Joule-Thomson Expansion of RN-AdS Black Hole Immersed in Perfect Fluid Dark Matter
JJoule-Thomson Expansion of RN-AdS Black Hole Immersed inPerfect Fluid Dark Matter
Yihe Cao, ∗ Hanwen Feng, † Wei Hong, ‡ and Jun Tao § Center for Theoretical Physics, College of Physics,Sichuan University, Chengdu, 610065, China
Abstract
In this paper, we study the Joule-Thomson expansion for RN-AdS black holes immersed inperfect fluid dark matter. Firstly, the negative cosmological constant could be interpreted asthermodynamic pressure and its conjugate quantity with the volume gave us more physical insightsinto the black hole. We derive the thermodynamic definitions and study the critical behaviour ofthis black hole. Secondly, the explicit expression of Joule-Thomson coefficient is obtained fromthe basic formulas of enthalpy and temperature. Then, we obtain the isenthalpic curve in T − P graph and demonstrate the cooling-heating region by the inversion curve. At last, we derive theratio of minimum inversion temperature to critical temperature and the inversion curves in termsof charge Q and parameter λ . ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: thphysics˙[email protected] § Electronic address: [email protected] a r X i v : . [ g r- q c ] J a n . INTRODUCTION Since Bekenstein and Hawking’s first study, black holes as thermodynamic systems havebeen an interesting research field in the theory of gravity [1–5]. The properties of AdS blackhole thermodynamics have been studied since the early work of Hawking-Page phase transi-tion [5]. Another notable thing is that the dark energy which corresponds to the cosmologicalconstant is introduced to the first law of black hole thermodynamics. Particularly, the neg-ative cosmological constant could be interpreted as thermodynamic pressure and treated asa thermodynamic variable [6, 7]. And its conjugate quantity could be regarded as volumeof black holes once the thermodynamic laws of black hole mechanics were generalized [8].Based on this idea, the charged AdS black hole thermodynamic properties and plenty ofother studies were successfully investigated [9–24].In the extended phase space (including P and V terms in the black hole thermodynamics),phase transition of charged AdS black holes is remarkably coincident with van der Waalsliquid-gas phase transition [18, 25, 26]. An interesting aspect of van der Waals system isJoule-Thomson expansion which indicates that the heating and cooling zone emerge throughthe throttling process. Since the phase structure and the critical behaviour of AdS blackholes are similar to van der Waals system, the Joule-Thomson expansion of AdS black holeswas firstly investigated by ¨Okc¨u and Aydıner [27]. In the extended thermodynamics, oneidentifies the enthalpy H with the mass of the black hole [8]. Joule-Thomson expansion ischaracterized by the invariance of enthalpy, so the throttling process is also an isenthalpicprocess. The slope of the isenthalpy curves equals the Joule-Thomson coefficient µ whichdetermines the final change of temperature in this system. One can use the sign of theJoule-Thomson coefficient to determine the heating and cooling zone. Subsequently, therewere many studies on Joule-Thomson expansion in various black holes [28–49].The standard model of cosmology suggests that our universe is compiled of dark matter,dark energy and baryonic matter, and the existence of dark matter and dark energy hasbeen proven by several experiments and observations [50]. There exists a greater velocitythan expected which means that more mass is required and it is thought to be provided bydark matter [51, 52]. As one of the dark matter candidates, the perfect fluid dark matter(PFDM) has been considered [53, 54]. While astrophysical observations show that thereexists a supermassive black hole surrounded by dark matter halo [55, 56] and many black hole2olutions within dark matter have been proposed. Particularly, spherically symmetric blackhole solutions surrounded by PFDM have been obtained [54, 57–59]. Plenty of propertiesabout this kind of black holes were discussed in [60–71]. Here we intend to study the Joule-Thomson expansion of RN-AdS black hole immersed in PFDM.This paper is organized as follows. In Section (II), we investigate the thermodynamicproperties of RN-AdS black holes immersed in PFDM. In Section (III), we discuss theJoule-Thomson expansion of this kind of black holes, which includes the Joule-Thomsoncoefficient, the inversion curves and the isenthalpic curves. Furthermore, we compare thecritical temperature and the minimum of inversion temperature, and the influence of thePFDM parameter λ and charge Q on inversion curves is discussed. Finally, we discuss ourresult in Section (IV). II. RN-ADS BLACK HOLE IMMERSED IN PFDM
The minimum coupling of dark matter field with gravity, electromagnetic field and cos-mological constant is described by the action [54, 59, 72], S = (cid:90) √− g (cid:18) L DM + F µν F µν − Λ8 πG + R πG (cid:19) d x, (1)where G is gravitational constant, Λ is cosmological constant, F µν is electromagnetic tensorand L DM is the Lagrangian which is related to the density of PFDM.The spacetime metric of static and spherically symmetric RN-AdS black holes immersedin PFDM is defined as [60, 72] ds = − f ( r ) dt + f ( r ) − dr + r d Ω , (2)where d Ω = dθ + sin θdφ and f ( r ) is given by f ( r ) = 1 − Mr + Q r − Λ3 r + λr ln (cid:18) r | λ | (cid:19) , (3)with Q being charge of the black hole and λ related to the dark matter density and pressure.This metric reduces to the RN-AdS black hole when λ = 0. For the given spacetime metricin the case of λ (cid:54) = 0 the stress energy-momentum tensor of the dark matter distribution isthat of an anisotropic perfect fluid reads as T µν = diag ( − ρ, p r , p θ , p φ ) , (4)3here density, radial and tangential pressures are given by [67] ρ = − p r = λ πr , p θ = p φ = λ πr . (5)Note that for a PFDM distribution we shall restrict ourselves to the case λ > l according toΛ = − ( d − d − l . (6)And the negative cosmological constant is related to thermodynamic pressure P , P = − Λ8 π . (7)The entropy of the black hole at the horizon is given by S = πr , (8)The event horizon r + is the solution of f ( r + ) = 0, which indicates the black hole mass M can be expressed in event horizon r + as M = r + πP r + Q r + + 12 λ ln (cid:16) r + λ (cid:17) . (9)The first law of thermodynamics can be expressed as [72] dM = T dS + Φ dQ + V dP + Adλ. (10)where the thermodynamic volume V of black hole, the electrostatic potential Φ and theconjugate quantity A are given by V = 43 πr , Φ = Qr + , A = 12 ln (cid:16) r + λ (cid:17) . (11)The Hawking temperature T is obtained as T = (cid:18) ∂M∂S (cid:19) Q,P,λ = λ πr + 2 P r + + 14 πr + − Q πr . (12)The Hawking temperature T versus event horizon r + and entropy S are shown in FIG.(1)for different values of Q with λ = 1 and P = 0 . T − S graph is related to4pecific heat capacity, it’s positive or negative values determine the stability of the systemwith respect to fluctuations. Hence the FIG. (1) shows that there exists a critical pointwhich indicates the phase transition [42]. In this case, we think that the asymptotic AdSblack hole in a charge fixed canonical ensemble shows a first-order phase transition similarto that of a van der Waals fluids. Q = = = = r + Q = = = = FIG. 1: The temperature T of RN-AdS black hole immersed in PFDM versus r + and S fordifferent values of Q , where λ = 1 and P = 0 . P ( r + , T ) = T r + − πr + Q πr − λ πr . (13)At the critical point, we have ∂P∂r + (cid:12)(cid:12)(cid:12)(cid:12) r = r c = ∂ P∂r (cid:12)(cid:12)(cid:12)(cid:12) r = r c = 0 , (14) ∂T∂r + (cid:12)(cid:12)(cid:12)(cid:12) r = r c = ∂ T∂r (cid:12)(cid:12)(cid:12)(cid:12) r = r c = 0 , (15)so the critical point r c is obtained as r c = 12 (cid:112) λ + 24 Q − λ . (16)By using the Eqs. (12), (13) and (16), we can determine the critical temperature T c andcritical pressure P c , T c = 16 Q − λ (cid:16)(cid:112) λ + 24 Q − λ (cid:17) π (cid:16)(cid:112) λ + 24 Q − λ (cid:17) , (17) P c = λ (cid:16) λ − (cid:112) λ + 24 Q (cid:17) + 6 Q π (cid:16)(cid:112) λ + 24 Q − λ (cid:17) . (18)5t is clear that the critical variables depend on the charge Q and parameters λ . Herewe focus on the critical temperature because it will be useful to analyze the Joule-Thomsonexpansion of the black hole in the next section. One can plot the critical point r c and thecritical temperature T c with varying Q and λ in FIG. (2). The critical temperature decreaseswhile charge Q increases. If the temperature is less than the critical temperature T c , theblack hole can undergo a first-order phase transition between the small black hole and thelarge black hole, which is similar to the van der Waals fluids. Furthermore, one can go backto the RN-AdS case as in [27] by taking the limit λ →
0. Thus, the influence of PFDM onthe critical behaviours of black hole is obvious. λ = λ = λ = λ = λ = r c λ = λ = λ = λ = λ = T c FIG. 2: Critical radius r c and critical temperature T c versus Q of RN-AdS black hole im-mersed in PFDM.We derive the thermodynamic definitions of RN-AdS black holes immersed in PFDM,and we study the critical behaviour of this black hole. In the next section, we investigatethe Joule-Thomson expansion of RN-AdS black holes immersed in PFDM. III. JOULE-THOMSON EXPANSION
The instability of the black holes appears in the process of Joule-Thomson expansion.And we introduce Joule-Thomson coefficient µ to determines the cooling and heating phasesof the isenthalpic expansion.The specific heat at constant pressure for the black holes is obtained according to thefirst law of thermodynamics, C p = T (cid:18) ∂S∂T (cid:19) P,Q,λ . (19)Regarding parameters Q, P, λ as constant, and substituting the Eqs. (11) and (12) into (19),6e have the isobaric heat capacity of this black hole, C p = 2 πr (cid:2) r + (cid:0) λ + 8 πP r + r + (cid:1) − Q (cid:3) r + ( − λ + 8 πP r − r + ) + 3 Q . (20)For the Joule-Thomson expansion of the van der Waals fluids, the fluids passes through aporous plug from one side to the other with pressure declining during throttling process.Therefore, we apply this corresponding concept to black hole thermodynamics and the en-thalpy M of the black hole keeps constant throughout this process. Besides, the partialdifferential of temperature versus pressure of the black hole is defined as Joule-Thomsoncoefficient µ , µ = (cid:18) ∂T∂P (cid:19) M = 1 C P (cid:20) T (cid:18) ∂V∂T (cid:19) P − V (cid:21) . (21)By substituting Eqs (12), (20) and (11) into (21), we obtain µ = 2 r + (cid:2) r + (cid:0) λ + 16 πP r + 4 r + (cid:1) − Q (cid:3) r + ( λ + 8 πP r + r + ) − Q ] . (22)The Joule-Thomson coefficient µ versus the horizon r + is shown in FIG. (3) where theparameter λ = 0 , . , P = 0 .
075 and charges Q = 0 .
7. There exist botha divergence point and a zero point at each different value of λ and the influence of PFDMon JT coefficient is apparent. The divergence point here reveals the information of Hawkingtemperature and corresponds to the extremal black hole, and it is clear that the divergencepoint of the blue curve in FIG. (3) is consistent with the zero point of Hawking temperaturein FIG. (1) in the case of Q = 0 . λ = λ = λ = λ = r + - μ FIG. 3: Joule-Thomson coefficient µ of RN-AdS black hole immersed in PDFM for Q = 0 . P = 0 .
075 and λ = 0 , . , µ >
0, the temperature of the black hole decreases in other words theblack hole is cooling down when pressure goes down in the process of throttling. On thecontrary, µ < µ = 0, and thetemperature of the black hole at µ = 0 is called the inversion temperature.One can obtain the inversion temperature T i = V ∂T∂V by µ = 0. While using the Eqs.(11) and (12), inversion temperature T i in terms of inversion pressure P i is expressed as T i = V (cid:18) ∂T∂V (cid:19) P = 2 P i r + Q πr − λ πr − πr + . (23)Bringing this equation into Eq.(13) at P = P i , we derive P i = 3 Q πr − λ πr − πr , (24) T i = Q πr − λ πr − πr − πr + . (25)From above results, the inversion curves T i − P i in FIG. (4) with different Q and λ is plotted,which is similar to the case of charged RN AdS black hole [27]. At low pressure, the inversiontemperature T i decreases with the increase of charge Q and increases with the increase of λ . This phenomenon is just the opposite of high pressure situation. In addition, there aresome numerical effects of PFDM on the inversion temperature, but no giant differences fromcharged AdS black hole. It can be observed that the inversion temperature still increasesmonotonically with the growth of inversion pressure, and the inversion curves are not closedwhich is different from the situation of the van der Waals fluids. Q = = = = (a) λ = 1, and Q = 0 . , . , . , . λ = λ = λ = (b) Q = 0 . λ = 0 . , , FIG. 4: Inversion curves of RN-AdS black hole immersed in PFDM.8he Joule-Thomson expansion occurs in the isenthalpic process. And for a black hole,enthalpy is mass M . The isenthalpic curves can be obtained from Eqs. (9) and (13) in FIG.(5) where inversion curves and isenthalpic curves are both presented. (a) λ = 1, Q = 0 . M = 2 , . , . , . , (b) λ = 1, Q = 0 . M = 2 , . , . , . , (c) λ = 1, Q = 0 .
7, and M = 2 , . , . , . , (d) λ = 0 . , Q = 0 . M = 2 , . , . , . , FIG. 5: The isenthalpic curves of RN-AdS black hole immersed in PFDM. From bottom totop, the isenthalpic curves correspond to increasing values of M .The intersections of the black inversion curves and the blue isenthalpic curves indicatingthe cooling-heating transition coincide with the extreme points of isenthalpic curves. And theisenthalpic curves have positive slope above the inversion curves indicating that the coolingoccurs. And the isenthalpic curves have negative slope under the inversion curves indicatingthat the heating occurs. We can conclude that the region above this inversion curve iscooling zone while the one below is heating zone. The temperature rises when the pressuregoes down in the heating zone. In contrast, the temperature is lower with reducing pressurein the cooling zone. Moreover, comparing the four graphs the temperature and pressuredecrease with larger charge Q or smaller λ , and PFDM statistically have an influence on theisenthalpic curves. 9urthermore, one can obtain the minimum inversion temperature T min i corresponding to P i = 0 as T min i = 4 (cid:104) λ (cid:16) λ − (cid:112) λ + 96 Q (cid:17) + 16 Q (cid:105) π (cid:16)(cid:112) λ + 96 Q − λ (cid:17) . (26)The ratio of inversion temperature to critical temperature T min i /T c is obtained according toEqs. (26) and (17), T min i T c = 4 (cid:16)(cid:112) λ + 24 Q − λ (cid:17) (cid:104) λ (cid:16) λ − (cid:112) λ + 96 Q (cid:17) + 16 Q (cid:105)(cid:16)(cid:112) λ + 96 Q − λ (cid:17) (cid:104) Q − λ (cid:16)(cid:112) λ + 24 Q − λ (cid:17)(cid:105) . (27)It has been shown in [27] that the ratio for the RN-AdS black holes equals 1 /
2, but in thecase of PFDM this value is corrected. In fact, the ratio grows with increasing
Q/λ , and theratio versus
Q/λ is shown in FIG. (6). When
Q/λ →
0, in other words Q → λ → ∞ ,the ratio has a leading term which is 25/54, T min i T c = 2554 + 4 Q λ + O (cid:2) ( Q/λ ) (cid:3) , (28)and it is close to the situation of Schwarzschild black hole within PFDM.For Q/λ → ∞ , T min i T c = 12 − λ Q + O (cid:2) ( λ/Q ) (cid:3) . (29)Furthermore, by taking the limit λ →
0, the ratio is approaching 1 / / λ T i min / T c FIG. 6: The ratio T min i /T c of RN-AdS black hole immersed in PFDM.10 V. CONCLUSION
In this paper, we discuss the Joule-Thomson expansion for RN-AdS black holes immersedin PFDM. In the extended phase space, the cosmological constant is identified with thepressure and its conjugate quantity as the thermodynamic volume. Since the black hole massis interpreted as enthalpy, an isenthalpic process can be applied to calculate the temperaturechange during the Joule-Thomson expansion. By computing the Joule-Thomson coefficient µ , the divergent point coincides with the extremal black hole while the zero point is theinversion point that determines the transition from heating/cooling phases. And we calculatethe inversion curves in the T − P plane as well as the corresponding isenthalpic curves. .And we can use inversion curve distinguish the cooling and heating regions for differentvalues of λ and Q . Our calculations show that the effect of PFDM density parameter λ onJoule-Thomson coefficient and the inversion temperature is obvious. In addition, one caneasily go back to the RN-AdS case by taking the limit λ →
0. We also discuss the isenthalpiccurves with different values of M during the throttling process.Furthermore, we derive the ratio of minimum inversion temperature to critical tempera-ture and the inversion curves in terms of charge Q and parameter λ . It has been shown in[27] that the ratio for the charged RN-AdS black holes is equal to 1 /
2, but in the case ofPFDM the ratio depends on the charge Q and parameter λ . In fact, the ratio of T min i /T c depends on Q/λ , which is shown in FIG. (6). The ratio is about 25/54 for small
Q/λ , andit is corresponding to the situation of RN-AdS black holes for λ = 0. Acknowledgments
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