#### Joule–Thomson expansion of Kerr–AdS black holes

Eur. Phys. J. C
Joule-Thomson expansion of Kerr-AdS black holes
Özgür Ökcü 0
Ekrem Aydıner 0
0 Department of Physics, Faculty of Science, Istanbul University , Vezneciler, 34134 Istanbul , Turkey
In this paper, we study Joule-Thomson expansion for Kerr-AdS black holes in the extended phase space. A Joule-Thomson expansion formula of Kerr-AdS black holes is derived. We investigate both isenthalpic and numerical inversion curves in the T - P plane and demonstrate the cooling-heating regions for Kerr-AdS black holes. We also calculate the ratio between minimum inversion and critical temperatures for Kerr-AdS black holes.
1 Introduction
Since the first studies of Bekenstein and Hawking [
1–6
],
black holes as thermodynamic system have been an
interesting research field in theoretical physics. The black hole
thermodynamics provides fundamental relations between
theories such as classical general relativity, thermodynamics and
quantum mechanics. Black holes as thermodynamic system
have many exciting similarities with general thermodynamic
system. These similarities become more obvious and precise
for the black holes in AdS space. The properties of AdS black
hole thermodynamics have been studied since the seminal
paper of Hawking and Page [7]. Furthermore, the charged
AdS black holes thermodynamic properties were studied in
[
8,9
] and it was shown that the charged AdS black holes have
a van der Waals like phase transition.
Recently black hole thermodynamics in AdS space has
been intensively studied in the extended phase space where
the cosmological constant is considered as the
thermodynamic pressure. Extended phase space leads to important
results: Smarr relation is satisfied for the first law of the black
holes thermodynamics in the presence of variable
cosmological constant. It also provides the definition of the
thermodynamic volume which is more sensible than the geometric
volume of the black hole. In addition to similar behaviours
with conventional thermodynamic systems, studying the AdS
black holes is another important reason for the AdS/CFT
correspondence [
10
]. Considering the cosmological constant as
thermodynamic pressure,
Λ
P = − 8π ,
V =
∂ M
∂ P
S,Q,J
(1)
(2)
lead us to investigate thermodynamic properties, rich phase
structures and other thermodynamic phenomena for AdS
black holes in a similar way to the conventional
thermodynamic systems.
Based on this idea, the charged AdS black hole
thermodynamic properties and phase transition were studied by
Kubiznak and Mann [
11
]. It was shown in this study that the charged
AdS black hole phase transition has the same characteristic
behaviors with van der Waals liquid–gas phase transition.
They also computed critical exponents and showed that they
coincide with exponents of van der Waals fluids. It was shown
in [
12
] that the cosmological constant as pressure requires
considering the black hole mass M as the enthalpy H rather
than as internal energy U . In recent years, thermodynamic
properties and phase transition of AdS black holes have been
widely investigated [
13–54
].1 The phase transition of AdS
black holes in the extended phase space is not restricted to
a van der Waals type transition, but also the reentrant phase
transition and the triple point for AdS black holes were
studied in [
31–34
]. The compressibility of rotating AdS black
holes in four and higher dimensions was studied in [
35,36
].
In [
37
], a general method was used for computing the critical
exponents for AdS black holes which have a van der Waals
like phase transition. Furthermore, heat engines behaviours
1 See [
50–52
] and the references therein for various black hole
solutions.
of the AdS black holes have been studied. For example, in
[38] two kind of heat engines were proposed by Johnson
for charged AdS black holes and heat engines were studied
for various black hole solutions in [
39–49
].2 More recently,
adiabatic processes [53] and Rankine cycle [54] have been
studied for the charged AdS black holes.
In [
55
], we also studied the well-known Joule–Thomson
expansion process for the charged AdS black holes. We
obtained inversion temperature to investigate inversion and
isenthalpic curves. We also showed heating–cooling regions
in T – P plane. However, so far, Joule–Thomson expansion
for Kerr–AdS black holes in extended phase space has never
been studied. The main purpose of this study is to investigate
Joule–Thomson expansion for Kerr–AdS black holes.
The paper is arranged as follows. In Sect. 2, we briefly
review some thermodynamic properties of Kerr–AdS black
holes which are introduced in [
14
].3 In Sect. 3, we first of all
derive a Joule–Thomson expansion formula for Kerr–AdS
black hole by using first law and Smarr formula. Then we
obtain the equation of inversion pressure Pi and entropy S to
investigate the inversion curves. We also show that the ratio
between minimum inversion and critical temperatures for
Kerr–AdS black holes is the same as the ratio of charged AdS
black holes [
55
]. Finally, we discuss our results in Sect. 4.
(Here we use the units G N = h¯ = kB = c = 1.)
2 Kerr–AdS black hole
In this section, we briefly review Kerr–AdS black hole
thermodynamic properties in the extended phase space. The line
element of Kerr–AdS black hole in four dimensional AdS
space is given by
Δ
ds2 = − ρ2
dt −
+
Δθ sin2 θ
ρ2
a sin2 θ
Ξ
adt −
dφ
2
+ ρΔ2 dr 2 + Δρ2θ dθ 2
r 2 + a2 dφ
Ξ
2
,
Finally, we obtain the angular velocity as follows:
π J
S,P = M S
1 +
respectively, and the Smarr relation can be derived by a
scaling argument [
12
]. From Eq. (7), one can obtain the
thermodynamic quantities. The expression for the temperature
is
(6)
(7)
(8)
(9)
(10)
(11)
(12)
In this section, we obtain some thermodynamic
quantities of Kerr–AdS black holes. In the next section, we will
use these quantities to investigate Joule–Thomson expansion
effects for Kerr–AdS black holes.
3 Joule–Thomson expansion
In this section, we will investigate Joule–Thomson expansion
for Kerr–AdS black holes. The expansion is characterized
by temperature change with respect to pressure. Enthalpy
remains constant during the expansion process. As we know
from [
12
], black hole mass is identified enthalpy in AdS
space. Therefore, the black hole mass remains constant
during expansion process. Joule–Thomson coefficient μ, which
characterizes the expansion, is given by [
57
]
μ =
Cooling–heating regions can be determined by sign of
Eq. (12). Change of pressure is negative since the pressure
always decreases during the expansion. The temperature may
decrease or increase during process. Therefore temperature
a2
Δθ = 1 − l2 cos2 θ ,
(3)
(4)
where
Δ =
(r 2 + a2)(l2 + r 2)
l2
ρ2 = r 2 + a2 cos2 θ ,
− 2mr,
a2
Ξ = 1 − l2 ,
and l represents AdS curvature radius. The metric parameters
m and a are related to the black hole mass M and the angular
momentum J by
m m
M = Ξ 2 , J = a Ξ 2 . (5)
2 See [52] and the references therein.
3 Indeed, Kerr–Newman–AdS black hole thermodynamics functions
are introduced in [
14
]. But one can easily obtain Kerr–AdS black holes
thermodynamic functions, when electric charge Q goes to zero.
.
T =
determines sign of μ. If μ is positive (negative), cooling
(heating) occurs. The inversion curve, which is obtained at
μ = 0 for infinitesimal pressure drops, characterizes the
expansion process and it determines the cooling–heating
regions in the T – P plane.4
We begin to derive Joule–Thomson expansion coefficient
formula for Kerr–AdS black holes. First, we differentiate
Eq. (8) to obtain
d M = 2(T d S + Sd T − V d P − Pd V + Ωd J
Since d M = d J = 0, Eqs. (7) and (13) can be written
T d S = −V d P,
T d S + Sd T − V d P − Pd V + J dΩ = 0,
respectively. If Eq. (14) can be substituted into Eq. (15), one
can obtain
Here we obtain the Joule–Thomson expansion formula in
terms of the Kerr–AdS black hole parameters. At inversion
pressure Pi , μ equals zero and therefore we obtain Pi from
Eq. (17),
Pi =
∂ P
∂ V
M
J
∂Ω
∂ P
M
− 2V .
From Eq. (6), we can obtain the pressure as a function of
mass, entropy and angular momentum,
3
P = 8
2√π √π 3 J 4 + M 2 S3 − 2π 2 J 2
S3
1
− S .
If we combine Eqs. (10), (11) and (19) with Eq. (18), we
obtain a relation between inversion pressure and entropy as
follows:
256 Pi3 S7 + 256 Pi2 S6 + 84 Pi S5 + (9 − 384π 2 J 2 Pi2)S4
−336π 2 J 2 Pi S3 − 72π 2 J 2 S2 − 72π 4 J 4 = 0.
(20)
The last equation is useful to determine inversion curves,
but first we will investigate minimum inversion temperature.
4 There are two approaches for the Joule–Thomson expansion process.
The differential and integral versions correspond to infinitesimal and
finite pressure drops, respectively. In this paper, we considered
differential version of Joule–Thomson expansion for Kerr–AdS black holes.
See [
57
].
− 2V +S
μ =
Eq. (20) can be given for Pi = 0
and we find four roots for this equation. However, one root
is physically meaningful. This root is given by
S =
2(2 +
One can substitute Eq. (22) into Eq. (9) and obtain the
minimum inversion temperature,
Timin = 4(916 + 37√43√6) 14 π √ J . (23)
For Kerr–AdS black holes, the critical temperature Tc is given
by [
16
]
64k12k24 + 32k1k23 + 3k22 − 12
4π k2 k2(8k1k2 + 3)(8k1k23 + 3k22 + 12)
1
√ J
− 22/3 225679003807 − 24183767608√87
3
√ √
103 − 3 87 − 17 103 − 3 87
2/3
× 484826973√87 − 5116133497
−
√32 68098470527 + 5855463275√87
(21)
(22)
(24)
(25)
(13)
(14)
(15)
(17)
(18)
(19)
Tc =
where
k1 =
×
×
Eq. (9), we obtain constant mass curves in the T – P plane.
As it can be seen from Fig. 2, the inversion curves divide
the plane into two regions. The region above the inversion
curves corresponds to cooling region, while the region under
the inversion curves corresponds to heating region. Indeed,
heating and cooling regions are already determined from the
sign of isenthalpic curves slope. The sign of the slope is
positive in the cooling region and it changes in the heating
region. On the other hand, cooling (heating) does not happen
on the inversion curve which plays the role of a boundary
between the two regions.
4 Conclusions
In this study, we investigated Joule–Thomson expansion for
Kerr–AdS black holes in the extended phase space. The Kerr–
AdS black hole Joule–Thomson formula was derived by
using the first law of black hole thermodynamics and the
Smarr relation. We plotted isenthalpic and inversion curves
in the T – P plane. In order to plot the inversion curves,
we solved Eq. (20) numerically. Moreover, we obtained the
minimum inversion temperature Ti and calculated the ratio
between inversion and critical temperatures for Kerr–AdS
black holes.
(Top-right) J = 2 and M = 2.5, 3, 3.5, 4. (Bottom-left) J = 10
and M = 10.5, 11, 11.5, 12. (Bottom-right) J = 20 and M =
20.5, 21, 21.5, 22
Cooling
Region
Heating
Region
Cooling
Region
Heating
Region
Similar results were reported for the charged AdS black
holes in [
55
] by us. For example, there is only a lower
inversion curve for Kerr–AdS and the charged AdS black holes.
Therefore, we only consider a minimum inversion
temperature T min at Pi = 0. Cooling regions are not closed for both
i
systems. The ratios between minimum inversion
temperatures and critical temperatures are nearly the same for the
two black hole solutions. The ratio may deviate from 0.5 for
other black hole solutions. The same ratio may be obtained
for other black hole solutions in the different limit cases.
Furthermore, we restricted the study to a four-dimensional
solution. Therefore the ratio may depend on the dimensions
of space-time.
In order to compare the charged AdS/Kerr–AdS black
holes with van der Waals fluids, we present schematic
inversion curves for van der Waals fluids and the charged
AdS/KerAdS black holes in Fig. 3. In contrast to the charged AdS and
Kerr–AdS black holes, there are upper and lower inversion
curves for van der Waals fluids [
55
]. Therefore the
cooling region is closed and we only consider both the minimum
inversion temperature Timin and the maximum inversion
temperature T max for this system. While cooling always occurs
i
above the inversion curves for both black hole solutions,
cooling only occurs in the region surrounded by the upper and
lower inversions curves for van der Waals fluids.
Acknowledgements We would like to thank the anonymous referees
for their helpful and constructive comments.
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Funded by SCOAP3.
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