Table Of ContentLogarithmic corrections to entropy for black holes
with hyperbolic horizon
Yun Soo Myung 1
Institute of Basic Science and School of Computer Aided Science
Inje University, Gimhae 621-749, Korea
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2 Abstract
n
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J We compute logarithmic corrections to entropy for black holes with hyperbolic horizon.
9 For this purpose, we introduce the topological black hole and MTZ black holes in four
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dimensions, while in five dimensions, the topological black holes and Gauss-Bonnet black
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c hole with negative coupling are considered. As they stand, logarithmic corrections are
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- problematic because small black holes with positive heat capacity have negative ener-
r
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gies. However, introducing the background state of extremal black hole, the logarithmic
[
corrections are performed for black hole with hyperbolic horizon.
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1e-mail address: [email protected]
1 Introduction
A number of authors have shown that for a large class of black holes, the Bekenstein-
Hawking entropy receives logarithmic corrections due to thermodynamic fluctuations
around thermal equilibrium [1, 2]. Up to now, a corrected-entropy formula takes the
form
1
S = S ln CT2 + , (1)
c
− 2 ···
h i
where C and T are the heat capacity and temperature of the given black hole, and S de-
notes the uncorrected Bekenstein-Hawking entropy. Here C should be positive for Eq.(1)
to be well-defined. We note that for C > 0 (C < 0), the system is thermodynamically
stable (unstable). A black hole with negative specific heat is in an unstable equilibrium
with the heat reservoir of the temperature T [3]. Its fate under small fluctuations will be
either to decay to hot flat space or to grow without limit by emitting or absorbing thermal
radiation in the heat reservoir [4]. Hence it is meaningless to apply Eq.(1) to the black
hole with C < 0. There exists a way to achieve a stable black hole in an equilibrium with
the heat reservoir. A black hole could be rendered thermodynamically stable by placing
it in AdS spacetimes. An important point is to understand how a black hole with positive
specific heat could emerge from thermal radiation through a phase transition. To this
end, one introduces the Hawking-Page phase transition between thermal AdS space and
Schwarzschild-AdS black hole [5, 6, 7].
It is well known that the entropy-corrected formula of Eq.(1) is universal [2], implying
that this prescription could apply to all black holes with positive specific heat. We note
that the canonical ensemble was used to derive Eq.(1). In this case, the positive energy
(mass) of the system is needed to define the canonical ensemble. However, small black
holes with hyperbolic horizon have the negative energy, even though they have positive
heat capacity. These may be counter examples, because they have negative masses in the
range M M 0 with M mass of extremal black hole. Hence, a direct application
e e
≤ ≤
of Eq.(1) to these black holes is not guaranteed for accounting their thermodynamic
fluctuations.
In this work, we address this issue and resolve this by introducing the background
state of extremal black hole, similar to the charged black holes with spherical horizon [8].
The difference is that for hyperbolic horizon, its background state energy is negative,
whereas its background state energy is positive for spherical horizon. In order to avoid
negative mass, we use the substraction scheme. The entropy-corrected formula is not
changed because the substraction scheme corresponds to a constant shifting from M to
E = M M . We will use Eq.(1) to study thermal fluctuations of black holes with
e
−
2
hyperbolic horizon.
The organization of this work is as follows. Section 2 is devoted to performing the
logarithmic corrections to topological black hole (4DTBH) and MTZ black holes in four
dimensions. We study the logarithmic corrections to topological black hole (5DTBH)
and Gauss-Bonnet black holes in five dimensions in section 3. We discuss the AdS/CFT
correspondences for these black hole in section 4. Finally, we discuss our results in section
5.
2 4DTBH and MTZ black holes
Topological black holes in asymptotically anti-de Sitter spacetimes were first found in
three and four dimensions [9]. Their black hole horizons are Einstein spaces of spherical
(k = 1), hyperbolic (k = 1), and flat (k = 0) curvature for higher dimensions more than
−
three [10, 11]. The standard equilibrium and off-equilibrium thermodynamic analyses are
possible to show that they are treated as the extended thermodynamic systems, even
though their horizons are not spherical. The topological black holes in four-dimensional
AdS spacetimes are given by
1
ds2 = g dxµdxν = f (r)dt2 + dr2 +r2dΣ2, (2)
4DTBH µν − T f (r) k
T
where the metric function f (r) is given by
T
m r2
f (r) = k + . (3)
T
− r l2
dΣ2 describes the 2D horizon geometry with a constant curvature
k
dΣ2 = dθ2 +f2(θ)dφ2, (4)
k k
where f (θ) is given by
k
f0(θ) = θ, f1(θ) = sinθ, f−1(θ) = sinhθ. (5)
Here we define k=1, 0, and 1 cases as the 4D Schwarzschild-AdS black hole (4DSAdS),
−
4D flat-AdS black hole, and 4D hyperbolic-AdS black hole (=4DTBH) [12], respectively.
In the case of k = 1,m = 0, we have an AdS -spacetime with its curvature radius l.
4
However, m = 0 generates the topological black holes. It is easy to check that the metric
6
(2) satisfies Einstein’s equations with a negative cosmological constant
3
R = g , (6)
µν −l2 µν
3
when the horizon is an Einstein space
R = kh . (7)
ij ij
In this work, we are interested in the negative curvature with k = 1 only. Then, the
−
horizon space is a hyperbolic manifold of Σk=−1 = H2/Γ, where H2 is 2D hyperbolic space
and Γ is a suitable discrete subgroup of the isometry group of H2 [13].
First of all, the 4DTBH provides thermodynamic quantities of Hawking temperature
T , mass M , entropy S , heat capacity C , and free energy F = M T S
T T T T T T T T
−
1 3ρ2 σ σρ ρ2
T = + 1 , M m = + + 1 , (8)
T 4πρ l2 − T ≡ 8πG 8πG l2 −
+ 4 4
(cid:16) (cid:17) (cid:16) (cid:17)
σρ2 3ρ2 l2 σρ ρ2
S = +, C = 2 + − S , F (ρ ) = + + +1 , (9)
T 4G4 T "3ρ2+ +l2# T T + −16πG4 l2
(cid:16) (cid:17)
where σ denotes the area of a unit 2D hyperbolic space Σk=−1 and ρ+ is the outer horizon
which satisfies f = 0 with k = 1. We note that the first-law of thermodynamics
T
−
dM = T dS holds for the 4DTBH. Also we observe that the entropy satisfies the
T T T
area-law.
Importantly, we observe that the temperature and heat capacity are positive for ρ >
+
ρ = l/√3, while the mass is positive only for ρ > l. This means that any canonical
e +
ensemble is not defined for ρ < l. Thus it seems that the entropy-correction formula
+
of Eq.(1) is useless for describing thermal fluctuations of the small 4DTBH. This arises
mainly because its horizon geometry is hyperbolic.
Inordertoresolvethisproblem, wehavetochooseanappropriatesubstractionscheme.
In the Reissner-Norstr¨om-AdS black hole, one has introduced the extremal black hole as
the background state to define an appropriate free energy in the canonical ensemble [8].
Similarly, we wish to introduce the extremal black hole at ρ = ρ as the background
+ e
state even though the charge is absent in the 4DTBH [14]. In this case, we define the
positive energy as
E = M Me (10)
T T − T
with respect to the negative background energy
σl
Me = M = . (11)
T T|ρ+=√l3 −12√3πG4
Although one uses E instead of M , all thermodynamics except F remain unchanged.
T T T
The free energy is an important quantity to discuss the global stability and phase transi-
tion. In this case, it is shifted by Fn = E T S . Then, for ρ > ρ , we may use the
M T − M M + e
4
corrected-entropy formula to study thermodynamic fluctuations around the equilibrium
4DTBH. We prove it by showing that the root-mean-square energy fluctuations defined
by the square root of
< (∆E )2 >= C T2 (12)
T T T
remains unchanged under the change from M to E . In deriving Eq.(1), one has used
T T
S = S 1 lnS′′ + with S′′ =< M2 > < M >2= CT2. It is easy to check
c − 2 ··· −
< E2 > < E >2=< M2 > < M >2 under the shifting M E = M M , which
e
− − → −
is also confirmed by noting that CT2 is an invariant quantity. Then, the mass (energy)
becomes positive and thus one could define the canonical ensemble to discuss its thermal
fluctuations around the equilibrium configuration. Importantly, the entropy-corrected
formula is not changed because thesubstraction scheme corresponds to a constant shifting
from M to E.
For a large 4DTBH with ρ l, we have approximate forms
+
≫
C 2S , T2 S (13)
T ∼ T T ∼ T
which leads to an approximately correct-entropy formula
Sc S ln S + . (14)
T ∼ T − T ···
h i
This is the same form as for large 4DSAdS [2]. Furthermore, we note that the Smarr
formula of M = TS/2 (Euler relation) is not satisfied for the choice of either M = M or
T
E .
T
On the other hand, the MTZ black hole dressed by scalar could be obtained from the
Einstein action minimally coupled to a scalar in AdS spacetimes [15]
R 2Λ 1
I [g,φ] = d4x√ g − 4 ( φ)2 V(φ) (15)
4
Z − " 16πG4 − 2 ∇ − #
where the potential V(φ) is given by
3 4πG
V(φ) = sinh2 4φ . (16)
−4πG4l2 "s 3 #
Here Λ = 3/l2 with l the curvature radius of AdS spacetimes. For φ = 0 case,
4 4
−
we obtain the action for the 4DTBH. In order to understand the role of scalar and its
potential, it would be better to go the conformal frame by introducing a conformal factor
Ψ = 3/4πG tanh[ 4πG /3φ]. Performing a conformal transformation as gˆ = (1
4 4 µν
−
4πGΨq2/3)−1g , theqaction I takes the from
µν 4
Rˆ 2Λ 1 1 2πG
I [gˆ,Ψ] = d4x gˆ − 4 ( Ψ)2 RˆΨ2 4Ψ4 , (17)
4
Z q− " 16πG4 − 2 ∇ − 12 − 3l2 #
5
where a conformally coupled scalar Ψ appears with the conventional potential Ψ4. In
connection with higher derivative terms, this potential may play a role of curvature-
squared term Rˆ2. It provides a black hole dressed by scalar with hyperbolic horizon. On
the other hand, for Ψ = 0, one has the action for 4DTBH, Eq.(2). Then, the solution of
the MTZ black hole is given by
dr2
ds2 = f (r)dt2 + +r2dΣ2 (18)
M − M f (r) k=−1
M
where the metric function f (r) is given by
M
r2 G µ 2
4
f (r) = 1+ (19)
M l2 − r !
and a conformally coupled scalar has the configuration
3 G µ
Ψ¯(r) = 4 . (20)
s4πG4r+G4µ
Thermodynamic quantities of the MTZ are given by Hawking temperature T , mass
M
M , entropy S , heat capacity C , and free energy F = M T S by [15, 16, 17]
M M M M M M M
−
1 2r 2G σ σr r
+ 3 + +
T = 1 = S , M µ = 1 , (21)
M 2πl l − σπl3 M M ≡ 4π 4πG l −
4
h i (cid:16) (cid:17) (cid:16) (cid:17)
σl2 2r σ 2r2
S = + 1 = C , F (r ) = + 2r +l ,
M M M + +
4G l − −8πG l −
4 4
(cid:16) (cid:17) (cid:16) (cid:17)
where r is the outer horizon which satisfies f = 0. We note that the first-law of
+ M
thermodynamics dM = T dS is satisfied for the MTZ. Also we observe that the
M M M
entropy S was obtained by using the Wald’s formula but it does not satisfy the area-
M
law. Importantly, the temperature and heat capacity are positive for r > r = l/2,
+ e
while the mass is positive for r > l only. This means that any canonical ensemble is
+
not suitable for describing the case of r < l and thus the corrected-entropy formula of
+
Eq.(1) may be useless for the small MTZ black hole.
In order to resolve this problem, we have to choose an appropriate background state.
Similarly, we introduce the extremal black hole as the background state even though the
charge is absent in the MTZ black hole, too. For the MTZ black hole, we define the
positive energy as [18]
E = M Me (22)
M M − M
with the background state energy
σl
Me = M = . (23)
M M|r+=2l −16πG4
6
Although one uses E instead of M , all thermodynamics of the MTZ except Fn =
M M M
E T S remain unchanged. Nicely, we check that the Smarr formula
M M M
−
T S
M M
E = (24)
M
2
is satisfied for r > r only when using the positive energy E . This implies that we
+ e M
could use the corrected-entropy formula to see thermodynamic fluctuations around the
equilibrium MTZ, even though its entropy does not satisfy the area-law. In this case, we
have a corrected-entropy formula for r > r
+ e
3
Sc = S ln S + (25)
M M − 2 M ···
h i
which is the same formula for the non-rotating BTZ black hole [2] whose thermodynamic
quantities are given by
r G r2 πr r2
T = + = 3 S , M = + = E , S = + = C , F = + . (26)
B 2πl2 π2l2 B B 8G l2 B B 2G B B −8G l2
3 3 3
h i
Here we have the Smarr formula of E = T S /2 for the non-rotating BTZ black hole.
B B B
Two are very similar in the sense that they satisfy the Smarr formula and have the same
corrected-entropy formula. A difference is that these formulae are valid for the outer
horizon r > r of MTZ black hole and for any size of non-rotating BTZ black hole.
+ e
Furthermore, we derive thermodynamic quantities for large MTZ black hole with r l
+
≫
as
r σr2 σlr σr2
T + , M + , S = C +, F + (27)
M ≃ πl2 M ≃ 4πlG M M ≃ 2G M ≃ −4πlG
4 4 4
which show the nearly same behavior as Eq. (26) is shown. Particularly, choosing σ = π/l
leads to close relations of T = 2T ,M = 2M ,F = 2F ,S = S ,C = C . This
M B M B M B M B M B
suggests that the MTZ black hole dressed by scalar is a corner stone for the 4D black
hole physics as the non-rotating BTZ black hole does play a key role in the 3D black hole
physics.
3 5DTBH and Gauss-Bonnet black holes
In five dimensions, the topological AdS black holes are given by
1
ds2 = h(r)dt2 + dr2 +r2dΣ2, (28)
5DTBH − h(r) k
7
where dΣ2 = dχ2 + h2(χ)(dθ2 + sin2θdφ2) describes the 3D horizon geometry with a
k k
constant curvature. Further h(r) and h (χ) are given by
k
m r2
h(r) = k − r2 + ℓ2, h0(χ) = χ, h1(χ) = sinχ, h−1(χ) = sinhχ. (29)
Here we define k=1, 0, and 1 cases as the 5D Schwarzschild-AdS black hole (5DSAdS)
−
[19, 20], 5D flat-AdS black hole, and 5D hyperbolic-AdS black hole (5DTBH) [21], respec-
tively. We are interested in the k = 1 case only. In this case, the location of the event
−
horizon is given by
ℓ2
r2 = 1+ 1+4m/ℓ2 . (30)
+ 2
(cid:16) q (cid:17)
The relevant thermodynamic quantities of Hawking temperature T , mass M , Bekenstein-
t t
Hawking entropy S , heat capacity C , and free energy F = M T S are given by
t t t t t t
−
1 2r2 3V m 3V r2 r2
T = + 1 , M 3 = 3 + + 1 , (31)
t 2πr ℓ2 − t ≡ 16πG 16πG l2 −
+ 5 5
(cid:16) (cid:17) (cid:16) (cid:17)
V r3 2r2 ℓ2 V r2 r2
S = 3 +, C = 3 + − S , F = 3 + + +1
t 4G5 t "2r+2 +ℓ2# t −16πG5 ℓ2
(cid:16) (cid:17)
where V3 is the volume of a unit 3D hyperbolic space Σk=−1 and G5 is the five-dimensional
Newton constant. Here we find that T , C > 0 for r > r = l/√2, while M > 0 for
t t + e t
r > l. It seems that any canonical ensemble is not defined for r < l and thus the
+ +
entropy-correction formula of Eq.(1) is meaningless for the small 5DTBH.
In order to cure this problem, we have to find an appropriate substraction scheme.
Similarly, we introduce the extremal black hole as the background state . In this case, we
define the positive energy as
E = M Me (32)
t t − t
with the background state energy
3V l2
Me = M = 3 . (33)
t t|r+=√l2 −64πG5
Although one uses E instead of M , all thermodynamics except Fn = E T S remain
t t t t − t t
unchanged. This implies that for r > r , we could use the corrected-entropy formula to
+ e
study thermodynamic fluctuations around the equilibrium 5DTBH.
For a large 5DTBH with r l, we have approximate forms
+
≫
2
C 3S , T2 S3 (34)
t ∼ t t ∼ t
which leads to an approximately correct-entropy formula
5
Sc S ln S + . (35)
t ∼ t − 6 t ···
h i
8
This is the same form as for large 5DSAdS [2, 22]. Furthermore, we note that the Smarr
formula is not satisfied for the choice of either M = M or E .
t t
In order to find a 5D version of MTZ black hole, we may consider the 5D action
like Eq.(17) with replacing the last two terms and Λ by (3/32)RΨ2 + α Ψ10/3 and Λ .
4 5 5
However, it was turned out that there is no solution like the MTZ black hole in AdS
5
spacetimes [18]. That is, there is no such a 5D black hole dressed by scalar. Instead,
we consider a similar black hole with hyperbolic horizon. This may be found from a 5D
gravitational action in the presence of a negative cosmological constant Λ = 6/l2 and
5
−
Gauss-Bonnet (GB) term as
1 c
I [g,c] = d5x√ g R 2Λ + L , (36)
5 5 GB
16πG − − 2
5 Z (cid:20) (cid:21)
where
L = R2 4R Rµν +R Rµνρσ. (37)
GB µν µνρσ
−
Here c is a GB coupling constant having mass dimension 2. For c = 0, this action
−
reduces to the action for the 5DTBH, Eq.(28). The solution of the TGBAdS black hole
is given by
dr2
ds2 = f (r)dt2 + +r2dΣ2. (38)
g − g f (r) k
g
The metric function is given by
r2 4c 2µ 3
f (r) = k + 1+ǫ 1+ . (39)
g 2c s 3 r4 − l2
(cid:18) (cid:19)
Hereafter, we consider the negative coupling of c < 0 with k = 1 and ǫ = 1 in
− −
accordance with the hyperbolic horizon.
For the TGBAdS black hole, we have relevant thermodynamic quantities [23, 24, 25]
r 2r2 V µ 3V r2 r2 c
T = + + 1 , M 3 = 3 + + 1+ , (40)
g 2π(r+2 2c) l2 − ! g ≡ 8πG5 16πG5 l2 − r+2 !
−
2
V r3 6c V r3 2r2 l2 1 2c/r2
S = 3 + 1 , C = 3 3 + + − − + ,
g 4G5 − r+2 ! g 4G5 "(2r+2(cid:16)+l2) (cid:17)(2(cid:16)c/r+2)(6r+2(cid:17) l2)#
− −
V r4 r2 3c 6r4
F = M T S = 3 + + +1 + r2 2c ,
g g − g g −16πG5(r+2 2c) " l2 − r+4 l2 − + − !#
− (cid:16) (cid:17)
where we point out that S was derived using the first-law of thermodynamics and thus
g
it does not satisfy the area-law. Here we find that T , C > 0 for r > r = l/√2. On
g g + e
the other hand, one has M > 0 for r > r where
t + 0
l
r = 1+ 1 4c/l2 > l (41)
0 √2r −
q
9
which is determined by the massless condition of M = 0. This means that any canonical
g
ensemble is not defined for r < r and thus the entropy-correction formula of Eq.(1)
+ 0
is useless for the small TGBAdS. In order to solve this problem, we have to find an
appropriate background state. We introduce the extremal black hole as the background
state. In this case, we define the positive energy as
E = M Me (42)
g g − g
with the background state energy
3V l2
Me = M = 3 c . (43)
g g|r+=√l2 −16πG5"4 − #
At this stage, we find an important equality for energies
3V r2 r2 l2
E = E = 3 + + 1+ . (44)
g t 16πG5" l2 − 4r+2 #
Although one uses E instead of M , all thermodynamics except Fn = E T S remain
g g g g − g g
unchanged. This implies that for r > r , we could use the corrected-entropy formula to
+ e
see thermodynamic fluctuations around the equilibrium TGBAdS. For a large TGBAdS
with r l,c, we have approximate forms
+
≫
2
C 3S , T2 S3 (45)
g ∼ g g ∼ g
which leads to an approximately correct-entropy formula
5
Sc S ln S + . (46)
g ∼ g − 6 g ···
h i
This is the same form as for large 5DSAdS [2]. Furthermore, we note that the Smarr
formula is not satisfied for the choice of either M = M or E .
g g
4 Boundary conformal field theories
The holographic principle means that the number of degrees of freedom associated with
the bulk gravitational dynamics is determined by its boundary spacetime. We start with
the bulk-relation between the entropy and energy for the non-rotating BTZ black hole
2E
B
S = πl (47)
B
s G
3
which is derived using E = (πlTB)2 and Smarr formula of E = TBSB.
B 2G3 B 2
10
