Joule-Thomson Expansion of d-Dimensional Charged Anti-de Sitter Black Holes Surrounded by Quintessence with a Cloud of Strings Background

FENG Yuanyuan(馮媛媛), NIE Weifu(聶維福), LAN Xiaogang(蘭小剛)

School of Physics and Astronomy, China West Normal University, Nanchong 637001, China

Abstract: Effects of the dimension on the Joule-Thomson expansion are investigated in details by considering the case of d-dimensional (d≥5) charged anti-de Sitter (AdS) black hole which is surrounded by the quintessence with a cloud of strings background. Firstly, the thermodynamic quantity of this black hole is reviewed. Secondly, three important features of the Joule-Thomson expansion in different dimensions are discussed, including the Joule-Thomson coefficients, inversion curves, and isenthalpic curves. Finally, the effects of the charge, the quintessence and strings cloud parameters on the Joule-Thomson expansion in the case of six-dimensional black hole are studied.

Key words: d-dimensional charged AdS black hole; quintessence; inversion curve; Joule-Thomson expansion; isenthalpic curve

Introduction

The thermodynamics of black holes has long been a fascinating topic. Based on the great contribution of Hawkingetal.[1], the black hole is regarded as a thermodynamic system which has temperature and entropy. Chamblinetal.[2-3]observed the close relationship between charged anti-de Sitter (AdS) black holes and liquid-gas systems, and the rich phase structure of Reissner Nordstr?m (RN)-AdS black hole. This relation was further enhanced by the discovery[4]in the extended phase space. By showing the existence of the reentrant phase transition[5-6]and triple point[7], it was even proposed that black holes have many features and properties similar to the classical thermodynamic systems. In particular, for black holes in AdS spacetime, these similarities become more pronounced.

Considering the cosmological constantΛas the thermodynamic pressureP, its conjugate variable is the thermodynamic volume[8-9].

(1)

(2)

whereMis the black hole mass,Sis the entropy of the black hole,Qis the charge of the black hole, andJis the angular momentum of the black hole. In this context, various interesting studies have been conducted[10-19], such as theP-Vcritical phenomenon of the black hole and the weak cosmic censorship conjecture. Recently, the Joule-Thomson expansion has been cleverly applied to black holes in the AdS spacetime, with the first study of the Joule-Thomson expansion of the charged Ads black hole in Ref. [20]. This research was soon extended to Kerr-AdS black holes[21]and RN-AdS black hole surrounded by quintessence with a cloud of strings[22].

As mentioned above, it is generally believed that there is a close relationship between AdS black holes and ordinary thermodynamic systems while Joule-Thomson expansion is an important process in classical thermodynamics and certainly deserves further investigation. Moreover, based on literature review, we found that thed-dimensional charged AdS black holes surrounded by quintessence with a cloud of strings background (AdS BHqcs) have not been studied in the extended phase space. Meanwhile, the Joule-Thomson expansion describes the temperature change of a real gas or liquid when it is forced through a valve or porous plug. So, to explore the Joule-Thomson expansion of charged AdS BHqcs in high-dimensional cases interests us. On the one hand, this study might be helpful to reveal the effect of the dimension on the Joule-Thomson expansion of the black hole. On the other hand, it might facilitate a further understanding of the thermodynamics ofd-dimensional (d≥5) charged AdS black holes. More importantly, the effects of the quintessence parameterαand the cloud of strings parameteraon the Joule-Thomson expansion in the six-dimensional case is worth studying.

In this paper, the current research of Joule-Thomson expansion in the case ofd-dimensional (d≥5) charged AdS BHqcs is generalized. The organization is as follows. In section 1, the thermodynamic properties ofd-dimensional charged AdS BHqcs are reviewed. In section 2, the Joule-Thomson expansion of the black hole is studied. Meanwhile, the effects of the parameters on six-dimensional charged AdS BHqcs are also investigated. In section 3, the conclusions are given in great details.

1 Thermodynamics of d-Dimensional (d≥5) Charged AdS BHqcs

In this section, the Hawking temperature ofd-dimensional (d≥5) charged AdS BHqcs is discussed though the first law of thermodynamics. Meanwhile, the equation of state for this black hole is given. Firstly, the solution which corresponds tod-dimensional charged AdS BHqcs is derived from Ref. [22] and expressed as

(3)

whereris the black hole radius,ωqis the quintessence state parameter，ais cloud strings parameter,mandqare integral constants proportional to the black hole mass and charge, respectively.

(4)

(5)

whereΩd-2is the volume of the unit (d-2) sphere.

αin Eq. (3) is a standardization factor always positive for the density of the quintessenceρqwhich is given by

(6)

(7)

wherer+is represented as the horizon radius.

Using Bekenstein-Hawking formula[25], the entropy is related to the black hole superficial areaAand the event horizonr+, which is given as

(8)

The Hawking temperature of the black hole can be estimated by

(9)

The black hole temperature calculated by using this method is consistent with the temperature in Ref. [22]. From classical thermodynamics, the black hole massMis interpreted as the analogue of enthalpy rather than total energy of the spacetime[26-28]. In the extended phase space, the first law of thermodynamics and the Smarr formula account for the cosmological constant effect, and the cloud of strings and the quintessence contributions are then expressed as

dM=TdS+VdP+ΦdQ+Ada+?dα，

(10)

(11)

The conjugate variables of the parametersP,Q,αandaare respectively written as

(12)

(13)

(14)

(15)

Meanwhile, through the expression for the temperature in Eq. (9), we can obtain the equation of state for the black hole, andPcan be estimated by

(16)

2 Joule-Thomson Expansion of d-Dimensional (d≥5) Charged AdS BHqcs

In this section, the Joule-Thomson expansion ofd-dimensional (d≥5) charged AdS BHqcs is studied. In classical thermodynamics, Joule-Thomson expansion is described as gas at a high pressure passing through a porous plug to a section with low pressure, during which the enthalpy remains constant. In the extended phase space, the black hole mass corresponds to the enthalpy. Thus, the Joule-Thomson expansion of the black hole in fact is an isenthalpic process. The slope of the isenthalpic curve is the Joule-Thomson coefficient, which can be used to determine whether the expansion process is heated or cooled. The Joule-Thomson coefficientμis defined as a change in temperature relative to the pressure, given by

(17)

It can be determined whether systemic cooling or heating occurs through Eq. (17). During the expansion process, the change of pressure ?Pis always negative. So, whetherμis positive or negative depends on the temperature. If heating (cooling) occurs,μis negative (positive).

Ifμ=0, the inversion temperatureTiis expressed as

(18)

The inversion temperature is used to make sure the cooling and heating areas on the temperature-pressure (T-P) plane. Furthermore, in addition to Eq. (17), there is a new expression describing the Joule-Thomson coefficient[29-30]which is given by

(19)

In accordance with this equation, we can also obtain the inversion temperature and draw out the inversion temperature curves. From Eqs. (7), (9), (14), and (19), we can obtain the Joule-Thomson coefficient:

(20)

wherea= 0.1,α=0.01,p=1,q=1. Figure 1 shows the effect of the dimension on the Joule-Thomson coefficient. There is a divergent point and a zero point in theμ-r+graph corresponding to different dimensions. It can be found that the horizon radiusr+corresponding to the divergent point increases with increasing the dimension.

Fig. 1 Joule-Thomson coefficients in 5 to 9 dimension

The divergent point here reveals the information of Hawking temperature and corresponds to the extremal black hole[23]. And then we present theμ-r+diagram andT-r+diagram in the case of six-dimension in Fig. 2 to further understand this issue. By the black straight line as shown in Fig. 2, we can find that the horizon radius corresponding to the divergent point of Joule-Thomson coefficient is consistent with the horizon radius corresponding to the zero point of black hole temperature.

Fig. 2 Joule-Thomson coefficient and temperature of six-dimensional black hole

Next, we make the Joule-Thomson coefficient equal to zero, and the inversion pressurePand the horizon radiusr+satisfy the following relation

(21)

Taking the real root of Eq. (21) into Eq. (9), combining with Eqs. (4) and (5), an expression for the inversion temperature can be obtained. Finally, the inversion curves of the black hole are drawn in Fig.3.

As seen in Fig.3, the inversion curve is not closed which is different from the van der Waals system. The inversion temperature increases monotonically with the inversion pressure, but the slope of the inversion curve decreases with increasing the electric charge. In different dimensions, it is observed that with increasing charge, the inversion temperature decreases at low pressure, while increases at high pressure.

Fig. 3 Inversion curves for the black holes: (a) d=5; (b) d=6; (c) d=7; (d) d=8; (e) d=9

Meanwhile, in order to more intuitively compare the influence of the dimension on the inversion curve, we also draw the inversion curve corresponding to different dimensions with certain charge and other parameters in Fig. 4. We find that with increasing dimension, the inversion temperature increases at the low pressure, and decreases at the high pressure.

Fig. 4 Effect of the dimension on the inversion curve

To compare more intuitively with the effects of parametersaandα, we show inversion curves of six-dimensional black hole with different parameters in Fig.5. It is obvious that the inversion temperature all decrease with increasingaorα, but it is worth noting thatαaffects more significantly on the inversion curve thana.

Fig. 5 Effects of the parameters on the inversion curve: (a) a=0.1, q=1; (b) α=0.01, q=1

The isenthalpic curves in different dimensions is computed and plotted in Fig. 6. Through the black hole mass expression of Eq. (7), we can obtain the horizon radius of the black hole in different dimensions and substitute the result into the black hole temperature in Eq.(9). By rewriting the black hole mass expression, we draw the isenthalpic curves in different dimensions. The isenthalpic curves in different masses and different dimensions are shown in Fig. 6, where the parametersq,aandαare keeping fixed values.

Fig. 6 Isenthalpic curves and inversion curves in different dimensions with q=1, a=0.1, α=0.01: (a) d=5; (b) d=6; (c) d=7; (d) d=8; (e) d=9

Similarly, the inversion curve (the slash line in black in Fig. 6) divides theT-Pplane into two regions, corresponding to the cooling region above the inversion curve and the heating region below the inversion curve of the black hole. When the pressure in the heating area drops, the temperature increases. Instead, when the pressure in the cooling area drops, the corresponding temperature decreases. In fact, the cooling and heating regions in Fig. 6 are determined by the slope of the isenthalpic curve. The slope is positive for the region of the isenthalpic curve corresponding to the cooling region, by contraries, the region with the negative slope corresponds to the heating region[31-33]. The inversion curve here is the boundary between the cooling region and the heating region. At the intersection of the inversion curve and the isenthalpic curve, the slope of the isenthalpic curve is zero, where no cooling or heating behavior occurs.

In Fig. 7, the isenthalpic graphs with identical parameters and only different dimensions show that the isenthalpic curve tends to expand towards higher pressure as the dimension increases. Meanwhile, the relationships between this special phenomena and each parameter in isenthalpic curves are also studied. We respectively setq,aandαas fixed values, and studied the isenthalpic curves of the black holes with different parameters. The relationship between isenthalpic temperature and pressure of the example of six-dimensional black hole with different parameter is shown in Fig. 8.

Fig. 7 Effects of the dimension on the isenthalpic curves: a=0.1, q=1, α=0.01, m =5

Fig. 8 Effects of different parameters on the isenthalpic curves: (a) a=0.1, α=0.01; (b) a=0.1, q=1; (c) α=0.01, q=1

3 Conclusions

In this paper, we discussed in details the effects of the dimension on the Joule-Thomson expansion by considering the case ofd-dimensional (d≥5) charged AdS BHqcs.

Firstly, we found that the horizon radius corresponding to the divergent point of Joule-Thomson coefficient was the same as the horizon radius corresponding to zero point of black hole temperature. And as the dimension increased, the horizon radius corresponding to the divergent point and the zero point also increased.

Secondly, we found that the inversion curve ofd-dimensional (d≥5) charged AdS BHqcs was different from that of the van der Waals system. The inversion curve was unclosed, and the inversion temperature increased monotonically with increasing pressure. Moreover, with increasing charge, the inversion temperature decreased at low pressure and increased at high pressure. While with increasing dimension, the inversion temperature increased at low pressure and decreased at high pressure. Meanwhile, we chose the six-dimensional case as an example, and studied the inversion curve corresponding to different values of parameteraandα. The inversion curves of different parameter values were not closed, and the influence ofαon them was more significant.

Finally, the isenthalpic curve tilted at high pressure as the black hole dimension increased. Meanwhile, the isenthalpic curve expanded with increasing dimension. We also plotted isenthalpic curves of six-dimensional black holes under different parametersq,a, andα. It was shown that the isenthalpic curve gradually shrank as the charge increased. Conversely, the isenthalpic curve expanded as the parametersa,αincreased and was more sensitive to the parametera.