A unified picture of phase transition: from liquid-vapour systems to AdS black holes
aa r X i v : . [ g r- q c ] S e p Preprint typeset in JHEP style - HYPER VERSION
A unified picture of phase transition: fromliquid-vapour systems to AdS black holes
Rabin Banerjee, Sujoy Kumar Modak and Dibakar Roychowdhury
S.N. Bose National Centre for Basic SciencesBlock-JD, Sector-III, Salt Lake City, Kolkata-700098, IndiaE-mail: [email protected]
E-mail: [email protected]
Email: [email protected]
Abstract:
Based on fundamental concepts of thermodynamics we examine phase tran-sitions in black holes defined in Anti-de Sitter (AdS) spaces. The method is in line withthat used a long ago to understand the liquid-vapour phase transition where the first orderderivatives of Gibbs potential are discontinuous and Clausius-Clapeyron equation is satis-fied. The idea here is to consider the AdS black holes as grand-canonical ensembles andstudy phase transition defined by the discontinuity of second order derivatives of Gibbs po-tential. We analytically check that this phase transition between the ‘smaller’ and ‘larger’mass black holes obey Ehrenfest relations defined at the critical point and hence confirma second order phase transition. This include both the rotating and charged black holes inEinstein gravity.
Keywords:
Black Holes, Classical Theories of Gravity. ontents
1. Introduction 12. Liquid to vapour phase transition 23. Phase transition in AdS black holes 34. Analytical check of Ehrenfest relations 6
5. Conclusions 8
1. Introduction
It is well known that black holes behave as thermodynamic systems. The laws of blackhole mechanics become similar to the usual laws of thermodynamics after appropriateidentifications between the black hole parameters and the thermodynamical variables [1].Nonetheless black holes show some exotic behaviors as a thermodynamic system, such as(i) entropy of black holes is proportional to area and not the volume, (ii) their temperaturediverges to infinity when mass tends to vanish due to Hawking evaporation. In fact theseexotic behaviors enhance the scope of alternative studies on black holes. The thermody-namic properties of black holes were further elaborated in early eighties when Hawking andPage discovered a phase transition between the Schwarzschild AdS black hole to thermalAdS space [2]. Thereafter this subject has been intensely studied in various viewpoints[3]-[15].The gibbsian approach to the phase transitions is based on the Clausius-Clapeyron-Ehrenfest’s equations [16, 17]. These equations allow for a classification of phase transitionsas first order or continuous (higher order) transitions. For a first order transition thefirst order derivatives (entropy and volume) of the Gibbs free energy is discontinuous andClausius-Clapeyron equation is satisfied. Liquid to vapour transition is an ideal exampleof such a phase transition. Similarly for a second order transition Ehrenfest’s relations aresatisfied. However, this specific tool concerning the nature of phase transition has neverbeen systematically used for black holes. Recently, in a series of papers [18]-[20] we havedeveloped an approach based on these concepts to study phase transitions in black holes.Despite of some conceptual issues mentioned in the earlier paragraph, we found that blackholes not only allow us to implement above basic ideas but also support them. In normalphysical systems a verification of the Clausius-Clapeyron or Ehrenfest relation is possible– 1 –ubjected to the design of a table top experiment. However black holes are found to belike a theorist’s laboratory, where these ideas on phase transition could be proved withoutperforming any table top experiment.In our earlier work for black hole phase transition we did not notice any discontinuityin the first order derivative of Gibbs energy and hence Clausius-Clapeyron equation wasnot needed. However, bacause of the discontinuity in second order derivatives we derivedEhrenfest’s relations for black holes in [18]. Because of infinite divergences of variousphysical entities at the critical point, we took recourse to numerical methods to check thevalidity of the Ehrenfest’s relations. A similar analysis was performed in [20] for the RNAdS black hole. In both cases we found that, away from the critical point, only the firstEhrenfest’s relation was satisfied. We further developed these ideas in [19] and were ableto verify both Ehrenfest’s relations infinitesimally close to the critical point. This analysiswas again based numerical computations. Nevertheless it should be emphasised that sucha numeric check has a drawback. Since in this method one assigns a numerical value forthe angular velocity or eletric potential beforehand and then check the Ehrenfest relations,everytime this value is changed it is necessary to repeat the numerical analysis. This adda limitation for generalising the result for other cases.In this paper we develop an elegant analytical technique to treat the singularities nearthe critical points of the phase transition curves. This is important for further studies onblack hole phase transition using our techniques. The example of the charged (RN) AdSand rotating (Kerr) black holes are worked out in details. We show analytically that bothEhrenfest’s relations are satisfied exactly. This confirms the onset of a second order phasetransition for these black holes. Explicit expressions for the critical temperature, criticalmass etc. are given. This phase transition is characterized by the divergence of the specificheat at the critical temperature. The sign of specific heat is different in two phases andthey essentially separate two branches of AdS black holes with different mass/ horizonradius. The branch with lower mass (horizon radius) has a negative specific heat and thusfalls in an unstable phase. The other branch with larger mass (horizon radius) is locallystable since it is associated with a positive specific heat and also positive Gibbs energy. Ourearlier results for the Kerr AdS [19], based on the numerical analysis, are also confirmed bythe present analytical scheme. Thus the present paper complets the developement initiatedin the earlier works [18]-[20].
2. Liquid to vapour phase transition
Liquid and vapour are just two phases of matter of a single constituent. When heatedupto the boiling point liquids vapourise and at this point the slope of the coexistence curve As a point of caution we would like to mention that the phase transition studied in this paper shouldnot be confused with the well known Hawking-Page (HP) phase transition[21]. The latter is a transitionbetween the AdS space and black hole phase [10]-[11]. As mentioned in earlier works [8],[11] such (HP)phase transition occurs at a temperature which is greater than the temperature where specific heat diverges.It is the latter case, which deals with a phase transition between two black hole phases, is considered inthis paper. – 2 –pressure P versus temperature T plot) obeys the Clausius-Clapeyron equation, given by dPdT = ∆ S ∆ V , (2.1)where ∆ S and ∆ V are the difference between the entropy and volume of the constituentin two different phases.Note that the derivation of the above relation comes from the definition of the Gibbspotential G = U − T S + P V ( U is the internal energy), where it is assumed that the firstorder derivatives of G with respect to the intrinsic variables ( P and T ) are discontinuousand it comes out that their specific values in two phases give the slope of the coexistencecurve at the phase transition point.
3. Phase transition in AdS black holes
As evident in the last section, the gibbsian approach is an effective tool for discussingphase transitions within the realm of conventional thermodynamics. Also, as we havealready highlighted in our works [18]-[19], this approach is equally powerful to study blackhole phase transitions. Here we further elaborate and extend such ideas and show thatthis formalism illuminates issues concerning the stability/ instability of AdS black holes.As a byproduct of the analysis, some key results obtained by Hawking and Page [2] arereproduced as well as generalized in a simple manner.Before we start giving our algebraic details, let us mention that throughout this letter,the various black hole parameters
M, Q, J, S, T should be interpreted as Ml , Ql , Jl , Sl , T l respectively, where the symbols have their standard meanings. Also, with this convention,the parameter l no longer appears in any of the equations [20].We start by considering the R-N AdS black hole. The Gibbs energy for RN-AdS blackhole is defined as G = M − T S − Φ Q where the last term is the analog of P V term (withasign difference) in conventional systems. The expressions for mass ( M ), temperature ( T ),charge ( Q ) and the entropy ( S ) are respectively given by [20] M = √ S [ π (1 + Φ ) + S ]2 π (3.1) T = π (1 − Φ ) + 3 S π / √ S , (3.2) Q = Φ r Sπ (3.3) S = πr . (3.4)where Φ is the electric potential and r + is the radius of the outer event horizon. Usingthese expressions one can easily write G as a function of ( S, Φ), G = √ S π / (cid:0) π (1 − Φ ) − S (cid:1) , (3.5)– 3 –onsidering a grand canonical ensemble (fixed Φ) it is now straightforward to findthe specific heat at constant potential ( C Φ ), which is the analog of C P (specific heat atconstant pressure) in conventional systems. This is found to be C Φ = T (cid:18) ∂S∂T (cid:19) Φ = 2 S π (1 − Φ ) + 3 S S − π (1 − Φ ) , (3.6)With this machinery we are in a position to describe phase transitions in AdS blackholes within the scope of standard thermodynamics. As a first step we plot Gibbs freeenergy, entropy and specific heat against temperature in figure 1. All these plots havea common feature; they exist only when the temperature of the system is greater thanthe critical temperature ( T ≥ T ). At T they all show nontrivial behavior and, as wesubsequently prove, there is a well defined phase transition at this temperature.First, consider the G − T plot. It has two wings which are joined at temperature T . Atthe upper wing, because of the positive definite values of G , this system falls in an unstablephase (Phase-1). At T Gibbs free energy is maximum which implies that the system ismost unstable at this point. Therefore it cannot stay at T for long and eventually passesto the locally stable phase (Phase-2) by minimizing its free energy. This is achieved byfollowing the other (lower) wing. In addition to that at T = T the entropy-temperatureplot is continuous and also changes its direction. The slope of this plot is negative forentropy lower than a critical value ( S ) while it is positive for all values higher than S .Entropy being proportional to the mass of a black hole, phase-1 in S − T plot correspondsto the lower mass (unstable) black holes while phase-2 belongs to black holes with highermass (locally stable). This behavior is further exemplified in the plot of specific heat withtemperature. The negative slope in S − T results in a negative heat capacity in phase-1. Asthe system approaches the critical point, C Φ diverges. Exactly at T it flips from negativeinfinity to positive infinity [8]. Through this phase transition the system emerges into alocally stable phase (phase-2) where the specific heat becomes positive. For T < T blackholes do not exist and what remains is nothing but the thermal radiation in pure AdSspace.Furthermore from the G − T plot we find that G changes its sign at temperature T = T . The free energy is thus always positive for T < T < T . For T > T the systemis globally stable. At T and we have the well known Hawking-Page transition [10]-[11].We now give algebraic details. As a first step, the minimum temperature T is calcu-lated for the RN AdS space. This is found from the temperature at which C Φ diverges (seefig.1). From (3.6) this occurs at the critical entropy, S = π − Φ ) . (3.7)Substituting this value in (3.2) we find the critical temperature T = 12 π p − Φ ) . (3.8)In the charge less limit (Φ = 0), this result goes over to, T = √ π (3.9)– 4 –his reproduce the result for the Schwarzschild AdS black hole found earlier in [2] . - - - C F Figure 1:
Gibbs free energy ( G ), entropy ( S ) andspecific heat ( C Φ ) plot for R-N AdS black holewith respect to temperature ( T ) for fixed Φ = 0 . . For detailed discussion see text. Observe that (3.8) also follows by following the original analysis of [2]. Followingthis approach we first calculate the minimum temperature T for the RN-AdS space-time.One can construct thermal states in RN-AdS space time by periodically identifying theimaginary time coordinate τ with period β = 4 π / √ Sπ (1 − Φ ) + 3 S (3.10)which is the inverse Hawking temperature ( T ). For the maximum value of β (i.e. minimumvalue of T ) at we set [ ∂β/∂r + ] Φ = 0. From this condition and by using (3.4) we find S = π (1 − Φ ). Substituting this into (3.2) we find the corresponding minimum temperaturefor RN-AdS space time to be identical with (3.8).The critical mass separating large and small black holes is now deduced. By substi-tuting (3.7) in (3.1) we obtain, M = √ − Φ (2 + Φ )3 √ . (3.11)In the charge less limit (Φ = 0) it yields, M = 23 √ Note that l does not appear since as explained before, it has been appropriately scaled out. – 5 –hich again reproduces the result given in [2]. Finally, the value of T is obtained from(3.5) and (3.2), T = √ − Φ π . (3.13)If we make Φ = 0, the temperature T simplifies to, T = 1 π (3.14)which is, once again, the result for the Schwarzschild AdS black hole [2].For the Kerr-AdS black hole one can easily perform a similar analysis and describephase transition for that case. The graphical analysis shown above for RN-AdS black holephysically remains unchanged for the Kerr-AdS case. In particular one can calculate theinverse temperature β Kerr = 4 π [ S ( π + S )( π + S − S Ω )] / π − πS (Ω − − S (Ω −
1) (3.15)and the condition (cid:2) ∂β Kerr /∂r + (cid:3) Ω = 0 gives( π + S ) (3 S − π ) − S ( π + S ) Ω + S (4 π + 3 S )Ω = 0 . (3.16)By substituting the positive solution of this polynomial in (3.15) one finds the minimumtemperature T Kerr . Furthermore for the Kerr-AdS black hole the heat capacity divergesexactly where (3.16) holds [19] which corresponds to the minimum temperature ( T Kerr ).In the irrotational limit T Kerr reproduces the known result of [2]. Results for M and T can also be calculated by a similar procedure. These expressions are rather lengthy andhence omitted. Moreover there are no new insights that have not already been discussedin the RN-AdS example.In the remaining part of this paper we shall analyze and classify the phase transitionat temperature T by exploiting Ehrenfest’s scheme. Since Hawking-Page transition attemperature T is already studied extensively in the literature we do not include this partin our paper.
4. Analytical check of Ehrenfest relations
Note that the ( S − T ) graph (fig.1) shows entropy is a continuous at temperature T .Consequently a first order transition is ruled out. However, the infinite discontinuity inspecific heat (see fig.1) strongly suggests the onset of a higher order (continuous) phasetransition. Under this circumstance, Ehrenfest’s equations are expected to play a role. Thederivation of these equations demands the continuity of specific entropy ( S ), specific charge( Q ) and specific angular momentum ( J ) at the critical point. As a result these relations aretruly local and only valid infinitesimally close to the critical point. For a genuine secondorder phase transition both equations have to be satisfied [16, 17].– 6 – .1 R-N AdS black hole We start by considering the RN-AdS black hole. Ehrenfest’s equations for this system aregiven by [20] − (cid:18) ∂ Φ ∂T (cid:19) S = C Φ − C Φ T Q ( α − α ) (4.1) − (cid:18) ∂ Φ ∂T (cid:19) Q = α − α k T − k T (4.2)where, α = Q (cid:16) ∂Q∂T (cid:17) Φ is the analog of volume expansion coefficient and k T = Q (cid:16) ∂Q∂ Φ (cid:17) T isthe analog of isothermal compressibility. Their explicit expressions are given by [20] Qα = 4 π Φ S S − π (1 − Φ ) (4.3) Qk T = r Sπ π Φ − π + 3 S S − π (1 − Φ ) (4.4)The L.H.S. of both Ehrenfest’s equations (4.1) and (4.2) are found to be identical at thecritical point S . Using the defining relations (3.2,3.3) the slope of the coexistence curveat the phase transition point is found to be [20], − (cid:20)(cid:18) ∂ Φ ∂T (cid:19) S (cid:21) S = S = − "(cid:18) ∂ Φ ∂T (cid:19) Q S = S = 2 √ π √ S Φ . (4.5)In order to calculate the right hand sides, Φ must be treated as a constant (this isanalogous to fixing pressure while performing an experiment). This would help us to re-express C Φ (3.6), Qα (4.3), and Qk T (4.4), which have functional forms f ( S ) g ( S ) , h ( S ) g ( S ) and k ( S ) g ( S ) respectively, infinitesimally close to the critical point ( S ). Note that they all havethe same denominator which satisfies the relation g ( S ) = 3 S − π (1 − Φ ) = 0. Thisobservation is crucial in the ensuing analysis.The expressions of C Φ , Qα and Qk T in the two phases ( i = 1 ,
2) are respectively givenby C Φ | S i = C Φ i , Qα | S i = Qα i and Qk T | S i = Qk T i . To obtain the R.H.S. of (4.1) we firstsimplify it’s numerator: C Φ − C Φ = f ( S ) g ( S ) − f ( S ) g ( S ) (4.6)Taking the points close to the critical point we may set f ( S ) = f ( S ) = f ( S ) since f ( S ) is well behaved. However since g ( S ) = 0 we do not set g ( S ) = g ( S ) = g ( S ). Thus C Φ − C Φ = f ( S ) (cid:18) g ( S ) − g ( S ) (cid:19) . (4.7)Following this logic one derives, C Φ − C Φ T Q ( α − α ) = f ( S ) T h ( S ) = 2 √ π √ S Φ (4.8)– 7 –nd, similarly, Q ( α − α ) Q ( k T − k T ) = h ( S ) k ( S ) = 2 √ π √ S Φ . (4.9)Remarkably we find that the divergence in C Φ is canceled with that of α in the firstequation and the same is true for the case of α and k T in the second equation. From(4.5,4.8,4.9) the validity of the Ehrenfest’s equations is established. Hence this phasetransition in RN-AdS black hole is a genuine second order transition. Ehrenfest’s set of equations for Kerr-AdS black hole are given by [18, 19] − (cid:18) ∂ Ω ∂T (cid:19) S = C Ω − C Ω T J ( α − α ) , (4.10) − (cid:18) ∂ Ω ∂T (cid:19) J = α − α k T − k T . (4.11)The expressions for specific heat ( C Ω ), analog of the volume expansion coefficient ( α )and compressibility ( k T ) are all provided in [19]. Once again, they all have the samedenominator (like the corresponding case for RN-AdS). Considering the explicit expressionsgiven in [19] and using the same techniques we find that both sides of (4.10) and (4.11)lead to an identical result, given by, l.h.s = r.h.s = 4 π ( π + S − S Ω ) ( π + S ) √ S Ω[3( π + S ) − S Ω (2 π + 3 S )] . (4.12)This shows that the phase transition for Kerr-AdS black hole is also second order. In ourearlier work [19] the above equality was shown numerically for various values of Ω. Thusthe analytical result (4.12) is compatible with our previous finding [19].
5. Conclusions
In the present paper we established that the concepts of thermodynamics which describethe well known liquid-vapour phase transitions are equally capable to understand black holephase transitions. This is true despite the fact that black hole entropy is non-extensive .In fact black holes play the role of a theorist’s laboratory to check the ideas based onClausius-Clapeyron-Ehrenfest schemes.This paper also completes a formulation to discuss phase transition in AdS black holesfollowing a conventional thermodynamical approach. This further clarify the interpreta-tion of black holes as thermodynamic objects. Considering a grand canonical ensemblethe critical temperature and mass of the charged (Reissner-Nordstrom) AdS black holesundergoing a phase transition were calculated. The phase transition discussed in this paperwas defined by the discontinuity of specific heat, volume expansion coefficient and com-pressibility. The two phases were identified with black holes having ‘smaller’ and ‘larger’mass/ horizon radius. The branch with smaller mass has negative specific heat and positive– 8 –ree energy, while, the other (larger mass) branch has positive specific heat and positivefree energy (less than the free energy of smaller mass black holes). The phase transitionoccurred from a lower mass black hole to a higher mass black hole. Thus it was a transitionbetween an unstable phase to a locally stable phase. Although various relevant thermody-namic variables diverged at the critical point, we were able to devise an analytical approach for checking the Ehrenfest’s relations appropriate for a second order transition. The ex-act validity of these relations confirmed the second order nature of the phase transition.Similar results for the rotating (Kerr) AdS black holes were outlined. These were foundto be compatible with our earlier work [19] which employed a numerical approach. As afuture work we plan to build a proper microscopic description of black hole phase phasetransitions which is still not known.
Acknowledgement:
S. K. M and D.R like to thank the Council of Scientific and Industrial Research (C. S. I.R), Government of India, for financial help.
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