Maxwell’s equal-area law for
Gauss-Bonnet-Anti-de Sitter Black holes
M. C HABAB1 ; H. E L M OUMNI1,2 ; K.M ASMAR1
1
LPHEA -C ADI AYYAD U NIVERSITY; 2 I BN Z OHR U NIVERSITY
Based on :
A BSTRAT
Eur. Phys. J. C (2015) 75:71
1
We study the Maxwell equal-area law of higher dimensional Gauss-BonnetAdS black holes .These solutions critically behave like van Der Waals systems.
It has been realized that below the critical temperature Tc the stable equilibrium is violated. We show through calculations that the critical behaviors for
the uncharged black holes only appear in d = 5. For the charged case, we
analyze solutions in d = 5 and d = 6 separately.
T HERMODYNAMICS
2
The action of Einstein-Maxwell theory in the presence of the Gauss-Bonnet
terms and a cosmological constant.
Z
1
d √
I=
d x −g[R−2Λ+αGB (Rµνγδ Rµνγδ −4Rµν Rµν +R2 )−4πFµν F µν ],
16π
(1)
where αGB is the Gauss-Bonnet coefficient. Fµν is the Maxwell field
,above action produces a static black hole solution with the following metric:
ds2 = −f (r)dt2 + f −1 (r)dr2 + r2 hij dxi dxj
(2)
1
r2
64παM
2αQ2
64παP
2)
f (r) = k+ (1−(1+
)
−
−
2α
(d − 2)Σk rd−1 (d − 2)(d − 3)r2d−4 (d − 1)(d − 2)
The mass and the Hawking temperature can be given in terms of the horizon
radius rh of the black hole :
T
=
M
=
16πP rh4
2Q2
2
2
[
+ (d − 3)krh + (d − 5)k α −
]
2
2d−8
4πrh (rh + 2kα) (d − 2)
(d − 2)rh
d−3 2
2
(d − 2)Σk rh
16πP rh
k α
Σk Q2
k+ 2 +
+
16π
rh
(d − 1)(d − 2)
8π(d − 3)rhd−3
1
P LOTS
5
To illustrate this effect, we plot these results:
T HE EQUAL - AREA LAW
To obtain the equal-area isobar, P = RP0 , one uses the following
ν
relations: 4=1 = P0 (ν2 − ν1 ) and 4=2 = ν12 P dν.
The equal-area law requires the following equality : 4=1 = 4=2
• Uncharged solutions: the points ν1 and ν2 should satisfy:
1
1
k(d − 3) 1
1
16αk
ν2
( 2 − 2)
( − )−
P0 (ν2 − ν1 ) = T0 ln( ) +
2
ν1
π(d − 2) ν2
ν1
π(d − 2) ν2
ν1
16α(d − 5)k 2 (ν12 + ν1 ν2 + ν22 )
+
3π(d − 2)3 ν13 ν23
we get a polynomial equation: aν24 + bν22 + c = 0
√
−b+ b2 −4ac
2
solving the above polynomial equation,we get : ν2 =
2a
ν1
In the limit x = ( ν2 ) → 1, one should have v1 = v2 = vc .
? k = 0 : flat topology, the limit diverges for any dimensions.
? k = −1 : hyperbolic topology, a negative value of v22 appears in d = 5. For
d ≥ 6, the limit also diverges.
? k = 1 : spherical topology, the limit diverges only when d ≥ 6. In the case of
d = 5, we have
r
r
y1
2
y1 =32α(x−1)3
νc = lim
=4
α where; {y =18x2 (x+1) ln(x)−36(x−1)x2
(3)
2
x−>1
y2
3
setting T0 = χTc ,with the critical temperature : Tc =
r
512
r
2048
2
3 y1
3
χx ( )(x − 1) +
(8
3α
y2
9
α=1
Figure 1:
α=2
Table 1:
(1, 1)
(1, 32 )
(2, 3)
Figure 3:
charge.
Numerical values of x, v1 , v2 and P0 at different values of α in six dimensional spherical topology with non vanishing
Table 2:
C ONCLUSION
we obtain :
y1 1/2
2α 3
(x − 1) + ( ) (1 − x2 )) = 0 (4)
3
y2
χ
1
0.8
0.7
1
0.8
0.7
1
0.8
0.7
x
1
0.07111
0.02543
1
0.07111
0.02543
1
0.07111
0.02543
v1
2.3094
0.85537
0.67313
3.2659
1.20969
0.95195
4.6188
1.71076
1.34627
v2
2.3094
12.0282
26.4629
3.26599
17.0104
37.4241
4.6188
24.0564
52.9257
P0
0.01326
0.00471
0.00213
0.00663
0.00235
0.00106
0.00331
0.00117
0.00053
Numerical solutions for x, v1 , v2 and P0 at different temperature with Q = 0 ,d = 5 ,k = 1.
(α, Q)
The P − v diagram charged Gauss-Bonnet-AdS black holes in five dimensions. The dashed blue curve corresponds to the
critical temperature Tc , the red one corresponds to isotherm with 0.8Tc and the green one corresponds to 0.7Tc .
6α
4
α = 0.5
Figure 2:
2π
1
√
R ESULTS
α
The P − v diagram uncharged Gauss-Bonnet-AdS black holes in five dimension. The dashed blue curve corresponds to
the critical temperature Tc , the red one is associated with the isotherm with 0.8Tc and the green one corresponds to 0.7Tc .
3
χ
1
0.8
0.7
1
0.8
0.7
1
0.8
0.7
x
1
0.11658
0.05765
1
0.13991
0.07388
1
0.13991
0.07388
v1
3.5447
1.69635
1.50503
3.77814
1.95332
1.759006
5.3431
2.76241
2.48768
v2
3.5447
14.5506
26.1024
3.77814
13.961
23.8078
5.3434
19.7438
33.6694
P0
0.00619
0.00254
0.00139
0.00582
0.002531
0.00146
0.00291
0.00126
0.00073
Numerical values of x, v1 , v2 and P0 at different temperature in d = 5, k = 1 in the presence of the charge.
6
In this paper, we have studied the Maxwell’s equal area law of higher dimensional Gauss-Bonnet Anti-de-Sitter black holes.We have shown that this construction can be used to eliminate the region of the violated stable equilibrium
∂P
> 0.
∂v
(α, Q)
R EFERENCES
References
(1, 2)
7
[1]
Z. De-Cheng, L. Yunqi, W. Bin, Critical behavior of charged Gauss?Bonnet AdS black holes in the grand canonical ensemble.
arXiv:1404.5194 [hep-th]
[2]
E. Spallucci, A. Smailagic, Maxwell?s equal area law for charged Anti-deSitter black holes. Phys. Lett. B 723, 436 (2013).
arXiv:1305.3379 [hep-th]
(1, 1)
Table 3:
χ
1
0.8
0.7
1
0.8
0.7
x
1
0.08212
0.03799
1
0.12462
0.06473
v1
1.91384
0.92344
0.85358
2.17689
1.18883
1.10356
v2
1.91384
11.2444
22.4677
2.17689
9.53966
17.0479
P0
0.0183735
0.006092
0.002983
0.002823
0.0066212
0.0036418
Numerical values of x, v1 , v2 and P0 at different values of α in d = 6 ,k = 1 with non vanishing charge.