Astrophys Space SciDOI 10.1007/s10509-014-1989-5

O R I G I NA L A RT I C L E

The equal area law of asymptotically AdS black holes in extendedphase space

Jun-Xin Zhao · Meng-Sen Ma · Li-Chun Zhang ·Hui-Hua Zhao · Ren Zhao

Received: 9 May 2014 / Accepted: 29 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract According to Maxwell’s equal area law we studythe “real” phase diagram of the charged AdS black hole andthe Kerr-AdS black hole. In the extended phase space con-structed by treating the cosmological constant as pressure,the two kinds of AdS black hole display themselves likevan der Waals system. Under the critical temperature Tc,there exists a regime where the condition of stable equilib-rium will be violated. We employ the equal area law to findan isobar which is the real two phase coexistence line. Asa byproduct we find a simple method to derive the criticalvalue of the thermodynamic quantities.

Keywords Black hole thermodynamics · Equal area law ·Phase transition

1 Introduction

Black hole is generally considered as a thermodynamic sys-tem. Except the intrinsic entropy of a black hole is propor-tional to the area of the event horizon, but not to the vol-ume, the other thermodynamic properties of black holes aresimilar to the ones of the usual thermodynamic system. Thepioneering works of Bekenstein and Hawking have openedmany interesting aspects of unification of quantum mechan-ics, gravity and thermodynamics (Bekenstein 1972, 1974;

J.-X. ZhaoCollege of Mechanical Engineering and Applied ElectronicsTechnology, Beijing University of Technology, Beijing 100124,China

M.-S. Ma (B) · L.-C. Zhang · H.-H. Zhao · R. ZhaoInstitute of Theoretical Physics, Shanxi Datong University,Datong 037009, Chinae-mail: xiaomaclever@gmail.com

Bardeen et al. 1973; Hawking 1974, 1975). Black holes notonly have temperature, but also have phase transition be-tween different states. Until now, there have been exten-sive studies on the phase transition in many kinds of blackholes.

Especially, the case with negative cosmological constant(AdS case) has been concerned by many physicists (Baner-jee and Roychowdhury 2011, 2012a,b; Banerjee et al. 2010,2011a,b; Kubiznak and Mann 2012; Dolan et al. 2013; Gu-nasekaran et al. 2012; Cvetic et al. 2011; Wei and Liu 2013;Fatima and Saifullah 2011; Akbar et al. 2011; Belhaj et al.2012; Dehghani and Asnafi 2011; Bellucci and Tiwari 2012;Shen et al. 2007). Asymptotically AdS black hole space-times admit a gauge duality description and are describedby dual thermal field theory. Correspondingly one has amicroscopic description of the underlying degrees of free-dom at hand. This duality has been recently exploited tostudy the behavior of quark-gluon plasmas and for the qual-itative description of various condensed matter phenomena(Kubiznak and Mann 2012). In some models, the cosmo-logical constant may be considered to be a time-variablequantity (Peebles and Ratra 1988; Horvat 2004), or as somethermodynamic quantities, such as thermodynamic pressure(Brown and Teitelboim 1987, 1988), which should be a con-jugate quantity of thermodynamic volume.

Recently, the idea is employed in the black hole thermo-dynamics. The studies on the phase transition of black holesby considering cosmological constant as a variable have gotmany attentions (Kubiznak and Mann 2012; Dolan et al.2013; Gunasekaran et al. 2012; Cvetic et al. 2011; Wei andLiu 2013; Dolan 2011a,b,c). For AdS black holes in fourdimensional spacetime, the cosmological constant is con-nected to the pressure in general thermodynamic system, the

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relation is

P = − 1

8πΛ = 3

8π

1

l2, (1.1)

and the corresponding thermodynamic volume is

V =(

∂M

∂P

)S,Qi,Jk

. (1.2)

In Wei and Liu (2013) the relation between cosmologi-cal constant and pressure is given in the higher dimensionalAdS spherically symmetric spacetime, which supplies thebasis for the study on the black hole thermodynamics in AdSspherically symmetric spacetime. Theoretically, if consider-ing black holes in AdS spacetime as a thermodynamic sys-tem, the critical behaviors and phase transitions should alsoexist. It has been shown that this transition is superficiallyanalogous to the phase transition of Van der Waals (VdM)liquid–gas system. It is well known that for the VdW sys-tem above the critical temperature Tc the isothermal curvesdisplay similar behavior to the experimental result. How-ever, below the critical temperature Tc, in the coexistenceline there will be a regime where (

∂p∂V

)T > 0. Therefore, inthis region the condition of stable equilibrium violates. Ac-cording to Maxwell’s equal area law, one can replace the os-cillatory part with an isobar which represents the liquid–gascoexistence line. Thus the experimental result is consistentwith the theoretical one. Since the AdS black hole is similarto the VdW system, in the process of phase transition of theblack holes there should be the similar isobar to representthe two-phase coexistence line. In Spallucci and Smailagic(2013), the authors have studied the equal area law in theT –S plane for the charged AdS black hole. In this paper weutilize a different method to construct the equal area law inthe P –v plane. Specifically, our method can supply an ef-ficient way to derive the critical thermodynamic quantities.We discuss two kinds of black holes: RN-AdS one and theKerr-AdS one. According to equal area law we construct iso-bars, which express the coexistence phase, to eliminate theunphysical region.

2 The construction of equal area law

For the VdW system, below the critical temperature Tc, onevalue of the pressure P corresponds to three different valuesof V in the isothermal curves. Experiments show that thereshould be a horizontal isobar in the isotherm to representthe condensation line, here the gas coexists with the liquid.On crossing the line from the gas side the system begins toform droplets of liquid on the condensation line, which growand coalesce until all the material has passed to the liquidphase.

Fig. 1 A P –v curve below the critical temperature

The chemical potential of the thermodynamic systemshould satisfy

dμ = −SdT + V dP (2.1)

Thus the difference of chemical potential between two statesin the isotherm with pressure P and P0 should be

μ − μ0 =∫ P

P0

V dP (2.2)

In some AdS black holes, there are the similar behaviors,Fig. 1.

At the point B , the black hole lies at “gas” phase, whereasat point A, the black hole lies at “liquid” phase completely.The region between A and B should be a coexistence phase.But the oscillating part of the curve between A and B cannotbe the coexistence line, because the part CE violate the con-ditions of equilibrium. At the two points A and B , the chem-ical potentials are the same, μA(T ,P ) = μB(T ,P ) whichis just the thermodynamic condition for phase equilibrium.According to Eq. (2.2) this is equivalent to∫

BEDCAvdP = 0 (2.3)

or

area(BED) = area(ACD) (2.4)

Below we will discuss the RN-AdS and Kerr-AdS blackholes respectively and employ the relation (2.4) to get theposition of A and B .

2.1 RN-AdS black hole

The spherically symmetric solution of the RN-AdS blackhole is:

ds2 = −f dt2 + f −1dr2 + r2(dθ2 + sin2 θdφ2), (2.5)

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where

f = 1 − 2M

r+ Q2

r2+ r2

l2, l2 = − 3

Λ(2.6)

The Hawking temperature of the RN-AdS black hole can bederived easily

T = f ′(r+)

4π= 1

4πr+

(1 + 3r3+

l2− Q2

r2+

). (2.7)

By identifying the pressure

P = 3

8πl2, (2.8)

and the corresponding conjugate thermodynamic volume

V = 4

3πr3+, (2.9)

the equation of state for the RN-AdS black hole can be ob-tained from Eq. (2.7)

P = T0

2

(4π

3V

)1/3

− 1

2π

(4π

3V

)2/3

+ Q2

4π

(4π

3V

)4/3

(2.10)

Taking the specific volume v = 2l2P r+, one can obtain the

Van der Waals-like equation of state:

P = T

v− 1

2πv2+ 2Q2

πv4(2.11)

To find the equal area isobar, P = P0, one can employthe relations:

�S1 = P0(v2 − v1), (2.12)

and

�S2 =∫ v2

v1

Pdv (2.13)

where S1 and S2 are the areas. And v1, v2 are the horizontalcoordinates of the points A, B respectively.

According to the equal area law, there should be

�S1 = �S2 (2.14)

namely,

P0(v2 − v1) = T0 ln

(v2

v1

)+ 1

2π

(1

v2− 1

v1

)

− 2Q2

3π

(1

v32

− 1

v31

)(2.15)

The points v1, v2 in the isothermal curves should satisfy

P0 = T0

v1− 1

2πv21

+ 2Q2

πv41

(2.16)

P0 = T0

v2− 1

2πv22

+ 2Q2

πv42

, (2.17)

from which one can derive

T0v31v3

2 − 1

2πv2

1v22(v1 + v2) + 2Q2

π

(v2

1 + v22

)(v1 + v2) = 0

(2.18)

and

2P0 = T0

(1

v1+ 1

v2

)− 1

2π

(1

v21

+ 1

v22

)+ 2Q2

π

(1

v41

+ 1

v42

)

(2.19)

Setting x = v1v2

, thus 0 ≤ x ≤ 1, Eqs. (2.15), (2.18) and (2.19)turn into

P0v42x3(1 − x)

= −T0v32x3 lnx − v2

2x2

2π(1 − x) + 2Q2

3π

(1 − x3), (2.20)

0 = T0v32x3 − 1

2πv2

2x2(1 + x) + 2Q2

π

(1 + x2)(1 + x)

(2.21)

and

2P0v42x4 = T0v

32x3(1 + x) − 1

2πv2

2x2(1 + x2)

+ 2Q2

π

(1 + x4) (2.22)

respectively.Utilizing Eqs. (2.21) and (2.22), P0 and T0 can be re-

moved in Eq. (2.20). We obtain

v22

[2x2(1 − x) + x2(1 + x) lnx

]

= Q2[

16

3(1 − x)

(1 + x + x2) + 4(1 + x)

(1 + x2) lnx

]

(2.23)

One can set

y1(x) = 16

3(1 − x)

(1 + x + x2)

+ 4(1 + x)(1 + x2) lnx

y2(x) = 2x2(1 − x) + x2(1 + x) lnx

(2.24)

From Eq. (2.23) we can get an extra bonus. When x → 1,there must be v1 = v2 = vc. One can easily obtain

v2c = Q2 lim

x→1

y1(x)

y2(x)= 24Q2 (2.25)

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Table 1 Numerical solutions for x, v1, v2, P0 at different temperatureswith Q = 1

χ x v1 v2 P0

1 1 4.89898 4.89898 0.00331573

0.8 0.237789 2.98066 12.5349 0.00177739

0.7 0.149406 2.73504 18.3061 0.00118710

In Table 1, we give the numerical values of x, v1, v2, P0 forRN-AdS black hole. Therefore

vc = 2√

6Q (2.26)

which is the same as the one derived by Kubiznak and Mann(2012).

Substituting Eq. (2.25) into Eq. (2.21) to eliminate v2 and

setting T0 = χTc = χ√

618πQ

, we can obtain

χ√

6x3(

y1(x)

y2(x)

)3/2

= 9x2(1 + x)

(y1(x)

y2(x)

)

− 36(1 + x)(1 + x2). (2.27)

2.2 Kerr-AdS black hole

For a slowly rotating Kerr-AdS black hole, when omittingthe higher order terms of J , one can derive the equation ofstate (Kubiznak and Mann 2012):

P = T

v− 1

2πv2+ 48J 2

πv6, (2.28)

where the specific volume is v = 2( 3V4π

)1/3. V is the ther-modynamic volume conjugate to the pressure. Similarly, wealso set x = v1

v2. From two equations below:

P0 = T0

v1− 1

2πv21

+ 48Q2

πv61

(2.29)

P0 = T0

v2− 1

2πv22

+ 48J 2

πv62

, (2.30)

we can obtain two relations:

0 = T0v52x5 − 1

2πv4

2x4(1 + x)+ 48J 2

π

(1 + x3)(1 + x + x2)

(2.31)

and

2P0v62x6 = T0v

52x5(1 + x) − 1

2πv4

2x4(1 + x2)

+ 48J 2

π

(1 + x6) (2.32)

Table 2 Numerical solutions for x, v1, v2, P0 at different temperatureswith J = 1

χ x v1 v2 P0

1 1 6.16014 6.16014 0.00279607

0.8 0.320674 4.22613 13.1789 0.00159591

0.7 0.218238 3.96328 18.1604 0.00111124

In Table 2, we give the numerical values of x, v1, v2, P0

for Kerr-AdS black hole. Combing them with the equationbelow derived from the equal area law

P0(v2 − v1) = T0 ln

(v2

v1

)+ 1

2π

(1

v2− 1

v1

)

− 48J 2

5π

(1

v52

− 1

v51

)(2.33)

to eliminate P0 and T0, we get

v42 = J 2 y1(x)

y2(x), (2.34)

where

y1(x) = 96(1 + x3)(1 + x + x2) lnx + 96

5

(1 − x5)

+ 96(1 − x)(1 + x + x2 + x3 + x4) (2.35)

and

y2(x) = x4[2(1 − x) + (1 + x) lnx]

(2.36)

One can also employ the two functions and Eq. (2.34) toderive the critical value of the thermodynamic quantities.

v4c = J 2 lim

x→1

y1(x)

y2(x). (2.37)

Therefore, vc = 2×901/4√

J . This method is more effectivethan the usual method to calculate the critical point of thethermodynamic quantities of some complicated black holes.

From Figs. 2 and 3, it is shown that the isobar in theisotherm will shorten with increasing temperature. Whentemperature reaches a limit, namely the critical tempera-ture Tc, the left and right ends of the isobar will coincide.In the coexisting phases the evolution of black holes is verydifferent from the Hawking behavior. Due to Hawking radi-ation, the temperature of black holes should be higher andhigher. In the coexisting phases the Hawking radiation doesnot influence the temperature and the pressure/the cosmo-logical constant.

3 Conclusion

We studied the thermodynamic behavior of RN-AdS blackhole and Kerr-AdS black hole in the extended phase space. If

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Fig. 2 Plots of the pressure P

at constant temperature versusthe specific volume v for theRN-AdS black hole. The curvesfrom top to below correspond tothe T = Tc , T = 0.8Tc ,T = 0.7Tc respectively. We setQ = 1 here

Fig. 3 Plots of the pressure P

at constant temperature versusthe specific volume v for theKerr-AdS black hole. Thecurves from top to belowcorrespond to the T = Tc ,T = 0.8Tc , T = 0.7Tc

respectively. We set J = 1 here

the black holes really express themselves like van der Waalsliquid–gas system, in the isotherms of the AdS black holesthere must be regimes where the condition of stable equi-librium is violated. Analogous to the VdW system the un-physical oscillating part in the isotherm should be replacedwith an isobar, which represent a coexistence line. Accord-ing to the equal area law, we find out the position of theisobars in the P –v plane at different temperature T < Tc .It is shown that the length of the isobar indeed depends onthe temperature. The higher the temperature is, the shorterthe isobar will be. If the phase diagrams of the black holesare indeed the case, the evolution of black holes will be very

different from the usual Hawking behavior during the coex-istence regime, where the temperature and the pressure areboth constant.

Although the equal area law has been used to constructthe phase diagram of the charged AdS black hole in theT –S plane in Spallucci and Smailagic (2013). The methodwe used is different. We start from the chemical potentialor the Gibbs potential to obtain the equal area law. In theP –v plane we derive the position of the coexistence isobarby introducing two functions y1(x), y2(x). Taking the limit,x → 1, one can easily obtain the critical volume vc accord-ing to y1(x), y2(x). Thus the other critical thermodynamic

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quantities can be calculated easily. For some complicatedblack holes, such as Gauss-Bonnet-AdS one, the methodwill be more effective to obtain the critical value of the ther-modynamic quantities.

Acknowledgements This work is supported by NSFC under GrantNos. (11175109; 11075098) and by the Shanxi Datong University doc-toral Sustentation Fund Nos. 2011-B-03, China.

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