arX

iv:1

906.

0555

7v1

[gr

-qc]

13

Jun

2019

Heat engine efficiency and Joule-Thomson expansion of non-linear charged

AdS black hole in massive gravity

Cao H. Nam∗

High Energy Physics and Cosmology Laboratory,

Phenikaa University, Hanoi 100000, Vietnam and

Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea

(Dated: June 14, 2019)

In this paper, we have considered the heat engine and Joule-Thomson expansion for the

charged AdS black hole in the context of the non-linear electrodynamics and massive gravity.

For the black hole heat engine, we obtained the analytical expression for the efficiency in

terms of either the entropies or the temperatures and pressures in various limits. For the

Joule-Thomson expansion of the black hole, we derived the isenthalpic curves in T − P

diagram, the Joule-Thomson coefficient, and the inversion curves. We also indicated in

detail the effects of the non-linear electrodynamics and massive gravity on the heat engine

efficiency and the Joule-Thomson expansion of the black hole.

I. INTRODUCTION

Within the framework of General Relativity (GR), under general physical conditions, the space-

time would admit curvature singularities, i.e., the boundaries of spacetime beyond which an exten-

sion of spacetime is impossible and at which the curvatures and densities become infinite [1]. In the

cosmology, the beginning of Universe was at big bang singularity. The gravitational collapse of the

matters leads to the formation of the black holes with the curvature singularities surrounded by

the event horizon. It is widely believed that the presence of these curvature singularities are a sign

indicating the breakdown of GR and thus requiring a more fundamental theory of the gravitation,

e.g. quantum gravity. But, such a complete theory has not been achieved so far. Since, whether the

spacetime singularity can be resolved at the level of the classical gravity is still an open question

in current research.

The non-linear electrodynamics may appear as the low-energy limit of the heterotic string

theory [2–6]. On the other hand, the classical gravity with the nonlinear electrodynamics can be

realized as the effective description arising from quantum gravity coupled to the matter. Thus, it

∗Electronic address: hncao@yonsei.ac.kr

2

is natural to expect that the non-linear properties of the fundamental theory may be exhibited in

the physics of the black holes. In particular, Ayon-Beato and Garcia has indicated that a source of

the non-linear electrodynamics causes a regular black hole which is free of a curvature singularity

at the origin but possesses an event horizon [7, 8]. In addition, the Bardeen black hole, which is

the first regular black hole found by Bardeen [9] but whose physical source was not realized at that

moment, was later reobtained as a gravitational collapse of some magnetic monopole in the context

of the non-linear electrodynamics [10]. Since the pioneering works of Ayon-Beato and Garcia, many

regular black hole solutions with various non-linear electromagnetic sources have been found in the

literature [11–31]. Other regular black holes were also constructed [32–38]. Studying the regular

black holes has thus drawn many attentions [39–76].

One of the straightforward extensions of GR is to consider that graviton is massive spin-2

particle with five degrees of freedom. A theoretical reason for this extension is that it could

provide the natural resolution for the acceleration of Universe without introducing dark energy.

Also, the recent observations of the gravitational waves by LIGO have constrained graviton mass

to m ≤ 1.2× 10−22 eV [77]. This means that the mass of graviton can be tiny but non-zero. The

first construction for the massive gravity theory, which is ghost-free, was done by Fierz and Pauli

(FP) [78], at which mass term at the inearized level is included as

SFP = −m2

4

∫

d4x(

hµνhµν − h2

)

, (1)

where hµν is a symmetric tensor field describing massive graviton, hµν = ηµσηνρhσρ, and h =

ηµνhµν . However, this massive gravity theory suffers from an important problem, which GR is

not recovered in the zero mass limit of graviton, well-known as the van Dam-Veltman-Zakharov

(vDVZ) discontinuity [79, 80]. The vDVZ discontinuity could be resolved in the nonlinear massive

gravity theories which include additionally the higher derivative terms [81–83]. Unfortunately, these

non-linear massive gravity theories lead to the appearance of the so-called Boulware and Deser

(BD) ghosts. A successful non-linear massive gravity theory, which is ghost-free and recovers

GR as graviton mass goes to zero, was proposed by de Rham, Gabadadze and Tolley (dRGT)

[84, 85]. Later, various black hole solutions have been found in dRGT massive theory and their

thermodynamics as well as critical phenomena have been investigated [30, 86–100].

The study of the black hole thermodynamics in anti-de Sitter (AdS) space has been an inter-

esting research topic because of its rich phase structure as well as AdS/CFT correspondence [101].

Hawking and Page discovered a phase transition between Schwarzschild AdS black hole and ther-

mal AdS space [102], called the Hawking-Page transition in the literature, which is interpreted as

3

the confinement/deconfinement phase transition in the boundary conformal field theory [103, 104].

In addition, Chamblin et al. studied the charged AdS black holes in both canonical and grand

canonical ensembles, and they found a first-order phase transition between small and large black

holes [105, 106]. Recently, the phase space of the AdS black hole thermodynamics has been ex-

tended at which the cosmological constant is treated as thermodynamic pressure corresponding to

the conjugate quantity as the thermodynamic volume [107–112]. A a result, the black hole mass is

most naturally considered as the enthalpy, rather than the internal energy. In the extended phase

space, P − V criticality was discovered in the charged AdS black hole [113] as well as in various

black holes [114–135].

With the extended phase space, it is possible to extract the mechanical work from the heat

energy via the term PdV . This suggests that the concept of the traditional heat engine can be

incorporated into the black holes at which the black holes play the role as the working substances

[136–161]. A heat engine is defined by a heat cycle which is a closed path in the P − V diagram.

It works between warmer and colder reservoirs which correspond to the temperatures TH and TC

(TH > TC), respectively. During the working process, the heat engine absorbs a heat amount

QH from the warmer reservoir and exhausts a heat amount QC to the colder reservoir. A total

mechanical work W , which is produced by the heat engine, is given by W = QH − QC . The

efficiency of the heat engine is defined by

η =W

QH= 1− QC

QH. (2)

It is clear that the efficiency of the black hole heat engine depends crucially on both the equation

of the state provided by the black hole and the the paths forming the heat cycle in the P − V

diagram. On the other hand, various black holes should produce the different heat engines.

One of the recent developments with respect to the black hole thermodynamics in the ex-

tended phase space is the Joule-Thomson expansion. In the traditional thermodynamics, the

Joule-Thomson expansion is that the enthalpy is constant during the expansion process of the gas

or fluid from a high pressure to a low pressure through a porous plug. Intrestingly, Okcu and

Aydiner incorporated the concept of the Joule-Thomson expansion into the charged AdS black

holes [162] and the Kerr-AdS black holes [163]. Later, the Joule-Thomson expansion has been

generalized to various black holes [164–174].

This work is organized as follows. In Sec. II, we review briefly the non-linear charged AdS black

hole solution and its thermodynamics in the massive gravity, obtained in Ref. [30]. In Sec. III, we

treat this black hole as the heat engine. In this section, we compute the efficiency of the black hole

4

heat engine, study how the non-linear parameter as well as the massive gravity couplings affect

the efficiency of the black hole heat engine, and then compare it to the Carnot efficiency. Sec. IV

is devoted to study the Joule-Thomson expansion of the non-linear charged AdS black hole in the

massive gravity. Finally, we make a conclusion in last section. Note that, in this paper, we use

units in GN = ~ = c = kB = 1 and the signature of the metric (−,+,+,+).

II. A BRIEF REVIEW OF NON-LINEAR CHARGED AdS BLACK HOLE

The aim in this present work is to generalize the recent developments of the heat engine and

Joule-Thomson expansion for the non-linear charged AdS black hole in massive gravity, which was

recently obtained in Ref. [30]. Thus, in this section we will review this black hole solution and

its thermodynamic properties. This black hole solution obtained from solving the equations of

motion for the system of the massive gravity coupled to the non-linear electromagnetic field in the

four-dimensional AdS spacetime background. The action of this system is given by

S =

∫

d4x√−g

1

16π

[

R− 2Λ +m24∑

i=1

ciUi(g, f)

]

− 1

4πL(F )

, (3)

where R refers to the spacetime scalar curvature, Λ is the negative cosmological constant expressed

in terms of the curvature radius l of the AdS spacetime background as

Λ = − 3

l2, (4)

m is graviton mass, ci are the couplings of the massive gravity, f is the reference metric which is kept

fixed, Ui are symmetric polynomials in terms of the eigenvalues of the 4×4 matrix Kµν =

√

gµλfλν

given as

U1 = [K],

U2 = [K]2 − [K2],

U3 = [K]3 − 3[K][K2] + 2[K3],

U4 = [K]4 − 6[K]2[K2] + 8[K][K3] + 3[K2]2 − 6[K4], (5)

with [K] = Kµµ. The Lagrangian L(F ) of the non-linear electrodynamics L(F ) is defined as

L(F ) = Fe−k2Q (2Q

2F)14

, F ≡ 1

4FµνF

µν , (6)

where Fµν = ∂µAν − ∂νAµ is the strength tensor of the non-linear electromagnetic field, Q is the

total charge of the system, and k is a fixed parameter by which the charge Q and the mass M of

the system are related as, Q2 = Mk.

5

With the spherically-symmetric and static spacetime and the reference metric taken in the

following form [88]

fµν = diag(0, 0, c2, c2 sin2 θ), (7)

where c is a positive constant, we find the equations of motion which are obtained from the variation

of the action (3) as

Gνµ −

[

3

l2+m2

(

cc1r

+c2c2r2

)]

δνµ = 2

[

∂L(F )

∂FFµρF

νρ − δνµL(F )

]

,

∇µ

(

∂L(F )

∂FF νµ

)

= 0.

∇µ ∗ F νµ = 0. (8)

A spherically-symmetric and static black hole solution of the mass M and magnetic charge Q is

obtained from solving these equations of motion, given by

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ22,

Fµν =(

δθµδϕν − δθνδ

ϕµ

)

B(r, θ), (9)

where

f(r) = 1− 2M

re−

k2r +

r2

l2+m2

(cc1r

2+ c2c2

)

,

B(r, θ) = Q sin θ. (10)

It is important to note that 1 + m2c2c2 plays the role of the effective horizon curvature which

can be positive (the sphere effective horizon), zero (the flat one), or negative (the hyperbolic one)

depending on the sign and the absolute value of the coupling parameter c2. With M = Q = 0, it

leads to the vacuum solution as

f(r) = 1 +r2

l2+m2

(cc1r

2+ c2c2

)

. (11)

In the region of the large distances k/r ≪ 1, the function f(r) becomes

f(r) ≃ 1− 2M

r+

Q2

r2+

r2

l2+m2

(cc1r

2+ c2c2

)

. (12)

This means that the black hole behaves asymptotically like the 4D RN-AdS black hole in the

massive gravity [88–91].

The black hole mass M is expressed in terms of the event horizon radius r+ and the pressure

P = − Λ8π = 3

8πl2 as

M =r+2e

k2r+

[

1 +8πPr2+

3+m2

(cc1r+2

+ c2c2

)

]

. (13)

6

The first law of the black hole thermodynamics is given by

dM = TdS + V dP + C1dc1 + C2dc2, (14)

where, because in general the massive gravity couplings can vary, they have been treated as the

thermodynamic variables corresponding to the conjugating variables C1 and C2, respectively. The

black hole temperature is identified by using the surface gravity at the event horizon as

T =f ′(r+)

4π=

(

2r+ − k

3

)

P +(2r+ − k)

8πr2+

[

1 +m2(cc1r+

2+ c2c2

)]

+m2cc18π

. (15)

The black hole entropy S, the thermodynamic volume V , and the conjugating quantities C1,2 are

obtained from the first law as

S =

∫

1

T

(

∂M

∂r+

)

dr+ = πr2+

(

1 +k

2r+

)

ek

2r+ − πk2

4Ei

(

k

2r+

)

, (16)

V =

(

∂M

∂P

)

S,c1,c2

=4πr3+3

ek

2r+ , (17)

C1 =

(

∂M

∂c1

)

S,P,c2

=m2cr2+

4e

k2r+ , (18)

C2 =

(

∂M

∂c2

)

S,P,c1

=m2c2r+

2e

k2r+ , (19)

where the function Ei(x) is given by

Ei(x) = −∫ ∞

−x

e−t

tdt. (20)

Clearly, the black hole entropy is modified by only the non-linear electrodynamics. In the regime

of the large horizon radius k/r+ ≪ 1, the black hole entropy is approximately given by

S = πk2[

(r+k

)2+

r+k

+3− 2γ

8− 1

4ln

(

k

2r+

)

+O(

k

r+

)]

, (21)

where γ ≈ 0.577216 is Euler’s constant. This expansion suggests that the black hole entropy

satisfies approximately the area law (S = πr2+) in the regime of the large horizon radius. The heat

capacity at constant pressure is

CP = T

(

∂S

∂T

)

P

=∂M

∂r+

(

∂T

∂r+

)−1

,

=πr2+6

h1(r+)

h2(r+)e

k2r+ , (22)

where

h1(r+) = 16πPr2+(6r+ − k) + 3m2cc1r+(4r+ − k) + 6(1 +m2c2c2)(2r+ − k),

h2(r+) = 8πPr3+ −(

1 +m2c2c2 −m2cc1k

4

)

r+ + k(1 +m2c2c2). (23)

7

The equation of state for the black hole can easily be obtained from Eq. (15) as

P =T

2r+ − k/3+

2(1 +m2c2c2)(k − 2r+) +m2cc1r+(k − 4r+)

16πr2+(2r+ − k/3), (24)

where r+ = r+(V ) is understood to be a function of the thermodynamic volume V , which is

determined by Eq. (17). If the effective horizon curvature 1 +m2c2c2 is positive, a critical point

appears when the isotherm in the P − r+ diagram has an inflexion point, determined by

(

∂P

∂r+

)

T

=

(

∂2P

∂r2+

)

T

= 0. (25)

From this, we can obtain the critical radius rc, the critical temperature Tc and the critical pressure

Pc as

rc =6k(1 +m2c2c2)

4(1 +m2c2c2)−m2cc1k,

Tc =13(1 +m2c2c2)

81πk+

37m2cc1216π

+(m2cc1)

2k(

1 +m2c2c2 +m2cc1k/96)

108π(1 +m2c2c2)2,

Pc =(1 +m2c2c2 −m2cc1k/4)

3

54πk2(1 +m2c2c2)2. (26)

If the temperature is smaller than the critical one, the black hole can undergo a first-order phase

transition between the small black hole and the large black hole, which is analogous to the van

der Waals phase transition. This phase transition disappears if the temperature is larger than the

critical one.

III. THE NON-LINEAR CHARGED ADS BLACK HOLE AS A HEAT ENGINE

We consider a heat cycle which consists of two isobaric paths and two isochoric paths, as given

in Fig. 1. Note that, from Eqs. (16) and (17), we can infer that the entropy S is a function of the

thermodynamic volume V . This suggests dS ∼ dV and thus the isochoric or adiabatic paths are

the same. It is easy to calculate the total work done in this heat cycle as

W =

∮

PdV = (P1 − P4)(V2 − V1),

=4π

3(P1 − P4)

(

r3+2ek

2r+2 − r3+1ek

2r+1

)

. (27)

Whereas, the amount of the input heat is calculated as

QH =

∫ T2

T1

CP (P1, T )dT =

∫ r+2

r+1

∂M

∂r+dr+,

=4π

3P1r+e

k2r+

[

r2+ +3

8πP1+

3m2

8πP1

(cc1r+2

+ c2c2

)

]

∣

∣

∣

r+2

r+1

. (28)

8

P

V

1 2

34

FIG. 1: A cycle of the black hole heat engine with two isobaric paths and two isochoric paths.

The heat engine efficiency of the black hole thus is given by

η =W

QH=

(

1− P4

P1

)

r3+2ek

2r+2 − r3+1ek

2r+1

r+ek

2r+

[

r2+ + 38πP1

+ 3m2

8πP1

( cc1r+2 + c2c2

)

]∣

∣

∣

r+2

r+1

. (29)

Here, by solving Eq. (16) we can express r+2 and r+1 as the functions of the corresponding entropies

as

r+2 = r+2(S2), r+1 = r+1(S1). (30)

Or, by solving Eq. (15) we can express them as the functions of the corresponding temperature

and pressure as

r+2 = r+2(TH , P1), r+1 = r+1(TC , P4), (31)

where we have used TC ≡ T4 and TH ≡ T2. In general, because the expressions for the black hole

entropy and the black hole temperature are quite complex, it is very difficult to get explicitly these

functions. However, we can do this in some limits. For the case k/r+ ≪ 1, we have the following

approximation for the functions r+2 = r+2(S2) and r+1 = r+1(S1) as

r+2 =k

2

(

√

1 +4S2

πk2− 1

)

, r+1 =k

2

(

√

1 +4S1

πk2− 1

)

. (32)

Let us consider the heat cycle in the limit of the high temperature and high pressure in which the

event horizon radius r+ of the black hole can be expanded as

r+ =

∞∑

n=0

ǫnrn, ǫ =1

8πP. (33)

Using this expansion, one can solve perturbatively for r+ from Eq. (15). At the lowest-order

approximation (n = 0), we have

r+ = r0 =T

2P+

k

6. (34)

9

At this approximation, the output work and the net inflow of the black hole heat engine are given

by

W =4π

3(P1 − P4)

(

r302ek

2r02 − r301ek

2r01

)

≡ W0,

QH =4π

3P1

(

r302ek

2r02 − r301ek

2r01

)

≡ QH0, (35)

where

r01 =TC

2P4+

k

6, r02 =

TH

2P1+

k

6. (36)

As a result, at the lowest-order approximation the heat engine efficiency of the black hole becomes

η =W

QH

= 1− P4

P1. (37)

At the leading-order approximation (n = 1), we have

r+ = r0 + ǫr1 = r0 +k − 2r016πP1r20

[

1 +m2(cc1r0

2+ c2c2

)]

− m2cc116πP1

. (38)

We can easily obtain the output work and the net inflow of the heat at this approximation as

W = W0 +W1

8πP1,

QH = QH0 +QH1

8πP1, (39)

where

W1 =2π(P1 − P4)

3

[

r02r12(6r02 − k)ek

2r02 − r01r11(6r01 − k)ek

2r01

]

,

QH1 =P1

P1 − P4W1 + 4πP1

r02ek

2r02

[

1 +m2(cc1r02

2+ c2c2

)]

− r01ek

2r01

[

1 +m2(cc1r01

2+ c2c2

)]

,

(40)

with

r12 =k − 2r022r202

[

1 +m2(cc1r02

2+ c2c2

)]

− m2cc12

,

r11 =k − 2r012r201

[

1 +m2(cc1r01

2+ c2c2

)]

− m2cc12

. (41)

Then, the heat engine efficiency of the black hole corresponding to the leading-order approximation

is given by

η =

(

1− P4

P1

)[

1− 1

8πP1

(

QH1

Q0− W1

W0

)]

+O(

1

P 21

)

. (42)

10

k=1.0

k=0.8

k=0.6

k=0.4

k=0.1

k=1.6

10 20 30 40 50S20.74

0.75

0.76

0.77

0.78

0.79

0.80

0.81

0.82Η

c1=c2=1

k=1.0

k=0.8

k=0.6

k=0.4

k=0.1

k=1.6

10 20 30 40 50S20.800

0.802

0.804

0.806

0.808

0.810

0.812

0.814

Η

c1=c2=-1

c1=0.5

c1=-0.5

c1=-1.0

c1=1.0

10 20 30 40 50S20.74

0.75

0.76

0.77

0.78

0.79

0.80

0.81Η

k=0.5, c2=1

c2=0.5

c2=-0.5

c2=-1.0

c2=1.0

10 20 30 40 50S20.786

0.788

0.790

0.792

0.794

0.796

0.798

0.800Η

k=1, c1=1

FIG. 2: Heat engine efficiency η of the black hole as a function of the entropy S2, at S1 = P4 = m = c = 1

and P1 = 5.

Now we analyze the behavior of the heat engine efficiency η as a function of the entropy S2

and the pressure P1, corresponding to the heat cycle given in Fig. 1, for the various values of the

non-linear and massive gravity parameters. By employing Eqs. (16) and (29), one can generate

the parametric plots of the entropy S2 and the heat engine efficiency η , with the entropy S1 and

the pressures P1,2 kept fixed all, which are given in Fig. 2. From this figure, we observe that

the behavior of the heat engine efficiency is crucially dependent on the non-linear and massive

gravity parameters. With the proper parameters, the heat engine efficiency is a monotonously

increasing/decreasing function with the growth of the entropy S2. This suggests that the bigger

black holes have the larger/smaller efficiency of the heat engine. Whereas, with other proper

combinations of the parameters, they lead to the heat engine efficiency curve which has a global

maximum/minimum value. This means that there exits a finite value of the entropy S2 at which

the heat engine of the black hole works at the highest or lowest efficiency. In the limit of that

the entropy S2 goes to the infinity, the heat engine efficiency should approach η = 1 − P4/P1. In

addition, the heat engine efficiency is plotted against the pressure P1, with the entropies S1,2 and

11

k=1

k=2

k=3

10 20 30 40 50P1

0.75

0.80

0.85

0.90

0.95

1.00Η

c1=c2=2

c1=-10

c1=1

c1=10

10 20 30 40 50P1

0.75

0.80

0.85

0.90

0.95

1.00Η

k=c2=1

FIG. 3: Heat engine efficiency as a function of the pressure P1, at S1 = P4 = m = c = 1 and S1 = 15.

c1=c2=1

c2=-c1=1

c1=-c2=1

c1=c2=-1

2 4 6 8 10k0.78

0.79

0.80

0.81

0.82

0.83

0.84Η

FIG. 4: Heat engine efficiency as a function of the non-linear parameter k, at S1 = P4 = m = c = 1, S2 = 15

and P1 = 5.

the pressure P4 kept fixed all, given in Fig. 3. From this figure, we found that increasing the

pressure makes the increasing of the heat engine efficiency. In the limit of that the pressure P1

goes to the infinity, the efficiency should approach η = 1. From the mentioned two figures, we can

see that the heat engine efficiency almost decreases as the massive gravity parameters increase. In

order to see more explicitly the effect of the non-linear parameter k on the heat engine efficiency

η, we plot η as a function of k in Fig. 4. We observe that depending the value region of the

non-linear parameter k as well as the sign of the massive gravity parameters, increasing k leads to

the increasing or decreasing of the heat engine efficiency.

Let us compare the heat engine efficiency η with the Carnot efficiency ηC which is the maximum

value. The black hole as a heat engine with the Carnot efficiency is described by the heat cycle

including a pair of isothermal paths connected to each other by a pair of either isochoric paths, as

shown in Fig. 7. It is easily to calculate the amount of the input heat QH and the amount of the

12

k=0.6

k=0.4

k=0.1

k=0.8

10 20 30 40 50S20.78

0.80

0.82

0.84

0.86

0.88

0.90

Η

ΗC

c1=c2=1

k=0.6

k=0.4

k=0.1

k=0.8

10 20 30 40 50S20.80

0.82

0.84

0.86

0.88

0.90

Η

ΗC

c1=c2=-1

c1=0.5

c1=-0.5

c1=-1.0

c1=1.0

10 20 30 40 50S20.800

0.805

0.810

0.815

0.820

0.825

0.830

0.835

0.840

Η

ΗC

k=0.6, c2=1

c2=0.5

c2=-0.5

c2=-1.0

c2=1.0

10 20 30 40 50S20.79

0.80

0.81

0.82

0.83

0.84

Η

ΗC

k=0.8, c1=1

FIG. 5: The ratio η/ηC as a function of the entropy S2, at S1 = P4 = m = c = 1, and P1 = 5.

k=1

k=2

k=3

10 20 30 40 50P1

0.75

0.80

0.85

0.90

0.95

1.00

Η

ΗC

c1=c2=1

c1=-10

c1=1

c1=10

10 20 30 40 50P1

0.75

0.80

0.85

0.90

0.95

1.00

Η

ΗC

k=1, c2=1

FIG. 6: The ratio η/ηC as a function of the pressure P1, at S1 = P4 = m = c = 1, and S2 = 15.

exhaust heat QC as

QH = TH(S2 − S1),

QC = TC(S3 − S4). (43)

13

P

V

1

2

3

4

FIG. 7: A Carnot heat cycle with two isothermal paths and two isochoric paths.

From Eqs. (16) and (17), we can find S2 − S1 = S3 − S4 for the heat cycle given in 7. As a result,

the Carnot efficiency ηC is given by

ηC = 1− QC

QH

= 1− TC

TH

. (44)

In Figs. 5 & 6, we plot the ratio η/ηC as a function of the entropy S2 and the pressure P1. We find

that the heat engine efficiency η is close the Carnot efficiency ηC in the region of the entropy S2

near S1. The ratio η/ηc is a monotonously increasing function with the growth of the pressure P1.

This implies that the heat engine efficiency should approach the Carnot efficiency as the pressure

P1 goes to the infinity. Furthermore, the change of the non-linear and massive gravity parameters

affect significantly on the ratio η/ηC . More specifically, increasing the non-linear parameter k

makes either increasing or decreasing the ratio η/ηC , which are dependent on the value region of

k. Whereas, increasing the massive gravity parameters leads to a lower ratio η/ηC .

IV. JOULE-THOMSON EXPANSION OF NON-LINEAR CHARGED ADS BLACK

HOLE

In this section, we will study the isenthalpy process or the Joule-Thomson expansion of the non-

linear charged AdS black hole in the presence of graviton mass. The Joule-Thomson expansion of

the black hole is described by the constant mass curves in the T −P diagram. From Eqs. (13) and

(15), we can express the pressure and the temperature as the functions of the black hole mass M

and its event horizon radius r+ as

P (M, r+) =3

16πr3+

[

4Me− k

2r+ − 2(1 +m2c2c2)r+ −m2cc1r2+

]

, (45)

T (M, r+) =1

8πr3+

[

2M(6r+ − k)e− k

2r+ − 4(1 +m2c2c2)r2+ −m2cc1r

3+

]

. (46)

14

k=1.2

k=0.9

k=0.6

k=1.5

10 20 30 40P

1

2

3

4

5

6T

c1=c2=1

k=1.2

k=0.9

k=0.6

k=1.5

10 20 30 40 50 60 70P

2

4

6

8T

c1=c2=-1

c1=1

c1=-1

c1=-4

c1=4

2 4 6 8 10P

0.5

1.0

1.5

2.0

2.5T

k=c2=1

c2=1

c2=-0.5

c2=-1

c2=4

2 4 6 8 10 12 14P

0.5

1.0

1.5

2.0

2.5

3.0T

k=c1=1

FIG. 8: The constant mass curves are plotted for various values of the non-linear parameter k and the

massive gravity coupling parameters c1,2, at m = c = 1, and M = 5.

In principle, in order to find the function T = T (M,P ), one needs to solve Eq. (45) to obtain the

event horizon radius as a function of the pressure P and the black hole mass M then substituting

it into Eq. (46). Because of the complexity of Eq. (45), it is in general a difficult task. However,

employing these equations can generate the parametric plots of the pressure P and the temperature

T . In Fig. 8, we show the constant mass curves for various values of the non-linear parameter k

and the massive gravity coupling parameters c1,2. From this figure, we observe that with the same

mass, as the non-linear parameter k and the massive gravity coupling parameters c1,2 increase, the

constant mass curves tend to shrink towards the lower pressure and temperature.

The pressure always decreases in the Joule-Thomson expansion. But, the temperature can in-

crease or decrease. The increasing of the temperature in the Joule-Thomson expansion corresponds

to the appearance of the heating process, associated with the the negative slope of the constant

mass curves. Whereas, the decreasing of the temperature corresponds to the appearance of the

cooling process, associated with the the positive slope of the constant mass curves. An essential

15

physical quantity, which characterizes the Joule-Thomson expansion, is the change of the tem-

perature with respect to the pressure along the constant mass curves, called the Joule-Thomson

coefficient. This quantity not only describes that the Joule-Thomson expansion is fast or slow,

but also it allows us to determine whether the heating or the cooling process will appear in this

expansion. The Joule-Thomson coefficient µ is thus defined as

µ =

(

∂T

∂P

)

M

=

(

∂T

∂r+

)

M

/

(

∂P

∂r+

)

M

,

=

8r2+

[

ek

2r+ (1 +m2c2c2)r+ − 6M

]

+ 24kMr+ − 2k2M

6M(k − 6r+) + 3ek

2r+ [4(1 +m2c2c2) +m2cc1r+] r2+

. (47)

It is interesting that we can express the Joule-Thomson coefficient µ in terms of the temperature,

the thermodynamic volume and the heat capacity. For the Joule-Thomson expansion, we have

dM = 0, which leads to the following relation

T

(

∂S

∂P

)

M

+ V = 0. (48)

In addition, from Eqs. (15) and (16), the black hole entropy can be understood as a state function,

S = S(T, P ), and thus we find

(

∂S

∂P

)

M

=

(

∂S

∂P

)

T

+

(

∂S

∂T

)

P

(

∂T

∂P

)

M

. (49)

With these two relation, one can easily obtain

T

[(

∂S

∂P

)

T

+

(

∂S

∂T

)

P

(

∂T

∂P

)

M

]

+ V = 0. (50)

By using the Maxwell relation (∂S/∂P )T = − (∂V/∂T )P and the definition of the heat capacity

CP = T (∂S/∂T )P , we derive

µ =

(

∂T

∂P

)

M

=1

CP

[

T

(

∂V

∂T

)

P

− V

]

. (51)

The expressions for the Joule-Thomson coefficient µ are given at Eqs. (47) and (51) are equivalent.

The inversion points (Pi, Ti) form an inversion curve that divides the T − P graph into the

cooling and heating regions. The inversion points (Pi, Ti) are obtained by setting µ = 0, which

leads to the following equation

8r2+

[

ek

2r+ (1 +m2c2c2)r+ − 6M

]

+ 24kMr+ − 2k2M = 0, (52)

which is independent on the coupling parameter c1. Solving this equation for r+ and then substi-

tuting into Eqs. (45) and (46), one can derive the inversion curve Ti = Ti(Pi). In general, it is

16

difficult to solve exactly this equation. However, for 1 +m2c2c2 = 0, it is easily to get exactly two

solutions of Eq. (52) as

r+ =3±

√3

12k. (53)

With the positive sign, we have

T(+)i =

1

8π

[

96(2√3− 3)M

e3−√3k2

−m2cc1

]

,

P(+)i =

27[

96e√3M − (2 +

√3)e3m2cc1k

2]

2(3 +√3)3e3πk3

. (54)

For c1 > 0, we need the following condition

M

m2cc1k2≥ (2 +

√3)e3−

√3

96. (55)

With the negative sign, we obtain

T(−)i = − 1

8π

[

96(2√3 + 3)M

e3+√3k2

+m2cc1

]

,

P(−)i =

27[

96e−3−√3M − (2−

√3)m2cc1k

2]

2(3−√3)3πk3

, (56)

which only exits if c1 < 0 and

− M

m2cc1k2≤ e3+

√3

96(2√3 + 3)

. (57)

By eliminating the black hole mass M , we can obtain two inversion curves as

T(+)i =

√3 + 1

6kP

(+)i +

√3− 1

8πm2cc1,

T(−)i = −

√3− 1

6kP

(−)i −

√3 + 1

8πm2cc1. (58)

First, let us consider the case c1 > 0. In this case, it is easily to see that the curve T(−)i

(

P(−)i

)

disappears. Thus, the inversion points (Pi, Ti) are completely determined by the curve T(+)i

(

P(+)i

)

.

Here, one can obtain the global minimum inversion temperature Tmin corresponding to P(+)i = 0

as

Tmin =

√3− 1

8πm2cc1. (59)

Furthermore, in this case the inversion temperature is a monotonically increasing function of the

inversion pressure. And, with the inversion pressure kept fixed, the inversion temperature increases

17

with the growth of both the non-linear parameter k and the massive gravity coupling c1. In the

case c1 < 0, for the large enough inversion pressure, the inversion points (Pi, Ti) are determined

by the curve T(+)i

(

P(+)i

)

. Thus, the behavior of the inversion temperature under the change of the

inversion pressure, the non-linear parameter k and the coupling parameter c1 is like the case c1 > 0.

Whereas, for the small enough inversion pressure, the inversion points (Pi, Ti) are determined by

the curve T(−)i

(

P(−)i

)

. In contract to the large enough inversion pressure, in this case, increasing the

inversion pressure, the non-linear parameter k or the coupling parameter c1 leads to the decreasing

of the inversion temperature.

V. CONCLUSION

For the black hole thermodynamics in the extended phase space at which the cosmological

constant is considered as the pressure, one can introduce the concept of the heat engine for the

black hole. In this paper, we consider the non-linear charged AdS black hole in the massive gravity

as a heat engine. The heat cycle under consideration, given in the P − V diagram, consists of

two isobaric paths and two isochoric paths. It is shown that the non-linear parameter and massive

gravity couplings affect significantly the efficiency of the black hole heat engine. As the non-linear

parameter increases, the heat engine efficiency can increase or decrease dependently on the value

region of the non-linear parameter as well as the sign of the massive gravity couplings. In addition,

increasing the massive gravity couplings almost makes decreasing the heat engine efficiency.

In the extended phase space, the black hole mass is most naturally identified as the enthalpy.

Thus, one can consider the expansion process of the black hole during which the black hole mass is

kept fixed, well-known as the Joule-Thomson expansion process. We study how the presence of the

non-linear elctrodynamics and the massive gravity affect the isenthalpic curves of the black hole

in detail. Also, we calculate the Joule-Thomson coefficient whose sign allows to determine which

of the heating or cooling will appear. Finally, we derive analytically the inversion curves, which

separates the cooling region and heating region, for the case 1 +m2c2c2 = 0.

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