Critical phenomena of static charged AdS black holes in conformal gravity
CCritical phenomena of static charged AdSblack holes in conformal gravity
Wei Xu , and Liu Zhao School of Physics, Huazhong University of Science and Technology,Wuhan 430074, China School of Physics, Nankai University, Tianjin 300071, China email : [email protected] and [email protected]
Abstract
The extended thermodynamics of static charged AdS black holes in conformalgravity is analyzed. The P − V criticality of these black holes has some unusualfeatures. There exists a single critical point with critical temperature T c andcritical pressure P c . At fixed T > T c (or at fixed P > P c ), there are two zerothorder phase transition points but no first order phase transition points. Thesystems favors large pressure states at constant T , or high temperature states atconstant P . Recently the study of thermodynamics of black holes in AdS spacetime has been gener-alized to the extended phase space, in which the cosmological constant is identified asa thermodynamic pressure and its variations are included in the first law of black holethermodynamics [1–5]. Such studies are mainly motivated by the geometric derivationof the Smarr relation [1, 3, 5, 6], in which a term consisting of the pressure and its con-jugate “thermodynamic volume” appears. Once one takes the cosmological constantas thermodynamic pressure in the first law, the black hole mass should be explained asenthalpy rather than internal energy of the system. Besides, the criticality associated1 a r X i v : . [ g r- q c ] A ug ith the pressure was discussed in the extended phase space [7–25]. With the extendedstructure of the thermodynamic phase space, one can often find phase transitions andcritical points of various kinds. For instance, an interesting phenomena of black holereentrant phase transition is found recently in four-dimensional Born-Infeld-AdS blackhole [8], higher dimensional rotating AdS black holes [16], Kerr AdS black holes [19],GB-AdS black hole [21] and the third order Lovelock AdS black holes [26], which isobserved previously in multi-component fluids [27]. It is also possible to take other pa-rameters as novel dimensions of the thermodynamic phase space, e.g. the Born-Infeldparameter in the Born-Infeld-AdS black holes [8] and the GB parameter in GB-AdSblack holes [25].The reentrant phase transition includes both first and zeroth order phase transi-tions. However, the origin of zeroth order phase transition is still unclear. This makesit interesting to consider black hole systems containing only the zeroth order phasetransition. The thermodynamics of the AdS black holes in conformal gravity [28] tobe studied in this work provides precisely such an example.On the other hand, although taking the cosmological constant as a thermodynamicpressure is nowadays a common practice, such operations implicitly assume that grav-itational theories which differ only in the values of the cosmological constants areconsidered to fall in the “same class”, with unified thermodynamic relations. Thecommon excuse for doing this is that the classical theory of gravity may be an effectivetheory which follows from a yet unknown fundamental theory, in which all the presently“physical constants” are actually moduli parameters that can run from place to placein the moduli space of the fundamental theory. Given that the fundamental theory isyet unknown, it is preferable to consider the extended thermodynamics of gravitationaltheories involving only a single action, which requires that all variables included in thethermodynamical relations must be integration constants. It so happens that, in con-formal gravity, the cosmological constant comes as an integration constant rather thanas a bare parameter which appear explicitly in the classical action.In this paper, we consider the P − V criticality of the static charged AdS Blackhole in conformal gravity. There exists a single critical temperature, above which thereare two zeroth order phase transition points. In the next section, we first revisit thethermodynamics of static charged AdS black hole in conformal gravity in the extendedphase space. Section 3 is devoted to the P − V criticality, particular attention is paid2oward the appearance of critical points and the zeroth order phase transitions. Finally,some concluding remarks are given in the last section. We consider the static charged AdS Black hole in conformal gravity. The action ofconformal gravity reads S = α (cid:90) d x √− g (cid:18) C µνρσ C µνρσ + 13 F µν F µν (cid:19) , (1)where the unusual sign in front of the Maxwell term is inspired by critical gravity [29]and is required by requiring that the Einstein gravity emerges from conformal gravityin the infrared limit [30].The static black hole solution with AdS asymptotics for conformal gravity is foundin [28], which takes the formd s = − f ( r )d t + d r f ( r ) + r dΩ ,(cid:15) , (2)where f ( r ) = −
13 Λ r + c r + c + dr , (3)and A = − Qr d t (4)is the corresponding Maxwell field. dΩ ,(cid:15) represents the line element of a 2 d maxi-mally symmetric Einstein space with constant curvature 2 (cid:15) , with (cid:15) = 1 , − Q, c , c , d, Λ , (cid:15) in the solution, five of which areintegration constants, and the last one determines the spacial sectional geometry ofthe horizon. These parameters obey a constraint3 c d + (cid:15) + Q = c , (5)so there are actually only 5 independent parameters ( c is to be considered to bedetermined by other parameters as in (5)). Except the discrete parameter (cid:15) , the rest3 parameters Q, c , d, Λ are related to conserved charges: electric charge, charge ofmassive spin-2 hair, enthalpy and pressure, respectively. However, at fixed charges
Q, c , d , Λ and (cid:15) , there still exist a discrete freedom in choosing the integration constant c : c = ± (cid:112) c d + (cid:15) + Q . Under the point of view of taking the cosmological constant Λ as a thermodynamicpressure P = − Λ8 π , (6)the energy calculated by employing the Noether charge associated with the time-likeKilling vector [31] should be identified with the enthalpy H of the gravitational system.It reads H = α ( c − (cid:15) )(Λ r − c )72 πr + α (2Λ r − c + (cid:15) ) d πr , = α ( c c − c (cid:15) − πP d )24 π , (7)where r > f ( r ) which corresponds to the eventhorizon black hole , α is overall coupling which is present in the action (1). Becauseof the double-valuedness of the integration constant c , one immediately sees a double-valued behavior of the enthalpy. Such behaviors are also present in the four dimensionalcharged rotating black hole [32] and six dimensional static black holes [33] of conformalgravity. One can expect that the temperature and Gibbs free energy may also bedouble-valued. These double-valued variables must be all considered in order to havea holistic look at the thermodynamics in the extended phase space of the black hole.The thermodynamical conjugate of the pressure, i.e. the “thermodynamic volume”,is given by V = (cid:18) ∂H∂P (cid:19) S,Q e , Ξ = − α d . (8)It can be seen that the sign of V is determined by the sign of the parameter d . Besides P and V , all the other thermodynamic quantities are given in [28]. The temperatureis T = 8 πP r − c r − d π r , (9) We take Λ <
0, and so there is no cosmological horizon in the solution. S = α ( (cid:15)r − c r − d )6 r . (10)The electric charge and the conjugate potential are respectively Q e = α Q π , (11)Φ = − Qr . (12)The parameter c is a massive spin-2 hair which is now taken as a novel dimension inthe thermodynamic phase space. We label this novel dimension and its conjugate asΞ , Ψ: Ξ = c , (13)Ψ = α ( c − (cid:15) )24 π . (14)Throughout this work, we will take the normalization α = 2. It can be checked thatthe first law of thermodynamicsd H = T d S + Φ d Q e + Ψ dΞ + V d P (15)and the Smarr relation H = 2 P V + Ψ Ξ (16)hold in the extended thermodynamic phase space [28]. The absence of
T S and Φ Q terms in the Smarr relation can be explained by scaling arguments. The enthalpy canbe viewed as a homogeneous function of the extensive variables S, Q e , Ξ , P , i.e. H = H ( S, Q e , Ξ , P ) . Assuming each extensive variable has a scaling dimension which is denoted d S , d Q , d Ξ , d P respectively. If the enthalpy H itself has scaling dimension d H , then after a rescalingof the extensive variables we get λ d H H = H ( λ d S S, λ d Q Q e , λ d Ξ Ξ , λ d P P ) . Taking the first derivative with respect to λ and then setting λ = 1, we get d H H = d S T S + d Q Φ Q e + d Ξ ΨΞ + d P V P. c scales as [length] − ; c scales as [length] ;Λ scales as [length] − which is the same with P ; d scales as [length] ; Q scales as[length] , which is the same with Q e ; S scales as [length] ; H scales as [length] − ; d scales as [length] , which results in the Smarr relation (16), as expected.While considering critical behaviors, the Gibbs free energy will play an importantrole. It is given as follows: G = H − T S = − αP d − α π r (cid:18) ( c r − (cid:15)r + 3 d ) ( c r + 2 d ) + ( c − (cid:15) ) ( c r + d ) r (cid:19) . (17)On the other hand, the Helmholtz free-energy can be obtained from the Euclideanizedaction: F = α (cid:18) c − (cid:15) ) (cid:15) r + (3 (cid:15) + 8 πP r ) d (cid:19) π r . (18)A direct check yields F = H − T S − Φ Q e = G − Φ Q e . (19)This in turn justifies the explanation of H as thermodynamic enthalpy.Before proceeding, let us reveal a natural constraint of the black hole solutions.From f ( r ) = 0 we get P = − π ( r c + c r + d ) r . On the other hand, from eq. (5) we get d = − (cid:15) + Q − c c . Inserting both into (9), one can obtain T = (cid:15) + Q c π r − ( c + r c ) c π r , which leads to the constraint on T and r ( Q + (cid:15) ) − c π r T ≥ . (20)This constraint implies an upper bound of T at fixed horizon radius r , or an upperbound of r at fixed temperature T . 6 P − V criticality To consider the P − V criticality of the black hole, we should begin with the equation ofstate (EOS) in P − V plane at fixed conserved charges Q and c . The EOS arises fromthe expression (9) for the temperature T . However, to use (9) as a reasonable EOS, weneed to eliminate the parameters c and d . Assuming that both of these parametersare nonzero. Then the condition f ( r ) = 0 yields c = − πP r − c r − dr . (21)Inserting (21) into eqs.(5) and (9) respectively, one gets649 π P r + 163 πP r (cid:0) c r + d (cid:1) + c r + d r − Q − (cid:15) − c d = 0 (22) T = 43 P r + c π − d πr . (23)Then from eq.(23), we find d = 163 π P r − T π r + c r . Inserting this into eq.(22), we get the EOS64 π r P − π (4 πT − c ) r P + 16 π r T − π c r T + c r − ( Q + (cid:15) ) = 0 . Solving this equation for P we get P = T r − c r ± (cid:112) ( Q + (cid:15) ) − c π r T πr , and thanks to the condition (20), both branches of solutions should be consideredphysical. Comparing the above expressions for the pressure with the van der Waalsequation P = Tv − b − av (cid:39) Tv + bTv − av + O ( v − ) , we are tempted to use the variable v = 2 r (24)7s an effective specific volume for the black hole system under consideration. Thus werewrite the full EOS as4 π v P − πT − c ) π v P + 4 π v T − c π v T + c v − ( Q + (cid:15) ) = 0 . (25)The critical points ( P c , v c , T c ) results from the conditions ∂P∂v | v = v c ,T = T c = 0 , (26) ∂ P∂v | v = v c ,T = T c = 0 , (27) P c = P | v = v c ,T = T c . (28)The partial derivative in (26) and (27) can be evaluated directly using (25), which read ∂P∂v = − Pv + 4 T π − c π v + c Tv (4 T π − c − π P v ) (29) ∂ P∂v = 6 Pv − T π − c v π − c T (4 T π − c − π P v ) v + 4 π T c (4 T π − c − π P v ) v . (30)Solving these two equations and substituting into (28), we get two sets of critical pointparameters as follows.1. When c > v c = (cid:112) X ( Q + (cid:15) ) c , T c = (3 X − c π (27 X − , P c = − c (3 X − π (cid:112) X ( Q + (cid:15) ) X ;2. When c < v c = (cid:112) X ( Q + (cid:15) ) − c , T c = (3 X − c π (27 X − , P c = c (3 X − π (cid:112) X ( Q + (cid:15) ) X .
In the above, X can take two discrete values X or X : X = 403 + 163 √ (cid:39) . , (31) X = 403 − √ (cid:39) . . (32)However, the physical critical point must obey the constraints P c > r c > T c >
0, which lead to
X < , c > . (33)8his exclude the choice c < X = X , and we are left with a single critical pointwhich is characterized by the parameters v c = (cid:112) X ( Q + (cid:15) ) c , T c = (8 − X ) c π (8 − X ) , P c = c (8 − X )16 π (cid:112) X ( Q + (cid:15) ) X , (34)in which c >
0. These parameters satisfy the relation P c v c T c = 8 − X X (cid:39) . , (35)which gives a pure number and is independent of all parameters. Actually, this relationis very similar to the one of the van der Waals system, which behaviors as P c v c T c = atits critical point. However, if one replaces v in Eq.(35) by the thermodynamic volume V , the result of this relation will be no longer independent of parameters. In this sense,it is more natural to consider v as the specific volume instead of the thermodynamicvolume V . Note that the existence of critical point also exclude the possibility of c = 0. Also, the square sum of the charge Q and the signature (cid:15) must also be nonzero.Therefore, there is neither need to distinguish the case Q = 0 from the Q (cid:54) = 0 cases (aslong as (cid:15) (cid:54) = 0) nor need to distinguish the (cid:15) = 0 case from the (cid:15) = ± Q (cid:54) = 0).One may also be curious about the cases with d = 0 (the BPS black hole) or c = 0,both of which will result in an EOS of ideal gas after considering eq. (9) directly. Thusthey are all out of our discussion. Another degenerated case c = 0 , Q = 0 correspondsto the Schwarzschild-AdS black hole. In this case one can never find physical criticalpoints as is known in [7].The isothermal plots at generic parameters Q, c are depicted in Fig.1 (the rightplot is a magnification of a single isotherm at the temperature T = 1 . T c ). Whilecreating the plots, we take √ Q + (cid:15) c , c and c √ Q + (cid:15) respectively as units for v c , T c and P c . The pressure corresponding to the extremal specific volume is denoted P .It can be seen that on each isotherm there is an upper bound for the specific volume(black hole radius) v ex . For v < v ex , the isotherm can be subdivided into two segments,i.e. the lower branch and the upper branch, which reflect the double-valuedness of thepressure. The difference between the T < T c and T > T c curves lies in that, eachbranch of the isotherm in the former case is monotonic with respect to the specialvolume v , while the lower branch in the latter case is non-monotonic. Consequentlyphase transitions will occur only in the T > T c regime.9nother way of subdividing the isotherms is according to the sign of (cid:0) ∂P∂v (cid:1) T , whichis inversely proportional to the isothermal compressibility α ≡ v (cid:18) ∂v∂P (cid:19) T . According to the sign of the isothermal compressibility, each isotherm with
T > T c canbe subdivided into 4 segments, two with positive isothermal compressibility and twowith negative isothermal compressibility.Let us take a closer look at the magnified plot given on the right diagram in Fig.1.This is a curve corresponding to isotherm with T = 1 . T c . The lower and upperbranches of the curve is joined together at the point D which corresponds to theextremal specific volume v ex and the pressure P = P . On the lower branch (plottedin solid line) one can see that there is a local maximum P and local minimum P for P , the corresponding points on the isotherm are marked with B and C respectively.The 4 segments of the isotherm are IB (cid:95) , BC (cid:95) , CD (cid:95) and DJ (cid:95) respectively. Among these, IB (cid:95) and CD (cid:95) have negative isothermal compressibilities which imply that black hole statesfalling in these segments may be unstable. P v v T
T > T c . At such temperatures, we cansubdivide the range of the pressure into 4 regimes: P < P , P ≤ P < P , P ≤ P < P P ≥ P . When P < P , there is a single unstable black hole phase represented bythe segment IM (cid:95) . Due to its unstable nature, a perturbative increase in the pressurewould result in an increase of the black hole radius until it reaches the state M, thenthrough a phase transition it will enter the second regime for the pressure, P ≤ P < P .In this second regime for the pressure, there are 3 different black hole states for a singlepressure value, among these, only the one with intermediate sized specific volume (i.e.the state lying on the segment KC (cid:95) ) is stable. Therefore, as the pressure increases, theblack hole state will evolve from the state C until the state K is reached and thenthe pressure enters the next regime, P ≤ P < P . There are still 3 black hole statesat each pressure values in this regime, however, two of these have positive isothermalcompressibility (i.e. the states lying on the KB (cid:95) and DN (cid:95) segments), so it is hard to tellwhich is more stable by looking at the EOS alone. If the pressure enters the fourthregime, P ≥ P , then there is only a single stable phase which corresponds to stateson the segment NJ (cid:95) . From the above analysis, it is clear that there are two possiblephase transition points, the first one occurs at P = P , where the black hole willmost probably transit from the state M to the state C which is more stable uponperturbation. The second phase transition point occurs either at P = P (if the Gibbsfree energy on the segment KB (cid:95) is higher than that on DN (cid:95) ) or at P = P (if the Gibbsfree energy on the segment KB (cid:95) is lower than that on DN (cid:95) ). In the next subsection, itwill be clear that the Gibbs free energy on the segment KB (cid:95) is always lower than it ison DN (cid:95) , so the second phase transition point occurs at P = P , where the black holepicks the state B instead of N, because the state B has lower Gibbs free energy thanthe state N . In order to have a further look at the critical points, we need to plot the Gibbs freeenergy versus pressure at fixed temperature. The Gibbs free energy G as a function of T and P is quite complicated and cannot be given explicitly with ease. So we will tryto present the G − P relationship in terms of a pair of parametric equations.First we can invert the relation (23) to get P as a function of T and other parameters.Eliminating c in the resulting expression by use of (21), we get P ( v, d ) = 3 d π v + 3(4 π T − c )8 vπ , (36)11here we have also replaced r by v/
2. Inserting this equation together with (21) into(17), we get G ( v, d ) = (4 π T − c ) d vπ − (4 π T + c ) v π − (cid:15) (4 π T + c )12 π − Q + (cid:15) )9 vπ . (37)In (1) and (37), d cannot be taken as a free parameter, because there is an extraconstraint (22), which can be rewritten as9 d v + 3 d π T − c ) + v
16 (4 π T + c ) − ( Q + (cid:15) ) = 0 . (38)This quadratic algebraic equation gives two solutions for d as a function of T, v . Pickingeach solution and substituting into (36) and (37), the resulting pair of equations canbe taken as parametric definition for the function G ( P, T ) for constant T . So, we caneasily plot the G − P diagram at constant T , which is presented in Fig.2. T>T c T
Figure 3: The Gibbs free energy versus temperature at constant pressure: the curveson the upper-right has higher pressure 13
Concluding remarks
In this paper, we considered the P − V criticality of static charged AdS black holesin conformal gravity. Unlike the cases of Einstein gravity, the cosmological constantarises as an integration constant in conformal gravity, making the analysis for P − V criticality more self-contained, e.g. without need to consider systems with differentactions.The thermodynamics in the extended phase space for black hole in conformal gravitypossesses several unusual features: • there exists only one critical point but there are two phase transition points, bothof which corresponds to zeroth order phase transitions; • there is no first order phase transition in the system; • the phase transition can occur only when T > T c (or P > P c ) but not the otherway round; • at fixed T > T c , the system favors large pressure states, whilst at fixed P > P c ,the system favors high temperature states.We do not know of any other black hole or ordinary matter systems which exhibitsimilar thermodynamic behaviors. It would be interesting to find other examples whichyield similar behaviors, because otherwise the system under study would seem to betoo bizarre to understand. Acknowledgements
We would like to thank Hong Lu for suggesting this research.
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