Phase transition and heat engine efficiency of phantom AdS black holes
aa r X i v : . [ g r- q c ] M a r Phase transition and heat engine efficiency of phantom AdS black holes
Jie-Xiong Mo, Shan-Quan Lan
Institute of Theoretical Physics, Lingnan Normal University, Zhanjiang, 524048, Guangdong, China
Phase transition and heat engine efficiency of phantom AdS black holes are investigated withpeculiar properties found. In the non-extended phase space, we probe the possibility of T − S criticality in both the canonical ensemble and grand-canonical ensemble. It is shown that no T − S criticality exists for the phantom AdS black hole in the canonical ensemble, which is different fromthe RN-AdS black hole. Contrary to the canonical ensemble, no critical point can be found for neitherphantom AdS black holes nor RN-AdS black hole in the grand-canonical ensemble. Moreover, westudy the specific heat at constant electric potential. When the electric potential satisfies A > P − V criticality for phantom AdS black holes,contrary to the case of the RN-AdS black hole. Furthermore, we define a new kind of heat enginevia phantom AdS black holes. Comparing to RN-AdS black holes, phantom AdS black holes havea lower heat engine efficiency. However, the ratio η/η C of phantom AdS black hole is higher, thusincreasing the possibility of approaching the Carnot limit. This observation is obviously of interest.The interesting results obtained in this paper may be attributed to the existence of phantom fieldwhose energy density is negative. PACS numbers: 04.70.Dy, 04.70.-s
I. INTRODUCTION
Ever since the discovery of the well-known Hawking-Page phase transition [1] which describes the phase transitionbetween the Schwarzschild AdS black hole and the thermal AdS space, phase transition of AdS black holes has longbeen an interesting research topic. Chamblin et al. studied the phase transition of RN-AdS black holes [2, 3] andrevealed the close relation between charged AdS black holes and the liquid-gas system. Recently, Kubizˇn´ak andMann further enhanced this relation by probing the P − V criticality [4] in the extended phase space. The extendedphase space here refers to including the thermodynamic pressure and thermodynamic volume in the phase space[5]. Interpreting the cosmological constant as a variable [6]-[10], thermodynamics especially phase transition of blackholes in the extended phase space has received more and more attention. One can refer to the recent review [11] andreferences therein.Recent cosmological observations suggests the Universe appears to be expanding at an increasing rate. To explainthis phenomenon, varieties of models have been proposed. One kind of candidates may be the modified gravitytheories, such as f ( R ) gravity. Another way is to introduce an effective field generating repulsive gravity. Among thelatter approach, exotic fields represented by a distribution with negative energy density can be introduced to solvethe problem of cosmic acceleration. Phantom fields serve as such candidates that explain the observational data ofthe cosmic acceleration quite well [12, 13]. So it is of great interest to investigate the physical properties of phantomfield, especially from the context of gravity theories.The first motivation of this paper is to present a unified phase transition picture of phantom AdS black holes, notonly in the non-extended phase space but also in the extended phase space. For the sake of comparison, we choosethe phantom AdS black hole solution proposed in Ref. [14] as our research subject. This solution was obtained whileEinstein-Hilbert action with the cosmological constant is coupled with the Maxwell field or the phantom field. When η = −
1, it represents phantom AdS black holes and it reduces to RN-AdS black holes when η = 1. So it is veryconvenient to compare the results with those of RN-AdS black holes and disclose the effects of phantom fields.Concerning the phase transition of phantom black holes, many efforts have been made. Ref. [14] obtained thebasic quantities of phantom AdS black holes in the Einstein-Maxwell theory, such as the Hawking temperature, theentropy, the specific heat at constant charge and the free energy. Moreover, it pointed out the inconsistency betweenthe classical analysis and geometrothermodynamics. Although it claimed that local and global stabilities have beenestablished through the specific heat and the canonical and gran-canonical ensembles, the specific heat at constantelectric potential (an important tool to probe the stability in the grand-canonical ensemble) was missed in theiranalysis. Ref. [15] shows that the classical analysis and geometrothermodynamics can be compatible if one considerthe cosmological constant as a thermodynamic variable. Instead, they introduced the term Ldl into the first law,where L is the intensive variable dual to l . This treatment is in accordance with the spirit of the extended phasespace, although the definition of the physical quantities is varied. Moreover, their analysis was concentrated on thegeometrothermodynamics, leaving the phase transition in the extended phase space (especially the existence of P − V criticality) unexplored. Ref. [16] studied the asymptotically flat black hole solutions of the Einstein-Maxwell-Dilatontheory (where the effect of dilaton field is included) and their thermodynamics. Their critical phenomena wereinvestigated in [17] and geometrothermodynamics was revisited in [18] with a novel type of phase transition found.As stated in the former paragraph, we aim at a unified phase transition picture of phantom AdS black holes.Besides the phase transition, we are also interested in the heat engine utilizing phantom AdS black holes as workingsubstance. The concept of holographic heat engine was creatively introduced in Ref. [19], allowing one to extractuseful mechanical work from black holes. It was argued that the engine cycle represents a journal through a familyof holographically dual large N field theories [19], thus having interesting holographic implications. Subsequently, alot of efforts have been devoted to this topic and varieties of interesting findings are emerging [20]-[43]. It is worthmentioning that effects of quintessence dark energy on the heat engine was studied in Ref. [35]. One may wonderwhether phantom dark energy exerts influence on the heat engine efficiency, just as the quintessence dark energy does.And this is partly why we carry out the work in this paper.The organization of this paper is as follows. We will present a short review of the thermodynamics of phantomAdS black holes in Sec.II. Phase transition of phantom AdS black holes in the non-extended phase space will becomplemented in Sec.III while phase transition will be probed in the extended phase space in Sec.IV. Moreover, wewill study the heat engine efficiency of phantom AdS black holes in Sec.V. And the paper will be end with a briefconclusion presented in Sec.VI. II. REVIEW OF THERMODYNAMICS OF PHANTOM ADS BLACK HOLES
The action of Einstein theory with cosmological constant minimally coupled to the electromagnetic field reads [14] S = Z d x √− g ( R + 2 ηF µν F µν + 2Λ) , (1)where Λ denotes the cosmological constant and η is a constant which indicates the nature of the electromagnetic field.For the Maxwell field, η = 1 while η = − ds = f ( r ) dt − f ( r ) dr − r (cid:0) dθ + sin θdφ (cid:1) ,F = − qr dr ∧ dt , f ( r ) = 1 − Mr − Λ3 r + η q r , (2)where M, q denote the mass of the black hole and the electric charge of the source respectively. When all the parametersvanish, this solution reduces to Minkowski spacetime. It was argued that it is asymptotically anti-de Sitter for Λ < η = 1 [14] .Solving the equations f ( r ) = 0, one can obtain two positive real roots for the case η = 1 (corresponding to theevent horizon and the internal horizon) and only one positive root for the case η = − M = r + (cid:18) − Λ3 r + η q r (cid:19) , (3) T = 14 πr + (cid:18) − Λ r − η q r (cid:19) . (4) S = 14 A = πr . (5) A = qr + . (6)With the above physical quantities, the first law of the thermodynamics was presented as [14] dM = T dS + ηA dq . (7)For the case η = −
1, the second term in the right hand side of the above equation changes its sign implying itcontributes negative energy to the system. Ref. [15] argued that it is necessary to consider the cosmological constantas a variable and proposed the extended version of the first law as dM = T dS + Ldl + ηA dq , (8)where l = − / Λ. L is the intensive variable conjugate to l related by L = − ( r + /l ) [15]. In this paper, we wouldlike to probe the phase transition of phantom AdS black holes in both the non-extended phase space (where thecosmological constant is viewed as constant) and extended phase space ((where the cosmological constant is viewedas a thermodynamic variable). III. PHASE TRANSITION OF PHANTOM ADS BLACK HOLES IN THE NON-EXTENDED PHASESPACE
The specific heat at constant q was derived in Ref. [14] as C q = 2 S − πS + Λ S + ηπ q πS + Λ S − ηπ q . (9)And it was observed that only one phase transition point (also the divergent point of C q ) exists for the case η = − T − S criticality, which has become a hot topic in recent years since the originalwork of Spallucci and Smailagic [44](where T − S criticality was observed for RN-AdS black holes). One may wonderwhether there exists similar phenomena for the phantom AdS black holes. Utilizing Eqs. (4) and (5), one can obtain (cid:18) ∂T∂S (cid:19) q = − πS + Λ S − ηπ q π / S / , (10) (cid:18) ∂ T∂S (cid:19) q = 3 πS + Λ S − ηπ q π / S / . (11)It is not difficult to observe that the numerator of ∂T∂S coincides with the denominator of C q . One can soon draw theconclusion that the equation (cid:0) ∂T∂S (cid:1) q = 0 has only one positive root for η = −
1. The numerator of (cid:16) ∂ T∂S (cid:17) q turns out tobe 2 πS − ηπ q with the equation πS + Λ S = 3 ηπ q substituted. 2 πS − ηπ q is always positive when η = − (cid:16) ∂ T∂S (cid:17) q = 0 has no real root. And no T − S criticality exists for the case η = − η = 1 (the RN-AdS black hole).For the grand-canonical ensemble, the specific heat at constant electric potential is of interest. Utilizing Eqs. (4)and (6), one can reexpress the Hawking temperature into the function of A and S as T = π − S Λ − ηA π π / √ S . (12)Then the specific heat at constant electric potential can be derived as C A = T (cid:18) ∂S∂T (cid:19) A = 2 S ( − π + Λ S + ηπA ) π + Λ S − ηπA . (13)When S = − π + ηA π Λ , π + Λ S − ηπA = 0. For the case η = 1, this root is physical if the electric potential satisfies0 < A <
1. For the case η = −
1, this root is always physical. So the phantom AdS black holes undergo phasetransition in the grand-canonical ensemble. To gain an intuitive understanding, the behavior of C A is plotted in Fig.1for comparison. It can be witnessed that for A = 1 / , Λ = −
1, both the phantom AdS black hole and RN-AdS blackholes undergo phase transition although the divergent point of the specific heat differs from each other. However,for A = 2 , Λ = −
1, only phantom AdS black holes undergo phase transition. From the graph of the Hawkingtemperature, one can see that these choices of parameters lead to positive temperature and the results discussedabove make sense physically. (a) (b)(c) (d)
FIG. 1: Specific heat at constant electric potential for (a) η = 1 , − A = 1 / , Λ = − η = 1, A = 2 , Λ = − η = − A = 2 , Λ = − η = − A = 2 , Λ = − One can also probe the issue of T − S criticality in the grand-canonical ensemble. From Eq. (12), one can obtain (cid:18) ∂T∂S (cid:19) A = − π + Λ S − ηπA π / S / , (14) (cid:18) ∂ T∂S (cid:19) A = 3 π + Λ S − ηπA π / S / . (15)Solving the equations (cid:0) ∂T∂S (cid:1) A = 0 and (cid:16) ∂ T∂S (cid:17) A = 0, one can obtain the solution S = 0 , A = √ η . Obviously, thissolution is not physical for neither the case η = 1 nor the case η = −
1. So no critical point can be found for both theRN-AdS black hole and the phantom AdS black hole in the grand-canonical ensemble.
IV. PHASE TRANSITION OF PHANTOM ADS BLACK HOLES IN THE EXTENDED PHASE SPACE
In this section, we would like to probe the phase transition of phantom AdS black holes in the extended phasespace.Firstly, we define the thermodynamic pressure and thermodynamic volume as follows [4] P = − Λ8 π , (16) V = (cid:18) ∂M∂P (cid:19) , (17)where cosmological constant is view as a variable and identified as pressure.Substituting Eq. (16) into Eq. (3), one can reexpress the mass into the function of P as M = r + (cid:18) πP r + η q r (cid:19) . (18)Then the thermodynamic volume can be calculated as V = 4 πr . (19)The expression above is independent of η , implying that the phantom AdS black hole shares the same thermodynamicvolume with the RN-AdS black hole.The equation of state can be derived as P = T r + ηq πr − πr . (20)The difference of phantom AdS black holes and RN-AdS black holes is reflected in the sign of the second term in theequation of state. We will show below just this minor difference leads to completely different results in the existenceof P − V criticality.Based on Eq. (20), it is straightforward to derive (cid:18) ∂P∂r + (cid:19) T,q = r − πr T − ηq πr , (21) (cid:18) ∂ P∂r (cid:19) T,q = r ( − πr + T ) + 10 ηq πr . (22)Then the roots of the equations (cid:16) ∂P∂r + (cid:17) T = T c ,r + = r c = 0 and (cid:16) ∂ P∂r (cid:17) T = T c ,r + = r c = 0 can be analytically solved as T c = r c − ηq πr c , r c = q √ η . Obviously, this solution of r c is not physical when η = −
1. So there is no P − V criticalityfor phantom AdS black holes, contrary to RN-AdS black holes. For an intuitive understanding, see Fig.2(a).One can further investigate the Gibbs free energy of phantom AdS black holes in the extended phase space, wherethe mass should be interpreted as enthalpy rather than the internal energy. And the definition of Gibbs free energyreads G = H − T S = M − T S . Utilizing Eqs. (5), (18) and (20), one can obtain G = r + − P πr ηq r + . (23)The difference of phantom AdS black holes and RN-AdS black holes lies in the sign of the third term. The behaviorof Gibbs free energy of phantom AdS black holes is shown in Fig.2(b). For each choice of P , the curve of the Gibbsfree energy can be divided into two branches and the lower branch is more stable due to low free energy. No swallowtail behavior (two stable branches and one unstable branch) can be observed from the Gibbs free energy graph andthus no first order phase transition exists for phantom AdS black holes. V. PHANTOM ADS BLACK HOLES AS HEAT ENGINES
A new kind of heat engine via phantom AdS black holes would be built in this section. To probe its heat engineefficiency, we consider a rectangle cycle consisting of two isochores and two isobars in the P − V plane. See Fig.3 (a) (b) FIG. 2: (a) The thermodynamic pressure for η = − q = 1 with various choices of T (b)The Gibbs free energy of phantomAdS black holes for η = − q = 1 with various choices of P (a) FIG. 3: The heat engine cycle considered in this paper for a sketch of the cycle. Subscripts 1 , , , C V equals to zero. Then no heat flows along the isochores.Substituting Eq. (5) into (18), one can further express the mass into the function of SM = 3 S + 8 P S + 3 ηπq √ πS . (24)The heat input Q H along the isobar 1 → Q H = Z T T C P dT = Z S S C P (cid:18) ∂T∂S (cid:19) dS = Z S S T dS = Z H H dH = M − M . (25)With Eq. (24), Q H can be obtained as Q H = 12 √ π ( p S − p S ) + 4 P √ π ( S / − S / ) + √ πηq S − / − S − / ) . (26)Utilizing Eqs. (5) and (19), one can obtain the work done along the cycle as W = ( V − V )( P − P ) = 4( P − P )( S / − S / )3 √ π . (27)Via Eqs. (26) and (27), the heat engine efficiency can be derived as η = WQ H = (cid:18) − P P (cid:19) × P ( S / − S / )4 P ( S / − S / ) + ( √ S − √ S ) (cid:16) − πηq √ S S (cid:17) . (28)From Eqs.(26) and (27), one can conclude that the existence of phantom field only exerts influence on the heatinput Q H , leaving the work unchanged. For the same work done along the cycle, the heat input for the phantomblack holes is more than that of RN-AdS black holes. So phantom AdS black holes have a lower heat engine efficiencycomparing to RN-AdS black holes. And this may be attributed to the existence of phantom field whose energy densityis negative. Moreover, with the increasing of q , the heat engine efficiency of phantom AdS black holes decreases whilethat of RN-AdS black holes increases.To gain an intuitive understanding, we plot the heat engine efficiency for a specific example P = 2 , P = 1 , S =5 , S = 10 in Fig. 4. The solid line shows the case η = − η = 1. Note thatthe positivity of the Hawking temperature and the mass puts on constraints on the parameter q . Considered theseconstraints (by demanding T > M > q should not exceed q π (for the parameters chosen for the cycle)for the phantom AdS black holes while it should not exceed q π for RN-AdS black holes. And we have consideredthese constraints in the graph. It can be witnessed from Fig. 4(a) that the heat engine efficiency of phantom AdS blackholes decreases with q while the heat engine efficiency of RN-AdS black holes increases with q . And this observationis well in accord with the theoretical derivation above.One can also compare the heat engine efficiency with the well-known Carnot efficiency η C and investigate the ratio η/η C . The highest temperature of the heat engine cycle T H should be T and the lowest temperature of the heatengine cycle T C should be T for our cycle. So η C = 1 − T C T H = 1 − S / ( S + 8 P S − ηπq ) S / ( S + 8 P S − ηπq ) . (29)For the case η = −
1, both the numerator and denominator of the second term in the right hand side of the aboveexpression increase. To see the combined effect, one can compare the value of η C for both cases. η C | η =1 − η C | η = − = 2 πq S / ( S − S + 8 P S − P S ) S / ( − πq + S + 8 P S )( πq + S + 8 P S ) > . (30)So the carnot efficiency η C of phantom AdS black holes is lower than that of RN-AdS black holes. Fig. 4(b) showsthe behavior of ratio η/η C for the case P = 2 , P = 1 , S = 5 , S = 10. One can see that the ratio increases with q for the case η = − q for the case η = 1. So the existence of phantom field increases thepossibility of approaching the Carnot limit, which is of interest to researchers. VI. CONCLUSIONS
We study the phase transition and the heat engine efficiency of phantom AdS black holes in this paper. Firstly,we consider the phase transition in the non-extended phase space. Except the specific heat at constant charge whichhas already been studied in the former literature, we probe the possibility of T − S criticality in both the canonicalensemble and grand-canonical ensemble. In the canonical ensemble, it is shown that no T − S criticality exists for thecase η = − η = 1 (the RN-AdS black hole).Contrary to the canonical ensemble, no critical point can be found for neither phantom AdS black holes nor RN-AdSblack hole in the grand-canonical ensemble. Moreover, we study the specific heat at constant electric potential C A .It is shown that when the electric potential satisfies 0 < A <
1, both phantom AdS black holes and RN-AdS blackholes undergo phase transition in the grand-canonical ensemble. However, when the electric potential satisfies A > (a) (b) FIG. 4: A specific example with the parameters chosen as P = 2 , P = 1 , S = 5 , S = 10 (a) The heat engine efficiency η vs. q (b) the ratio η/η C vs. q Secondly, we investigate the phantom AdS black holes in the extended phase space. We identify the cosmologicalconstant as thermodynamic pressure and define its conjugate quantity as the thermodynamic volume. Then we showthat there is no P − V criticality for phantom AdS black holes. This observation is contrary to RN-AdS black holes.We further study the behavior of Gibbs free energy of phantom AdS black holes. It can be witnessed that the curveof the Gibbs free energy can be divided into two branches and the lower branch is more stable due to low Gibbs freeenergy. No swallow tail behavior can be observed from the Gibbs free energy graph and thus no first order phasetransition exists for phantom AdS black holes.Thirdly, we define a new kind of heat engine via phantom AdS black holes. It is shown that the existence ofphantom field only exerts influence on the heat input Q H , leaving the work unchanged. For the same work donealong the cycle, the heat input for the phantom black holes is more than that of RN-AdS black holes. So phantomAdS black holes have a lower heat engine efficiency comparing to RN-AdS black holes. And this may be attributed tothe existence of phantom field whose energy density is negative. Moreover, with the increasing of q , the heat engineefficiency of phantom AdS black holes decreases while that of RN-AdS black holes increases. Moreover, we comparethe heat engine efficiency with the well-known Carnot efficiency η C and investigate the ratio η/η C . We show that thisratio increases with q for phantom AdS black holes while it decreases with q for RN-AdS black holes. So the existenceof phantom field increases the possibility of approaching the Carnot limit, which is of interest to researchers. Acknowledgements