
flat black holes. After some algebraic manipulation, one obtains
HM = 4 2q2(3n 4)r4n−4 r2 2α′ +
S,Q − + +−
(cid:12)(cid:12) (cid:12)(cid:12) 4q(cid:8)4r+2n−2 2α′(11n−(cid:2)24)−(7n(cid:3)−8)r+2 β
(cid:2) (cid:3)
8
FIG. 3: Asymptotically flat solutions: (cid:16)∂∂2SM2 (cid:17)Q (left figure) and (cid:12)(cid:12)HMS,Q(cid:12)(cid:12) (right figure) versus r+ for n=5, q =1, α=0.1 and
β =0.001 (solid line), β=0.003 (dotted line) and β =0.005 (dashed line). ”bold lines represent the temperature”
FIG. 4: Asymptotically flat solutions: (cid:16)∂∂2SM2 (cid:17)Q (left figure) and (cid:12)(cid:12)HMS,Q(cid:12)(cid:12) (right figure) versus r+ for n=5, q =1, α=0.1 and
β =0.01 (solid line), β =0.03 (dotted line) and β =0.05 (dashed line). ”bold lines represent the temperature”
(n 8)α′r2
+12q2(n 1)(n 2)2r4n−8 − +
− − + (n 2)
(cid:20) −
2(n 4)α′2
+ − +r4 β
(n 2) +
− (cid:21)
(n 8)α′r2
(n 1)(n 2)(3n 4)r6n−10 − +
− − − − + (n 2)
(cid:20) −
2(n 4)α′2
+ − +r4 / r8n−14
(n 2) + +
− (cid:21)(cid:27)
r2 +2α′ 3(n 1)2(3n(cid:2) 4)(n 2) +O β2 . (39)
+ − − −
(cid:0) (cid:1) i (cid:0) (cid:1)
9
FIG. 5: Asymptotically flat solutions: (cid:16)∂∂2SM2 (cid:17)Q (left figure) and (cid:12)(cid:12)HMS,Q(cid:12)(cid:12) (right figure) versus r+ for n=5, β =0.001, α=0.7
and q=1.2 (solid line), q=1.3 (dotted line) and q=1.4 (dashed line). ”bold lines represent the temperature”
FIG. 6: Asymptotically flat solutions: (cid:16)∂∂2SM2 (cid:17)Q (left figure) and (cid:12)HMS,Q(cid:12) (right figure) versus r+ for n=5, β =0.001, α=0.7
(cid:12) (cid:12)
and q=0.5 (solid line), q=0.7 (dotted line) and q=0.9 (dashed line). ”bold lines represent the temperature”
In addition, it is worthwhile to mention that for small and large values of r , we obtain
+
4(11n−24)βq4 >0, α′ =0,β =0
π(n−1)2(3n−4)α′2r6n−12 6 6
+
 16(7n−8)βq4 <0, α′ =0,β =0
(cid:12)HMS,Q(cid:12)Smallr+ = −−r+6(nn−−81()n2−(n1−)22q2(2)nα−′22r)+4(n3n−−104)<0, α′ 6=0,β =6 0 , (40)
(cid:12) (cid:12) 8q2 >0, α′ =0,β =0
HM =  (4n−1)2(n<−20)r,+4n−6 (41)
S,Q Larger+ −π(n−1)r+n
(cid:12) (cid:12)
(cid:12) (cid:12)
10
which indicate that, for nonvanishing α′ and β, asymptotically flat black holes with a large horizon radius are not
stable. In other words, there is an upper limit, r , for the asymptotically flat stable black holes (r < r ).
+max + +max
In the absence of the GB gravity case (α′ = 0,β = 0), there is a lower limit for stable solutions. It means that for
α′ = 0 the results of the stability conditions for th6 e canonical and grand canonical ensembles are identical and both
ensembles show that small and large black holes are unstable. We should note that in the presence of GB gravity
thermal stability is ensemble dependent. These results are shown in Figs. 1 – 6.
Now, we need to discuss ensemble dependency [15]. As we know in the canonical ensemble the internal energy is
allowedto fluctuate with fixed electric charge and, therefore,the black hole and the heat reservoirremain in thermal
equilibrium with a certaintemperature. While in the grandcanonicalensemble the black hole is inboth thermaland
electrical equilibrium, with its reservoir held at a constant temperature and a constant potential. In other words,
different boundary conditions lead to different ensembles.
In the usual discussions of the stability criterion of black holes, one expects ensemble independence of the system.
Indeed, different ensembles which imply different boundary conditions, should lead to similar stability conditions in
the usual thermodynamical systems.
Ensemble dependency of a system may come from two subjects. One of them is real ensemble dependency, which
mayoccasionallyoccurinsomethermodynamicalsystems. Theexistenceofensembledependencywasseenintheusual
thermodynamical systems [16]. Another one, which is not real, can be removed by improving our thermodynamic
view point. In other words, our lack of knowledge may lead to ensemble dependency and we should improve our
thermodynamicalunderstandingtoobtainensembleindependency. Inourblackholemodel(withasphericalhorizon),
we find that the nonlinearity of electrodynamics results in the same changes for the behavior of both canonical and
grand canonical ensembles. One can see the ensemble dependency comes from the contribution of Gauss-Bonnet
gravity. ThismeansthatintheabsenceoftheGBparameter,bothensembleshavethesamestabilityconditions. This
shows that the GB parameter makes more of a contribution to thermodynamical behavior of the black hole systems.
In other words, one may consider the GB parameter not only as a fixed parameter, but also as a thermodynamical
parameter [17]. By doing so, the Hessian matrix for this system will be modified and, therefore, we may expect to
see that this modification solves the ensemble dependency of the black hole system.
Anotherwaytosolveensembledependencyisthroughconsideringthefactthatthermodynamicalsystemsdescribed
by a thermodynamical potential must be invariant under the Legendre transformation. To do so, one can use the
method that was introduced in [18] and can use geometrothermodynamics to solve ensemble dependency of the
system[18, 19].
IV. ASYMPTOTICALLY ADS SPINNING BLACK BRANES (k=0 & Λ6=0)
In this section, we will investigate rotating adS space-time. In order to investigate the asymptotic behavior of the
solutions for k = 0 and Λ = 0, one can use the series expansion of the metric function for a large value of r. We
6
obtain
2Λ m 2q2 1
f(r)= eff r2 + +O , (42)
−n(n−1) − 1+ n4α(n′Λ−e1ff) rn−2 (n−1)(n−2) 1+ n4α(n′Λ−e1ff) r2n−4 (cid:18)r2n−2(cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
where
n(n 1) 1+ 8α′Λ 1
− n(n−1) −
Λ = . (43)
eff (cid:16)q4α′ (cid:17)
Eq. (42)showsthatfork =0thesolutionsareasymptoticallyadSwithaneffectivecosmologicalconstantΛ . When
eff
constructing a rotating space-time one can apply the following transformation in the static space-time with k =0
t Ξt a φ ,
i i
−→ −
a
i
φ Ξφ t. (44)
i
−→ − l
Usingthistransformation,themetricof(n+1)-dimensionalasymptoticallyadSrotatingspace-timewithprotation
parameters is
ds2 = f(r) Ξdt p a dφ 2+ dr2 + r2 p a dt Ξl2dφ 2
− − i i! f(r) l4 i − i
i=1 i=1
X X(cid:0) (cid:1)