P − V criticality of AdS black hole in f (R) gravity Songbai Chen∗ , Xiaofang Liu, Changqing Liu
arXiv:1301.3234v2 [gr-qc] 31 May 2013
Institute of Physics and Department of Physics, Hunan Changsha, Hunan 410081, People’s Republic Key Laboratory of Low Dimensional Quantum and Quantum Control of Ministry of Education, Hunan Changsha, Hunan 410081, People’s Republic
Normal University, of China Structures Normal University, of China
Abstract We study thermodynamics of a charged AdS black hole in the special f (R) correction with the constant Ricci scalar curvature. Our results show that the the f (R) correction influences the Gibbs free energy and the phase transition of system. The ratio ρc occurred at the critical point increases monotonically with the derivative term f ′ (R0 ). We also disclose that the critical exponents are the same as those of the liquid-gas phase transition in the Van der Waals model, which does not depend on the f (R) correction considered here. PACS numbers: 04.70.-s, 95.30.Sf, 05.70.Ce
∗
[email protected]
2 I.
INTRODUCTION
f (R) gravity is a kind of important modified gravity theory, which generalizes Einstein’s gravity by adding higher powers of the scalar curvature R, the Riemann and Ricci tensors, or their derivatives in the usual lagrangian formulation [1–3]. Without unknown forms of dark energy or dark matter, f (R) gravity can explain the accelerating expansion of the current Universe. It can also mimic the inflation and structure formation in the early Universe. Thus, a lot of efforts have been focused in the study of f (R) gravity including not only the applications of f (R) theories on gravitation and cosmology, but also many observational and experimental methods to distinguish them from general relativity [2, 3]. It is of interest to extend the study of f (R) gravity to the black hole physics, which could provide us some features of black holes differed from that obtained in Einstein’s gravity. In general, the black hole solution is difficult to be obtained in the f (R) theory since the field equations of f (R) gravity become more complicated than that in Einstein’s gravity, even without the presence of a matter field. With the condition that the energy-momentum tensor is traceless for the matter field, Moon et al [4] obtain an exact analytical four-dimensional solution from R + f (R) theory coupled to a matter field, which is later extended to the rotating cases [5]. The higher dimensional charged solution [6] are obtained only in the case of power-Maxwell field with d = 4p, where p is the power of conformally invariant Maxwell lagrangian. Moreover, some other solutions of black hole in f (R) gravity have been also constructed in [7–11]. Thermodynamical properties of the AdS black hole has been a subject of intense study for the past decades because that it is dual to a thermal state on the conformal boundary in terms of the AdS/CFT correspondence. The duality has been recently applied to study the strongly correlated condensed matter physics from the gravitational dual. Recently, Kubiznak et al [12] investigated the critical behavior of charged AdS black holes by treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume [13]. It is found that in the system of charged AdS black hole the small-large black hole phase transition possesses the same critical behavior of liquid-gas phase transitions in the Van der Waals model. The similar critical behaviors are disclosed in the spacetimes of a rotating AdS black hole [14] and a high dimensional Reissner-Nordstr¨ om-AdS black hole [15]. The same qualitative properties are also found in the AdS black hole spacetime with the Born-Infield electrodynamics [14, 16], with the power-Maxwell field [17] and with Gauss-Bonnet correction [18]. The aim of this Letter is to study the thermodynamics of a charged AdS black hole [4] in the special f (R)
3 correction in which the Ricci scalar curvature remains the constant (R = R0 ) and to probe the effects of the f (R) correction on thermodynamical quantities and the small-large black hole phase transition. Finally, we discuss the analogy of black hole with Van der Waals liquid-gas system. We firstly review the four-dimensional charged AdS black hole obtained by Moon et al [4] in the R + f (R) gravity with the constant Ricci scalar curvature. The action describing a four-dimensional charged AdS black hole in the R + f (R) gravity can be expressed as [4] S=
Z
M
√ d4 x −g[R + f (R) − Fµν F µν ],
(1)
where R is the Ricci scalar curvature and f (R) is an arbitrary function of R. Fµν is electromagnetic field tensor which is related to the electromagnetic potential Aµ by Fµν = ∂µ Aν − ∂ν ∂µ . Varying the action (1), one can obtain the equations of motion for gravitational field gµν and the gauge field Aµ 1 Rµν [1 + f ′ (R)] − gµν [R + f (R)] + (gµν ∇2 − ∇µ ∇ν )f ′ (R) = Tµν . 2
(2)
√ ∂µ ( −gF µν ) = 0,
(3)
and
respectively. In order to obtain an analytical solution of the equation (2), Moon et al [4] consider only the case of the constant Ricci scalar curvature R = R0 = const for the sake of convenience. In this simple case, Eq. (2) can be rewritten as Rµν [1 + f ′ (R0 )] −
gµν R0 [1 + f ′ (R0 )] = Tµν , 4
(4)
since the Maxwell energy-momentum tensor is traceless in the four-dimensional spacetime. The general metric for a four-dimensional charged static spherically symmetric black hole can be expressed as ds2 = −N (r)dt2 +
dr2 + r2 (dθ2 + sin2 θdφ2 ). N (r)
(5)
Solving Eq. (3) in the four-dimensional static spacetime (5), one can get the Maxwell field Ftr =
q2 . r2
(6)
Inserting Eqs.(5) and (6) into Eq.(4), one can find that the metric (5) has the form [4] N (r) = 1 −
2m R0 2 q2 + 2− r . r br 12
(7)
4 Here b = [1 + f ′ (R0 )]. The parameters m and q are related to the ADM mass M and the electric charge Q of the black hole by [6] q Q= √ , b
M = mb,
(8)
and the electric potential Φ at horizon r+ are Φ=
√
bq . r+
(9)
Thus, the f (R) correction modifies the mass M and the electric charge Q of black hole and the corresponding thermodynamic potentials. Treating the Ricci scalar curvature as R0 = − 12 l2 = −4Λ, one can find that the spacetime described by solution (7) is asymptotically AdS. In other words, the Ricci scalar curvature R0 is negative in the AdS background. The Hawking temperature T with the outer event horizon r = r+ is 2 R0 r+ q2 1 N ′ (r+ ) , 1− 2 − = T = 4π r=r+ 4πr+ r+ b 4
(10)
and the entropy of the black hole is [4, 6]
2 S = πr+ b.
(11)
For an AdS black hole, one can get both the differential and integral formulas of the first law of thermodynamics by treating the cosmological constant as a variable related to the thermodynamic pressure [13]. Motivated by this spirit, we here interpret R0 as a thermodynamic pressure P , which is given by bR0 , 32π
(12)
3 4πr+ . 3
(13)
P =− and the corresponding volume V is V =
Thus, for the charged AdS black hole (5), the Smarr formula can be expressed as M = 2T S + ΦQ − 2P V,
(14)
which is consistent with the usual charged AdS black hole without f (R) correction. The differential form the first law of thermodynamics becomes S Φ P M = Td + dQ + V d . d b b b b
(15)
Obviously, it recovers the standard first law of thermodynamics for the black hole as b = 1. From Eqs. (12) and (15), we find that R0 and f ′ (R0 ) modifies the thermodynamic pressure P of the system and the
5 differential form the first law of thermodynamics. Thus, it is expected that the f (R) correction will influence the thermodynamical phase transition of the AdS black hole and could lead to some new features for the thermodynamics of black hole. From the Hawking temperature (10) and the pressure (12) of the system, one can get the equation of state P = P (T, V ), P =
bT b q2 − + 2 4 , 2r+ 8πr+ 8πr+
(16)
for a fixed charge q. Here r+ is a function of the thermodynamic volume V . As doing in Refs.[12], one can find that the geometric quantities P and T can be translated into physical pressure and temperature of system by using dimensional analysis and lP2 = G~/c3 [Press] =
~c P, lP2
[Temp] =
~c T. k
(17)
And then the physical pressure and physical temperature are given by Press =
kTemp ~c T ~c + ··· = P = 2 + ··· . 2 lP lP 2r+ 2lp2
(18)
In order to compare it with the Van der Waals equation [12, 14], we must identify the specific volume v with the horizon radius r+ of the black hole by v = 2r+ lP2 .
(19)
After finishing these operations, one can find that the equation of state (16) can be rewritten as P =
bT 2q 2 b + 4. − 2 v 2πv πv
(20)
In Fig.(1) , we plot the P − V diagram1 for a four-dimensional charged AdS black hole in the f (R) with the different f ′ (R0 ) for fixed q = 1. Obviously, there exist a small-large black hole phase transition in this system. Fig.(1) tells us that with increase of f ′ (R0 ) the critical temperature Tc increases. The critical points in P − V diagram can be obtained from ∂P = 0, ∂v
∂ 2P = 0, ∂v 2
(21)
which yields √ 2q 6 vc = √ , b
√ 6b , Tc = 18πq
Pc =
b2 . 96πq 2
(22)
6 0.010
0.010
0.008
0.008
0.008
0.006
0.006
T>TC
0.004
T>TC
0.006
0.004
P
T>TC
P
P
0.010
T=TC
0.004
T=TC
T=TC 0.002
0.002
0.002
0.000
0.000 0
5
TPC
0.8
0.4
0.8
G
0.6
PPC G
P=PC
P=PC
0.6
1.0
P>PC
0.04
0.06
0.4
P
