aainstitutetext: Department of Physics, Faculty of Science, Istanbul
University
Vezneciler 34134 Istanbul, Turkey

# Joule-Thomson Expansion of Kerr$-$AdS Black Holes

Özgür Ökcü111Corresponding author. a and Ekrem Aydıner
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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 Bek1972 ; Bek1973 ; Bard1973
; Hawk1974 ; Bek1974 ; Hawk1975 , 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 Hawk1983 .
Furthermore, the charged AdS black holes thermodynamic properties were studied
in Cham1999a ; Cham1999b  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 Juan1999 .
Considering the cosmological constant as thermodynamic pressure,

$P=-\frac{\Lambda}{8\pi}\,,$ (1)

and its conjugate quantity as thermodynamic volume

$V=\left(\frac{\partial M}{\partial P}\right)_{S,Q,J}$ (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 Kub2012 . 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 Kast2009  that 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
Dolan2011 ; Dolan2012 ; Mo2013 ; Wei2016 ; Guna2012 ; Spal2013 ; Bel2013 ;
Cai2013 ; Zha2013 ; Ma2014 ; Belhaj2015 ; Dutta2013 ; Li2014 ; Liang2016 ;
Hendi2013 ; Hendi2016 ; Sadeghi2014 ; Mome2017 ; Alta2014b ; Frass2014 ;
Henn2015 ; Wei2015 ; Dolan2011b ; Dolan2014 ; Maj2017 ; John2014 ; John2016 ;
John2016b ; Bel2015b ; Caceres2015 ; Jaf2017 ; Setare2015 ; John2016c ;
Bham2017 ; Liu2017 ; Mo2017 ; Zhang2016 ; Dolan2015 ; Alta2014 ; Kub2017 ;
Lan2017 ; Wei2017 .222See Dolan2015 ; Alta2014 ; Kub2017  and references
therein for various black hole solutions. The phase transition of AdS black
holes in the extended phase space is not restricted to van der Waals type
transition, but also reentrant phase transition and triple point for AdS black
holes were studied in Alta2014b ; Frass2014 ; Henn2015 ; Wei2015 . The
compressibility of rotating AdS black holes in four and higher dimensions was
studied in Dolan2011b ; Dolan2014 . In Maj2017 , a general method was used for
computing the critical exponents for AdS black holes which have van der Waals
like phase transition. Furthermore, heat engines behaviours of the AdS black
holes have been studied. For example, in John2014  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 John2016 ; John2016b ; Bel2015b ;
Caceres2015 ; Jaf2017 ; Setare2015 ; John2016c ; Bham2017 ; Liu2017 ; Mo2017 ;
Zhang2016 .333See Kub2017  and references therein. More recently, adiabatic
processes Lan2017  and Rankine cycle Wei2017  have been studied for the
charged AdS black holes.

In Okcu2017 , 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
Dolan2012 444Indeed, Kerr-Newman-AdS black hole thermodynamics functions are
introduced in Dolan2012 . But one can easily obtain Kerr$-$AdS black holes
thermodynamic functions, when electric charge $Q$ goes to zero.. 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 equation of
inversion pressure $P_{i}$ 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 Okcu2017 . Finally, we discuss our results in Sect. 4. (Here
we use the units $G_{N}=\hbar=k_{B}=c=1$.)

##  2 Kerr$-$AdS Black Holes

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

$\displaystyle
ds^{2}=-\frac{\Delta}{\rho^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right)^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left(adt-\frac{r^{2}+a^{2}}{\Xi}d\phi\right)^{2},$
(3)

where

$\displaystyle\Delta$
$\displaystyle=\frac{(r^{2}+a^{2})(l^{2}+r^{2})}{l^{2}}-2mr\,,\qquad\Delta_{\theta}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta\,,$
(4) $\displaystyle\rho^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,,\qquad\Xi=1-\frac{a^{2}}{l^{2}}\,,$

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

$M=\frac{m}{\Xi^{2}}\,,\qquad J=a\frac{m}{\Xi^{2}}\,.$ (5)

The mass of Kerr$-$AdS black hole in terms of $S$, $J$ and $P$ Cald2000 ;
Dolan2012  is given by

$M=\frac{1}{2}\sqrt{\frac{\left(S+\frac{8PS^{2}}{3}\right)^{2}+4\pi^{2}\left(1+\frac{8PS}{3}\right)J^{2}}{\pi
S}}\,.$ (6)

The first law and the corresponding Smarr relation of Kerr-AdS black hole are
given by

$dM=TdS+VdP+\Omega dJ\,,$ (7) $\frac{M}{2}=TS-VP+\Omega J\,,$ (8)

respectively, and the Smarr relation can be derived by a scaling argument
Kast2009 . From Eq.(7), one can obtain the thermodynamic quantities. The
expression for the temperature is

$T=\left(\frac{\partial M}{\partial S}\right)_{J,P}=\frac{1}{8\pi
M}\left[\left(1+\frac{8PS}{3}\right)\left(1+8PS\right)-4\pi^{2}\left(\frac{J}{S}\right)^{2}\right]\,.$
(9)

Thermodynamic volume is defined by

$V=\left(\frac{\partial M}{\partial P}\right)_{S,J}=\frac{2}{3\pi
M}\left[S\left(S+\frac{8PS^{2}}{3}\right)+2\pi^{2}J^{2}\right]\,.$ (10)

Finally, we obtain angular velocity as follows:

$\Omega=\left(\frac{\partial M}{\partial J}\right)_{S,P}=\frac{\pi
J}{MS}\left(1+\frac{8PS}{3}\right)\,.$ (11)

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 Kast2009 , black hole mass is identified enthalpy in AdS space.
Therefore, the black hole mass remains constant during expansion process.
Joule$-$Thomson coefficient $\mu$, which characterizes the expansion, is given
by Maytal1997

$\mu=\left(\frac{\partial T}{\partial P}\right)_{J,M}\,.$ (12)

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
determines sign of $\mu$. If $\mu$ is positive (negative), cooling (heating)
occurs. The inversion curve, which is obtained at $\mu=0$ for infinitesimal
pressure drops, characterizes the expansion process and it determines the
cooling$-$heating regions in $T-P$ plane 555There 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 Maytal1997 . .

We begin to derive Joule$-$Thomson expansion coefficient formula for
Kerr$-$Ads black holes. First, we differentiates Eq.(8) to obtain

$dM=2(TdS+SdT-VdP-PdV+\Omega dJ+Jd\Omega)\,.$ (13)

Since $dM$=$dJ$=0, Eqs.(7) and (13) can be written

$TdS=-VdP\,,$ (14) $TdS+SdT-VdP-PdV+Jd\Omega=0\,,$ (15)

respectively. If Eq.(14) can be substituted into Eq.(15), one can obtain

$-2V+S\left(\frac{\partial T}{\partial P}\right)_{M}-P\left(\frac{\partial
V}{\partial P}\right)_{M}+J\left(\frac{\partial\Omega}{\partial
P}\right)_{M}=0\,,$ (16)

which can be rearranged to give Joule$-$Thomson formula as follows:

$\mu=\left(\frac{\partial T}{\partial
P}\right)_{M}=\frac{1}{S}\left[P\left(\frac{\partial V}{\partial
P}\right)_{M}-J\left(\frac{\partial\Omega}{\partial
P}\right)_{M}+2V\right]\,.$ (17)

Here we obtain Joule$-$Thomson expansion formula in terms of Kerr$-$AdS black
hole parameters. At inversion pressure $P_{i}$, $\mu$ equals zero and
therefore we obtain $P_{i}$ from Eq.(17),

$P_{i}=\left(\frac{\partial P}{\partial
V}\right)_{M}\left[J\left(\frac{\partial\Omega}{\partial
P}\right)_{M}-2V\right]\,.$ (18)

From Eq.(6), we can obtain pressure as function of mass, entropy and angular
momentum,

$P=\frac{3}{8}\left[\frac{2\sqrt{\pi}\sqrt{\pi^{3}J^{4}+M^{2}S^{3}}-2\pi^{2}J^{2}}{S^{3}}-\frac{1}{S}\right]\,.$
(19)

If we combine Eqs.(10), (11) and (19) with Eq.(18), we obtain a relation
between inversion pressure and entropy as follows:

$\displaystyle
256P_{i}^{3}S^{7}+256P_{i}^{2}S^{6}+84P_{i}S^{5}+(9-384\pi^{2}J^{2}P_{i}^{2})S^{4}$
$\displaystyle-336\pi^{2}J^{2}P_{i}S^{3}-72\pi^{2}J^{2}S^{2}-72\pi^{4}J^{4}=0\,.$
(20)

The last equation is useful to determine inversion curves, but first we will
investigate minimum inversion temperature. Eq.(3) can be given for $P_{i}=0$

$S^{4}-8\pi^{2}J^{2}S^{2}-8\pi^{4}J^{4}=0\,,$ (21)

and we find four roots for this equation. However, one root is physically
meaningful. This root is given by

$S=\sqrt{2(2+\sqrt{6})}\pi J\,.$ (22)

One can substitute Eq.(22) into Eq.(9) and obtain minimum inversion
temperature,

$T_{i}^{min}=\frac{\sqrt{3}}{4(916+374\sqrt{6})^{\frac{1}{4}}\pi\sqrt{J}}\,.$
(23)

For Kerr$-$AdS black holes, critical temperature $T_{c}$ is given by Wei2016

$\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{64k_{1}^{2}k_{2}^{4}+32k_{1}k_{2}^{3}+3k_{2}^{2}-12}{4\pi
k_{2}\sqrt{k_{2}(8k_{1}k_{2}+3)(8k_{1}k_{2}^{3}+3k_{2}^{2}+12)}}\frac{1}{\sqrt{J}}$
(24) $\displaystyle\simeq$ $\displaystyle\frac{0.041749}{\sqrt{J}}\,,$

where

$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{64\left(103-3\sqrt{87}\right)^{17/3}}$
$\displaystyle\times$
$\displaystyle\bigg{(}-2^{2/3}\left(225679003807-24183767608\sqrt{87}\right)$
$\displaystyle\sqrt[3]{103-3\sqrt{87}}-17\left(103-3\sqrt{87}\right)^{2/3}$
$\displaystyle\left(484826973\sqrt{87}-5116133497\right)$
$\displaystyle-\sqrt[3]{2}\left(68098470527+5855463275\sqrt{87}\right)\bigg{)},$
$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\left(2+\sqrt[3]{206-6\sqrt{87}}+\sqrt[3]{206+6\sqrt{87}}\right).$

The ratio between minimum inversion and critical temperature is given by

$\frac{T_{i}^{min}}{T_{c}}\approx 0.504622\,,$ (25)

Figure 1: Inversion curves of Kerr$-$Ads black hole in $T-P$ plane. From
bottom to top,the curves correspond to $J=1,2,10,20$.

Figure 2: Inversion and isenthalpic (constant mass) curves of Kerr$-$AdS black
holes. Red and black lines present isenthalpic and inversion curves,
respectively. From bottom to top, isenthalpic curves correspond to increasing
values of $M$. (Top-left) $J=1$ and $M=1.5,2.2.5,3$. (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 $M=20.5,21,21.5,22$.

which is the same as the value of charged AdS black holes Okcu2017 .

Solving Eq.(3) may not be analytically possible. Therefore, we use numerical
solutions to plot inversion curves in $T-P$ plane. In Fig.(1), we plotted
inversion curves for various angular momentum values. In contrast to van der
Waals fluids, it can be seen from Fig.(1) the inversion curves are not closed
and there is only one inversion curve. We found the similar behaviours for the
charged AdS black holes in our previous work Okcu2017 .

In Fig.(2), we plot isenthalpic (constant mass) and inversion curves for
various values of angular momentum in $T-P$ plane. If entropy from Eq.(6) can
be substituted into Eq.(9), we obtain constant mass curves in $T-P$ 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 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 role as a boundary between
two regions.

Figure 3: (Left) Inversion curves for van der Waals fluids. Red thick line and
black solid line correspond to lower and upper inversion curves, respectively.
(Right) Lower inversion curves for the charged AdS and Kerr-AdS black holes.

##  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
Smarr relation. We plotted isenthalpic and inversion curves in $T-P$ plane. In
order to plot inversion curves, we solved Eq.(3) numerically. Moreover, we
obtained minimum inversion temperature $T_{i}$ and calculated ratio between
inversion and critical temperatures for Kerr$-$AdS black holes.

Similar results were reported for the charged AdS black holes in Okcu2017  by
us. For example, there is only a lower inversion curve for Kerr$-$AdS and the
charged AdS black holes. Therefore, we only consider minimum inversion
temperature $T_{i}^{min}$ at $P_{i}=0$. Cooling regions are not closed for
both systems. The ratios between minimum inversion temperatures and critical
temperatures are nearly the same for the both black hole solutions. The ratio
may deviates 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 four dimensional solution. Therefore
the ratio may depend on 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/Ker-AdS 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 Okcu2017 . Therefore the cooling region is closed and we only
consider both the minimum inversion temperature $T_{i}^{min}$ and maximum
inversion temperature $T_{i}^{max}$ for this system. While cooling always
occurs 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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