Joule-Thomson Expansion of Charged Ads Black Holes

Joule-Thomson Expansion of Charged AdS Black Holes

Ozgür¨ Okcü¨ ∗ and Ekrem Aydıner† Department of Physics, Faculty of Science, Istanbul˙ University, Istanbul,˙ 34134, Turkey (Dated: January 24, 2017) 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 T − P plane and determine the cooling-heating regions.

I. INTRODUCTION Variable cosmological constant notion has some nice features such as phase transition, heat cycles and com- It is well known that black holes as thermodynamic pressibility of black holes [41]. Applicabilities of these systems have many interesting consequences. It sets deep thermodynamic phenomena to black holes encourage us and fundamental connections between the laws of clas- to consider Joule-Thomson expansion of charged AdS sical general relativity, thermodynamics, and quantum black holes. In this letter, we study the Joule-Thompson mechanics. Since it has a key feature to understand expansion for chraged AdS black holes. We find sim- quantum gravity, much attention has been paid to the ilarities and differences with van der Waals fluids. In topic. The properties of black hole thermodynamics have Joule-Thomson expansion, gas at a high pressure passes been investigated since the first studies of Bekenstein and through a porous plug to a section with a low pressure Hawking [1–6]. When Hawking discovered that black and during the expansion enthalpy is constant. With holes radiate, black holes are considered as thermody- the Joule-Thomson expansion, one can consider heating- namic systems. cooling effect and inversion temperatures. Black hole thermodynamics shares similarities with The paper arranged as follows. In Section II, we general thermodynamics systems. Specifically, black briefly review the charged AdS black hole. In Section holes in AdS space have common properties with general III, we firstly review Joule-Thompson expansion for van systems. The study of AdS black hole thermodynamics der Waals gases and then we investigate Joule Thom- began with pioneering paper of Hawking and Page [7]. son expansion for charged AdS black holes. Finally, we They found a phase transition between Schwarzschild discuss our result in Section IV. (Here we use the units AdS black hole and thermal AdS space. Up to now, GN = ~ = kB = c = 1.) thermodynamic properties of AdS black holes have been widely studied in the literature [8–41]. In [8, 9], authors studied the thermodynamics of charged AdS black holes II. CHARGED ADS BLACK HOLES and they found analogy between phase diagrams of black hole and van der Waals fluids. When cosmological con- In this section, we briefly review charged AdS black stant and its conjugate quantity are, respectively, con- hole and we present its thermodynamic properties. sidered as a thermodynamic pressure Charged black hole in four dimensional space is defined with the metric Λ P = − , (1) 2 2 −1 2 2 2 8π ds = −f(r)dt + f (r)dr + r dΩ , (2) ∂M where dΩ2 = dθ2 + sin2(θ)dφ2 and f(r) is given by and thermodynamic volume V = ( ∂P )S,Q,J , this analogy gains more physical meaning. Particularly, in the ex- 2 2 tended phase space (including P and V terms in the first 2M Q r arXiv:1611.06327v2 [gr-qc] 21 Jan 2017 f(r) = 1 − + + . (3) law of black hole thermodynamics), charged AdS black r r2 l2 holes phase transition is remarkable coincidence with van In these equations, l, M and Q are the AdS radius, mass, der Waals liquid-gas phase transition [16]. This type of and charge of the black hole, respectively. One can obtain transition is not limited with charged AdS black holes, black hole event horizon as largest root of f(r+) = 0. The various kind of black holes in AdS space show the same mass of black hole in Eq. (3) is given by phase transitions [17–27]. 2 2 It is also possible to consider heat cycle for AdS black r+ Q r+ M = 1 + 2 + 2 (4) holes [33–40]. In [33, 40], author suggested two kind 2 r+ l of heat cycles and obtained exact efficiency formula for which satisfies the first law of black hole thermodynamics black holes. dM = T dS + ΦdQ + V dP (5) and corresponding Smarr relation is given by ∗ [email protected] † [email protected] M = 2(TS − PV ) + ΦQ. (6) 2

One can derive Smarr relation by scaling argument [15]. III. JOULE-THOMSON EXPANSION The first law of black hole thermodynamic includes P and V , when the cosmological constant is considered as In this section, we review the well-known Joule- a thermodynamic variable. The cosmological constant Thomson expansion [42, 43]. In Joule-Thomson expan- corresponds to the pressure, sion, gas at a high pressure passes through a porous plug or small valve to a section with a low pressure in a ther- 1 3 1 P = − Λ = (7) mally insulated tube and enthalpy remains constant dur- 8π 8π l2 ing the expansion process. One can describe temperature change with respect to pressure and this change is given and cosmological constant’s conjugate quantity corre- by sponds to thermodynamic volume. The expression for entropy is given by ∂T µ = . (15) A ∂P S = = πr2 ,A = 4πr2 (8) H 4 + + Here µ is called the Joule-Thomson coefficient. It is possi- and the corresponding Hawking temperature ble to determine whether cooling or heating will occur by evaluating the sign of Eq. (15). In Joule-Thomson expan- ∂M l2(r2 − Q2) + 3r4 sion, pressure decreases so change of pressure is negative T = = + + . (9) ∂S 4πl2r3 but change of temperature may be positive or negative. P,Q + If the change of temperature is positive (negative) µ is negative (positive) and so gas warms (cools). On the other hand, the electric potential is given by Φ = It is also possible to express Eq. (15) in terms of volume Q and equation of state P = P (V,T ) for charged AdS r+ and heat capacity at constant pressure. From the first black hole is obtained from Eqs. (7) and (9) as law of thermodynamics, one can write the fundamental relation for constant particle number N 2 T 1 Q 3V 1 3 P = − 2 + 4 , r+ = ( ) . (10) dU = T dS − P dV . (16) 2r+ 8πr+ 8πr+ 4π Using the relation H = U + PV , Eq. (16) is given by The critical points [16] obtained from dH = T dS + V dP . (17) ∂P ∂2P = 0, 2 = 0, (11) Since dH = 0, Eq. (17) is given by ∂r+ ∂r+ ∂S which leads to 0 = T + V. (18) ∂P √ H 6 √ 1 Tc = , rc = 6Q, Pc = . (12) Since entropy is a state function, the differential dS is 18πQ 96πQ2 given by

Other thermodynamic properties can be obtained by us- ∂S ∂S ing above relations. For example, heat capacities at dS = dP + dT (19) ∂P T ∂T P constant pressure and constant volume are, respectively, given by which can be rearranged to give 4 2 2 2 2 ∂S ∂S ∂S ∂T ∂S 2 3r+ − l Q + l r+ = + . (20) CP = T = 2πr , (13) 4 2 2 2 2 ∂P H ∂P T ∂T P ∂P H ∂T P,Q 3r+ + 3l Q − l r+ If one can substitute this expression into Eq. (18), one and can obtain the following expression:

∂S ∂S ∂S ∂T CV = T = 0 . (14) 0 = T + + V. (21) ∂T V,Q ∂P T ∂T P ∂P H

∂S ∂V In this section, we give thermodynamic definitions for Substituting Maxwell relation ∂P T = − ∂T P and ∂S charged AdS black hole. In the next section, we will CP = T ∂T P into Eq. (21) gives review Joule-Thomson expansion for van der Waals fluids and investigate Joule-Thomson expansion for charged ∂V ∂T 0 = −T + CP + V (22) AdS black holes. ∂T P ∂P H 3 and it can be rearranged to give the Joule-Thomson co- where Pi denotes the inversion pressure. From Eq. (25), efficient [42] as follows: one can get ∂T 1 ∂V 1 a µ = = T − V . (23) Ti = Pi + 2 (v − b) . (31) kB v ∂P H CP ∂T P Subtracting Eq. (30) from Eq. (31) yields At the inversion temperature, µ equals zero and inversion 2 temperature is given by bPiv − 2av + 3ab = 0 (32)

∂T and, solving this equation for v(Pi), one can obtain two Ti = V (24) roots ∂V P √ a ± a2 − 3ab2P which is useful to determine the heating and cooling re- v = i . (33) gions in the T − P plane. bPi Substituting these roots into Eq. (31), one can obtain √ 2 5a − 3b2P − 4 a2 − 3ab2P A. van der Waals Fluids T lower = i i , (34) i 9bk The van der Waals equation is a generalized form of √ 2 5a − 3b2P + 4 a2 − 3ab2P ideal gas equation, which usually describes the liquid-gas upper i i Ti = (35) phase transition behaviours for real fluids [43, 44]. It 9bk takes into account the size of molecules and attraction which give lower and upper inversion curves, respectively. between them. It is given by In Fig. (1), lower and upper inversion curves are presented. At the point P = 0, we can obtain the minimum k T a i P = B − . (25) and maximum inversion temperatures v − b v2 min 2a max 2a V Ti = ,Ti = . (36) Here v = N , P , T and kB denote the specific volume, 9bk bk pressure, temperature, and Boltzmann constant. a > 0 The critical temperature for van der Waals fluids is given constant is a measure of attraction between particles and 8a by Tc = and hence b > 0 is a measure of molecule volume. a and b constants 27bk are determined from experimental data. T min 3 T max 27 i = , i = . (37) Before more proceeding to the Joule-Thomson expan- Tc 4 Tc 4 sion, it is useful to give some thermodynamic properties of van der Waals equation. Following [16, 45], free energy is given by

" 3 #! (v − b)T 2 a F (T, v) = −k T 1 + ln − . (26) B Φ v

Here φ is a constant characterizing the gas. Now, entropy can be obtained from Eq. (26)

" 3 #! ∂F 5 (v − b)T 2 S(T, v) = − = kB + ln . ∂T v 2 Φ (27) Using Eqs. (26) and (27), we can calculate the internal energy 3k T a U(T, v) = F + TS = B − (28) 2 v FIG. 1. Lower (dashed blue line) and upper (solid orange line) inversion curves. We ﬁx a = b = k = 1. and from Eqs. (25) and (28), enthalpy is B 3k T k T v 2a Using Eqs. (25) and (29), we can obtain the isenthalpic H(T, v) = U + PV = B + B − . (29) 2 v − b v curves in T − P plane. In Fig. (2), isenthalpic and inversion curves are presented. When the isenthalpic curves Now, let us calculate the inversion temperature for van cross inversion curves, their slopes change sign. Isen- der Waals equation. Using Eq. (24), inversion tempera- thalpic curves have positive slopes inside the inversion ture is given by curves, otherwise their slopes are negative. As a result 1 a the Joule-Thomson coeﬃcient is positive inside the in- Ti = Piv − 2 (v − 2b) (30) kB v version curves and cooling occurs inside this region. 4

FIG. 2. Dashed blue line and orange line are inversion curves. FIG. 3. Inversion curves for charged AdS black hole.From Red lines are isenthalpic curves. The enthalpies of isenthalpic bottom to top, the curves correspond to Q = 1, 2, 10, 20. curves increase from bottom to top and correspond to H = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. We fix a = b = kB = 1. and solving this equation for r+ gives us four roots but only one root is physically meaningful, other roots are B. Charged AdS Black Holes complex or negative. A positive and real root is s p 2 In this section, we will consider Joule-Thomson expan- 1 1 + 24PiπQ 1 sion for charged AdS black holes. In [15], authors sug- r+ = √ − . (43) 2 2 Piπ Piπ gested that black hole mass is considered as the enthal- phy in AdS space. It means that our isenthalpic curves If we substitute this root into Eq. (41), the inversion tem- are actually constant mass curves in AdS space. We can perature is given by consider black hole mass not to change during the Joule- Thomson expansion. For a fixed charge, similar steps √ 2 p 2 1 + 16πPiQ − 1 + 24πPiQ in the previous section can be used to obtain the Joule- Pi Ti = √ 3 . (44) 2 Thomson coefficient. Hence 2π p 2 −1 + 1 + 24πPiQ

∂T 1 ∂V min µ = = T − V . (38) When Pi is zero, we have Ti ∂P M CP ∂T P 1 The charged AdS black hole equation of state can be T min = √ (45) i 6 6πQ given in terms of thermodynamic volume,

2 " 2 2 # and ratio between minimum inversion and critical tem- 3 3 3 2 1 3 1 3 1 4π Q 3 peratures is T = V 8π P + 2 − 4 3 4π 4π V 3 3 V 3 (39) T min 1 i = . (46) and evaluating this in the right hand side of Eq. (38), the Tc 2 inversion temperature is given by

1 1 1 In Fig (3), inversion curves are presented for various val- 1 h Q2 i 1 6 3 3 π 3 6 3 1 Ti = 3 π V 6 4 − π 2 + Pi ues of charge Q. There is only a lower inversion curve. In V 3 12V 3 2 contrast to van der Waals ﬂuids, the expression inside the Q 1 2Pir+ = 3 − + . (40) square root in Eq. (44) is always positive, so this curve 4πr+ 12πr+ 3 does not terminate any point. From Eq. (39), one can get Now, we can plot isenthalpic, i.e. constant mass, 2 2 2 1 h Q2 i curves in T −P plane. From Eq. (4), one can obtain event 1 3 3 3 3 3 1 4π 3 Ti = 3 4π V 8π 4π Pi + 2 − 3 4 V 3 V 3 horizon and substituting event horizon into Eq. (10) gives Q2 1 isenthalpic curves in T − P plane. In Fig. (4), inversion = − 3 + 4πr + 2Pir+ . (41) 4πr+ + curves and isenthalpic curves are presented. Isenthalpic Subtracting Eq. (40) form Eq. (41) we can obtain curves have positive slope above the inversion curves so cooling occurs above the inversion curves. The sign of slope changes under the inversion curves and heat- 4 2 2 8πPir+ + 2r+ − 3Q = 0 (42) ing occurs in this region. It is also interesting to talk 5

FIG. 4. Inversion and isenthalpic curves for charged AdS black hole. From bottom to top, the isenthalpic curves correspond to increasing values of M. Red and black lines are isenthalpic and inversion curves, respectively. (a) Q = 1 and M = 1.5, 2, 2.5, 3 (b) Q = 2 and M = 2.5, 3, 3.5, 4 (c) Q = 10 and M = 10.5, 11, 11.5, 12 (d) Q = 20 and M = 20.5, 21, 21.5, 22 about naked singularities for charged AdS black holes. heating regions were shown for various values of charge In Fig. (5), we plot event horizon versus mass and pres- Q and mass M. We also denoted naked singularity which sure. We introduce four graphics, which correspond to is not sensible for Joule-Thomson expansion due to the Q = 1, 2, 10, 20. The regions can be seen that denote lack of event horizon. the naked singularities in Fig. (5). One cannot consider Both systems are not well behaved for low temper- Joule-Thomson expansion due to the lack of event hori- atures. Unfortunately, isenthalpic curves have positive zon for naked singularity. For example, we cannot deﬁne slopes under the lower inversion curves for both systems. event horizon for Q = 20 and M ≤ 20. For these values, It is also known that van der Waals equation does not event horizon is imaginary and it corresponds to naked too well agree with experiments. Thus Joule-Thomson singularity so isenthalpic curves in T −P plane are imag- expansion have been investigated for various equations inary. of state. In charged AdS case, it needs further investiga- tion.

IV. CONCLUSION ACKNOWLEDGMENTS In this paper, we studied the well known Joule- Thomson expansion for charged AdS black hole. The We thank Can Onur Keser with improving the figures black hole mass in AdS space is identified with enthalpy in this work. This work was sported by Scientific Re- due to variable cosmological constant notion, so one can search Projects Coordination Unit of Istanbul University. consider that mass does not change during the expansion. Project number is FYL-2016-20615. 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 6

FIG. 5. Event horizon of charged AdS black hole. (a) Q = 1 (b) Q = 2 (c) Q = 10 (d) Q = 20 7

[1] J. D. Bekenstein, Lett. Nuovo Cimento 4, 737 (1972) [22] S. Dutta, A. Jain, R. Soni, J. High Energy Phys. 12, 60 [2] J. D. Bekenstein, Phys. Rev. D 7, 97 (2333) (2013) [3] J. M. Bardeen, B. Carter, S. W. Hawking, Commun. [23] G. Q. Li, Phys. Lett. B 735, 256 (2014) Math. Phys. 31, 161 (1973) [24] N. Altamirano, D. Kubiznak, R. B. Mann, Z. Sherkat- [4] J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974) ghanad, Galaxies 2, 89 (2014) [5] S. W. Hawking, Nature 248, 30 (1974) [25] J. Liang, C. B. Sun, H. T. Feng, Europhys. Lett. 113, [6] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) 30008 (2016) [7] S. W. Hawking, D. N. Page, Commun. Math. Phys. 87, [26] S. H. Hendi, M. H. Vahidinia, Phys. Rev. D 88,084045 577 (1983) (2013) [8] A. Chamblin, R. Emparan, C. V. Johnson, R. C. Myers, [27] S. H. Hendi, S. Panahiyan, B. E. Panah, J. High Energy Phys. Rev. D 60, 064018 (1999) Phys. 01, 129 (2016) [9] A. Chamblin, R. Emparan, C. V. Johnson, R.C. Myers, [28] E. Spallucci, A. Smailagic, Phys. Lett. B 723, 436 (2013) Phys. Rev. D 60, 104026 (1999) [29] J. X. Mo, W. B. Liu, Phys. Lett. B 727, 336 (2013) [10] C. Niu, Y. Tian, X. N. Wu, Phys. Rev. D 85, 024017 [30] N. Altamirano, D. Kubiznak, R. B. Mann, Z. Sherkat- (2012) ghanad, Class. Quant. Grav. 31, 042001 (2014) [11] Y. D. Tsai, X. N. Wu, Y. Yang, Phys. Rev. D 85, 044005 [31] J. Sadeghi, H. Farahani, Int. J. Theor. Phys. 53, 3683 (2012) (2014) [12] R. Banerjee, S. K. Modak, Phys. Rev. D 84, 064024 [32] R. Zhao, M. Ma, H. Li, L. Zhang 2013 , Adv. High Energy (2011) Phys. 2013, 371084 (2013) [13] R. Banerjee, S. K. Modak, D. Roychowdhury, J. High [33] C. V. Johnson, Classical Quant. Grav. 31, 205002 (2014) Energy Phys. 10, 125 (2012) [34] C. V. Johnson, Classical Quant. Grav. 33, 135001 (2016) [14] M. M. Caldarelli, G. Cognola, D. Klemm, Class. Quant. [35] C. V. Johnson, arXiv:1511.08782 (2015) Grav. 17, 399 (2000) [36] A. Belhaj, M. Chabab, H. E. Moumni, K. Masmar, M. [15] D. Kastor, S. Ray, J. Traschen, Class. Quant. Grav. 26, B. Sedra, A. Segui, J. High Energy Phys. 05, 149 (2015) 195011 (2009) [37] E. Caceres, P. H. Nguyen, J. F. Pedraza, J. High Energy [16] D. Kubiznak, R. B. Mann, J. High Energy Phys. 07, 033 Phys. 1509, 184 (2015) (2012) [38] J. Sadeghi, K. Jafarzade, arXiv:1504.07744 (2015) [17] B. P. Dolan, Class. Quant. Grav. 28, 235017 (2011) [39] M. R. Setare, H. Adami, Gen. Relat. Gravity 47,132 [18] A. Belhaj, M. Chabab, H. E. Moumni, L. Medari, M. B. (2015) Sedra, Chin. Phys. Lett. 30, 090402 (2013) [40] C. V. Johnson, Entropy 18, 120 (2016) [19] R. G. Cai, L. M. Cao, L. Li, R. Q. Yang, J. High Energy [41] B. P. Dolan, Phys. Rev. D 84, 127503 (2011) Phys. 9, 005 (2013) [42] D. E. Winterbone, Advanced Thermodynamics for Engi- [20] A. Belhaj, M. Chabab, H. E. Moumni, K. Masmar, M. B. neers, 1st edn. (Butterworth-Heinemann, Oxford, 1997) Sedra, Int. J. Geom. Methods Mod. Phys. 12, 1550017 [43] D. C. Johnston, arXiv:1402.1205 (2014) (2015) [44] S. L. Vent, IJMEE 29, 257 (2001) [21] S. Gunasekaran, R. B. Mann, D. Kubiznak, J. High En- [45] N. Goldenfeld, Lectures on Phase Transition and the ergy Phys. 11, 110 (2012) Renormalization Group, 1st edn. (Westview Press, New York, 1992)