# Joule–Thomson expansion of Kerr–AdS black holes

## 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 [1, 2, 3, 4, 5, 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.

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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 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, 32, 33, 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 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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 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 \(P_{i}\) 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 [55]. Finally, we discuss our results in Sect. 4. (Here we use the units \(G_{N}=\hbar =k_{B}=c=1\).)

## 2 Kerr–AdS black hole

*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

*S*,

*J*and

*P*[14, 56] is given by

## 3 Joule–Thomson expansion

*T*–

*P*plane.

^{4}

which is the same as the value of charged AdS black holes [55].

Solving Eq. (20) may not be analytically possible. Therefore, we use numerical solutions to plot inversion curves in the *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 that the inversion curves are not closed and there is only one inversion curve. We found similar behaviours for the charged AdS black holes in our previous work [55].

In Fig. 2, we plot isenthalpic (constant mass) and inversion curves for various values of angular momentum in the *T*–*P* plane. If the entropy from Eq. (6) can be substituted into 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 \(T_{i}\) and calculated the ratio between inversion and critical temperatures for Kerr–AdS black holes.

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 [55]. Therefore the cooling region is closed and we only consider both the minimum inversion temperature \(T_{i}^{\mathrm{min}}\) and the maximum inversion temperature \(T_{i}^{\mathrm{max}}\) 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.

## Footnotes

- 1.
- 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. - 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].

## Notes

### Acknowledgements

We would like to thank the anonymous referees for their helpful and constructive comments.

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