# Joule–Thomson expansion of the charged AdS black holes

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## Abstract

In this paper, we study Joule–Thomson effects for charged AdS black holes. We obtain inversion temperatures and curves. We investigate similarities and differences between van der Waals fluids and charged AdS black holes for the expansion. We obtain isenthalpic curves for both systems in the *T*–*P* plane and determine the cooling–heating regions.

## 1 Introduction

It is well known that black holes as thermodynamic systems have many interesting consequences. It sets deep and fundamental connections between the laws of classical general relativity, thermodynamics, and quantum mechanics. Since it has a key feature to understand quantum gravity, much attention has been paid to the topic. The properties of black hole thermodynamics have been investigated since the first studies of Bekenstein and Hawking [1, 2, 3, 4, 5, 6]. When Hawking discovered that black holes radiate, black holes are considered as thermodynamic systems.

*P*and

*V*terms in the first law of black hole thermodynamics), charged AdS black holes phase transition is in remarkable coincidence with van der Waals liquid–gas phase transition [16]. This type of transition is not limited with charged AdS black holes, various kind of black holes in AdS space show the same phase transitions [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

It is also possible to consider heat cycle for AdS black holes [33, 34, 35, 36, 37, 38, 39, 40]. In [33, 40], author suggested two kind of heat cycles and obtained exact efficiency formula for black holes.

Variable cosmological constant notion has some nice features such as phase transition, heat cycles and compressibility of black holes [41]. Applicabilities of these thermodynamic phenomena to black holes encourage us to consider Joule–Thomson expansion of charged AdS black holes. In this letter, we study the Joule–Thompson expansion for chraged AdS black holes. We find similarities and differences with van der Waals fluids. In Joule–Thomson expansion, gas at a high pressure passes through a porous plug to a section with a low pressure and during the expansion enthalpy is constant. With the Joule–Thomson expansion, one can consider heating-cooling effect and inversion temperatures.

The paper is arranged as follows. In Sect. 2, we briefly review the charged AdS black hole. In Sect. 3, we firstly review Joule–Thompson expansion for van der Waals gases and then we investigate Joule–Thomson expansion for charged AdS black holes. Finally, we discuss our result in Sect. 4. (Here we use the units \(G_{\mathrm{N}}=\hbar =k_{\mathrm{B}}=c=1.\))

## 2 The charged AdS black holes

*f*(

*r*) is given by

*l*,

*M*, and

*Q*are the AdS radius, mass, and charge of the black hole, respectively. One can obtain the black hole event horizon as the largest root of \(f(r_{+})=0\). The mass of a black hole in Eq. (3) is given by

*P*and

*V*, when the cosmological constant is considered as a thermodynamic variable. The cosmological constant corresponds to the pressure,

## 3 Joule–Thomson expansion

*N*,

*T*–

*P*plane.

### 3.1 van der Waals fluids

*P*,

*T*, and \(k_{\mathrm{B}}\) denote the specific volume, pressure, temperature, and Boltzmann constant. \(a>0\) constant is a measure of the attraction between the particles and \(b>0\) is a measure of the molecule volume.

*a*and

*b*constants are determined from experimental data.

*T*–

*P*plane. In Fig. 2, isenthalpic and inversion curves are presented. When the isenthalpic curves cross inversion curves, their slopes change sign. Isenthalpic curves have positive slopes inside the inversion curves, otherwise their slopes are negative. As a result the Joule–Thomson coefficient is positive inside the inversion curves and cooling occurs inside this region.

### 3.2 Charged AdS black holes

*Q*. There is only a lower inversion curve. In contrast to van der Waals fluids, the expression inside the square root in Eq. (44) is always positive, so this curve does not terminate at any point.

Now, we can plot isenthalpic, i.e. constant mass, curves in the *T*–*P* plane. From Eq. (4), one can obtain the event horizon and substituting event horizon into Eq. (10) gives isenthalpic curves in the *T*–*P* plane. In Fig. 4, inversion curves and isenthalpic curves are presented. Isenthalpic curves have positive slope above the inversion curves so cooling occurs above the inversion curves. The sign of the slope changes under the inversion curves and heating occurs in this region. It is also interesting to talk about naked singularities for charged AdS black holes. In Fig. 5, we plot event horizon versus mass and pressure. We introduce four graphics, which correspond to \(Q=1, 2, 10, 20.\) The regions can be seen that denote the naked singularities in Fig. 5. One cannot consider Joule–Thomson expansion due to the lack of an event horizon for a naked singularity. For example, we cannot define event horizon for \(Q=20\) and \(M\le 20\). For these values, event horizon is imaginary and it corresponds to naked singularity so isenthalpic curves in the *T*–*P* plane are imaginary.

## 4 Conclusion

In this paper, we studied the well-known Joule–Thomson expansion for charged AdS black hole. The black hole mass in AdS space is identified with the enthalpy due to the variable cosmological constant notion, so one can consider that mass does not change during the expansion. First, we reviewed Joule–Thomson expansion for van der Waals fluids and then we investigated Joule–Thomson expansion for charged AdS black holes. We only found one inversion curve that corresponds to the lower curve. It means that black holes always cool above the inversion curve during the Joule–Thomson expansion. Cooling and heating regions were shown for various values of the charge *Q* and mass *M*. We also denoted the naked singularity which is not sensible for Joule–Thomson expansion due to the lack of an event horizon.

Both systems are not well behaved for low temperatures. Unfortunately, isenthalpic curves have positive slopes under the lower inversion curves for both systems. It is also known that van der Waals equation does not too well agree with experiments. Thus Joule–Thomson expansion have been investigated for various equations of state. In charged AdS case, it needs further investigation.

## Notes

### Acknowledgements

We thank Can Onur Keser with improving the figures in this work. This work was sported by Scientifc Research Projects Coordination Unit of Istanbul University. Project Number is FYL-2016-20615.

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