Black Hole Phase Transitions and the Chemical Potential
aa r X i v : . [ h e p - t h ] D ec Black Hole Phase Transitions and the ChemicalPotential
Reevu Maity, Pratim Roy, Tapobrata Sarkar ∗ Department of Physics,Indian Institute of Technology,Kanpur 208016,India
Abstract
In the context of extended phase space thermodynamics and the AdS-CFTcorrespondence, we consider the chemical potential ( µ ) dual to the num-ber of colours ( N ) of the boundary gauge theory, in the grand canonicalensemble. By appropriately defining µ via densities of thermodynamicquantities, we show that it changes sign precisely at the Hawking-Pagetransition for AdS-Schwarzschild and RN-AdS black holes in five dimen-sions, signalling the onset of quantum effects at the transition point. Suchbehaviour is absent for non-rotating black holes in four dimensions. ForKerr-AdS black holes in four and five dimensions, our analysis points tothe fact that µ can change sign in the stable black hole region, i.e above theHawking-Page transition temperature, for a range of angular frequencies.We also analyse AdS black holes in five dimensional Gauss-Bonnet gravity,and find similar features for µ as in the Kerr-AdS case. ∗ E-mail: reevum, proy, tapo @iitk.ac.in Introduction
The study of black hole thermodynamics has been an active field of research forthe past few decades. In this context, thermodynamics of anti de Sitter (AdS)black holes has been extensively studied, motivated by the gauge-gravity dual-ity. In these investigations, the black hole is considered as a bulk thermodynamicobject, with usual thermodynamic quantities like temperature, entropy etc. asso-ciated to it. Properties of black hole thermodynamics have been well establishedfor various classes of black holes in diverse dimensions, in the large amount ofexisting literature on the topic.One of the main features of black hole thermodynamics, in contrast withthermodynamics of ordinary matter is that the former depends on the ensemblein which the system is described. This has to do with the fact that black holeentropy is not extensive in the thermodynamic sense, i.e it scales as an area ratherthan a volume. Related to this issue is the fact that the definition of a volumeseems to be subtle in the context of black holes. Recent progress in this directionhas been made, with the proposal that the cosmological constant Λ be consideredas the pressure in the first law of black hole thermodynamics, and its conjugatequantity be identified as a volume [1]. For details about the consequences ofthis modification to black hole thermodynamics, see, e.g. [2], [3]. In terms ofthe pressure and volume, black holes show thermodynamic phase transitions,which resemble the liquid-gas phase transition, the re-entrant phase transitionetc., in a variety of AdS black hole systems. Examples of such phase transitions(in the canonical ensemble) have been known for a long time [4], but the explicitintroduction of the pressure makes the analogy with Van der Waals systems moretransparent. The analysis of black hole thermodynamics including the black holepressure via the cosmological constant has been dubbed as “extended phase spacethermodynamics” of black holes, and has seen a flurry of activity of late (see, forexample [5] and references therein).In fact, we can envisage two distinct approaches towards towards extendedphase space black hole thermodynamics. The first is, as mentioned, to relate thecosmological constant Λ to the black hole pressure P . Specifically, the relationthat one uses is P = − Λ / π [6] and a conjugate thermodynamic volume can alsobe derived [7]. In this approach, one usually holds the Newton’s constant (in thedimension in which the black hole lives) fixed. The second approach is to insteadconsider the relation between the cosmological constant and the AdS radius, andrelate the latter, by the AdS-CFT duality, to the number of colours N in thedual gauge theory. In this latter approach, one is allowed to “vary” the numberof colours. The first work in this direction appeared in [8] (see also [9], [10]).This issue might be quite subtle on the gauge theory side, as we are effectivelydescribing a “flow” in the theory space, parametrized by N . Importantly, when N is treated as a variable in the theory, a higher dimensional Newton’s constant hasto be held fixed (along with the Plank length in that dimension). For example,if we consider a theory on AdS × S , the AdS length is related to the number ofcolours of the dual gauge theory via the ten dimensional Newton’s constant G and the ten dimensional Planck length l . To vary N in such a theory, we fix1 = l = 1. From the point of view of the canonical ensemble, a varying N may not be ofparticular interest, as it simply redefines the critical values of the charges, andwe do not expect much interesting physics to emerge. As a concrete example, letus consider the five dimensional RN-AdS black hole, which is known to exhibit aliquid-gas type of phase transition up to a certain critical value of the charge Q and the horizon radius r h . An elementary calculation reveals that if we relate theAdS radius L via the gauge-gravity duality to N , by L = N / (we will not becareful about the precise pre-factors which can be absorbed in N ) the Hawkingtemperature of the black hole can be written as T = 12 πr h + r h π √ N − πQ N r h (1)It is then easy to check that the critical charge and horizon radius at the secondorder phase transition are simply rescaled : Q c = N √ π and r h,c = N / √ . Setting N = 1, we can recover the standard results for the situation where the five di-mensional Newton’s constant G is set to unity (without relating the cosmologicalconstant to the pressure). An entirely similar analysis follows for Kerr-AdS blackholes.However, as we have already said, from a bulk gravity point of view, one canconsider black hole thermodynamics in the canonical (fixed charges) or the grandcanonical (fixed potential) ensembles. If N is promoted to the level of a variablein the theory, we can either study a theory where this is held fixed, or one inwhich it can fluctuate. It is of course difficult to envisage a situation in whichthe system exchanges colour degrees of freedom with a surrounding bath, andthus one is naturally led to a description in which the number of colours is heldconstant (i.e it is a dialling and not a fluctuating variable). With this in mind, itis interesting to look at the system described by a grand canonical ensemble withrespect to the other charges (say the electric charge and the angular momentum)whose conjugate potentials are consequently held fixed. The concrete question that one can ask now is regarding the behaviour of thechemical potential conjugate to the number of colours in such an ensemble. It iswell known that in a grand canonical ensemble, a charged or rotating black holeexhibits the celebrated Hawking-Page phase transition (as opposed to the liquid-gas type phase transition in the canonical ensemble as discussed earlier), reachedat a temperature T HP where the Gibbs free energy is lower than a given referencebackground. In this ensemble, the chemical potential µ , conjugate to the numberof colours is an interesting object to study, since a vanishing chemical potentialindicates the onset of quantum effects in the system as it usually indicates thebreakdown of (particle) number conservation.In the original work that mooted the idea of a variable N [8], it was shownthat for the five dimensional AdS-Schwarzschild black hole, the chemical potential The Boltzmann’s constant is always set to unity in this paper, along with ~ and the speedof light, c . Throughout this work, by a slight abuse of notation, we will refer to this as the grandcanonical ensemble. It should be kept in mind however that this is really a mixed ensemble,with the number of colours held fixed, as are the other potentials. T HP , where the black hole is essentiallymetastable. Our observation here is that in the backdrop of AdS-CFT, it mightbe more natural (especially in the context of variable N ) to consider densities ofthermodynamic quantities at large N , and hence compute the chemical potential µ via these. One of the main purposes of this paper is to show that interestingphysics emerges when µ is computed via such densities of thermodynamic vari-ables (mass, entropy, charge, angular momentum), obtained by dividing theseby the volume of the space in which the dual gauge theory lives. In fact, as wewill show in sequel, for the five dimensional AdS-Schwarzschild background, thechemical potential calculated in this way changes sign precisely at T HP , indicat-ing that at this transition temperature, quantum effects might become important.We demonstrate that this fact remains valid for five dimensional RN-AdS blackholes as well.For four dimensional AdS-Schwarzschild and for the four dimensional RN-AdS black hole, this behaviour is however absent, and µ always changes sign ina metastable region. Interestingly, for rotating AdS black holes, we find that forsufficiently large values of the rotation parameter (close to its maximum value), µ can change sign in a stable black hole region, both in four and five dimensions.In the remainder of this paper, we will establish these facts. Towards the end,we also consider AdS black holes in five dimensions, modified by a Gauss-Bonnetterm. For such black holes, we show that the Gauss-Bonnet parameter behavesqualitatively like the rotation parameter, i.e for a sufficiently large parametervalues, one can expect important quantum corrections in a physical (i.e stable)black hole region. Let us begin with the well known example of the RN-AdS black hole. We firstrecord the expressions for a general d + 1 dimensional hole, and then specializeto the case of four and five dimensions [11]. In d + 1 dimensions, the Einstein-Maxwell theories in AdS space can be defined via the action S = 116 G d +1 π Z d d +1 x √− g (cid:18) R − F + d ( d − L (cid:19) (2)which is solved, along with an appropriate gauge potential, by the metric ds = − f ( r ) dt + 1 f ( r ) dr + r d Ω d − (3)where f ( r ) = 1 − mr d − + q r d − + r L . The parameters m and q appearing in eq.(3)are related to the ADM mass M and electric charge Q of the black hole by m = 16 πG d +1 M ( d − ω d − ; q = 8 πG d +1 Q p d − d − ω d − (4)3here ω d − = 2 π d/ / Γ( d/
2) is the volume of the unit d − f ( r ) = 1 − mr + q r + r L . Weremind the reader that we are working in natural units, ~ = c = 1, and will alsoset the Boltzmann’s constant to unity. Doing this, and setting the area of thehorizon A = 4 G S , where S is the entropy of the black hole and G is the fourdimensional Newton’s constant, the Smarr formula for the mass of the black holereads [12] M = G ( π L Q + S ) + πL S π / L √ G S (5)We will work in the grand canonical ensemble, and hence will express all thethermodynamic variables in terms of the electric potential and the temperature.To this end, noting that the electric charge of the black hole Q is related to itspotential Φ as Q = Φ √ S √ G π , we obtain the Hawking temperature of the black holeas T = 3 G S + L π (1 − Φ )4 π / L √ G S (6)We use eq.(6) to solve for the entropy in terms of the temperature, and obtaintwo solutions S ± = πL G (cid:16) − πLT √ π L T + 3Φ − π L T + 3(Φ − (cid:17) (7)These solutions define two distinct phases, with the positive sign correspondingto the large black hole phase and the negative sign to the small black hole phase.The small black hole phase is always unstable in the grand canonical ensemble,as can be checked.Now we will use the AdS-CFT dictionary to relate the four dimensional New-ton’s constant to the eleven dimensional one via the AdS radius, and the elevendimensional Planck length, l . Specifically, we use G = G L − , L = G N/l and also use the fact that G = l . Doing these substitutions, the mass of theblack hole simplifies to M = √ S (cid:0) πN / (Φ + 1) + S (cid:1) π / N / l (8)We also record the expression for the temperature and the Gibbs free energy interms N T = N / π (1 − Φ ) + 3 S π / N / √ Sl , G = √ S (cid:0) πN / (1 − Φ ) − S (cid:1) π / N / l (9)We will now set the eleven dimensional Planck length, l to unity. With this, wediscuss the chemical potential conjugate to the number of colours. Note that we have ignored some numerical factors in these definitions. These do notqualitatively affect the results, but their inclusion will make the expressions clumsy. N / , and hence it is natural to define the chemical potentialvia the first law of thermodynamics as dM = T dS + Φ dQ + µdN / (10)so that µ = ∂M∂N / . A simple calculation yields µ = √ S (cid:0) πN / (1 − Φ ) − S (cid:1) π / N / (11)As appropriate in the grand canonical ensemble, the Gibbs free energy and thechemical potential can be obtained by expressing G and µ in terms of the tem-perature T , for fixed values of Φ and N . This can be obtained for example, bysubstituting the entropy from eq.(7) in the second of eq.(9) or eq.(11). Alterna-tively, one can use the entropy as a parameter to obtain the Gibbs free energyor the chemical potential as a function of the temperature. Although entirelyequivalent, it is more illustrative to use the latter approach.The Hawking-Page temperature is defined via the vanishing of the Gibbs freeenergy of eq.(9), and the corresponding value of the entropy, S HP can be readoff from that equation. Similarly, the vanishing of the chemical potential occursat the value of S determined by setting µ = 0. These values are S HP = πN / (cid:0) − Φ (cid:1) , S µ =0 = 711 S HP (12)Using these values in the first of eq.(9), we obtain T HP = p − φ πN / , T µ =0 = 8 √ T HP (13)It is thus seen that the temperature at which the chemical potential becomeszero (i.e changes sign) is always less than the Hawking-Page phase transitiontemperature by a factor of 8 / √ ∼ .
91, and might seem to indicate that thisalways occurs in an unphysical region, where pure AdS is preferred over the blackhole.Now we show that the situation changes qualitatively if we consider the mass,entropy, and charge densities. For the four dimensional bulk that we consider,the dual CFT lives in a volume V ∼ L (again we will ignore some numerical prefactors that will not be important in our analysis). The Gibbs free energy is thenreplaced by the Gibbs free energy density, obtained by dividing the former by afactor of L and does not affect the Hawking-Page phase transition temperature.But the story is different as far as the chemical potential is concerned. This is As we have mentioned, this happens for the branch S + of the entropy defined in eq.(7). dρ = T ds + Φ dq + µdN / (14)where ρ , s and q denote the mass, entropy and charge densities respectively. Itis straightforward to check that in this case, the chemical potential (we will keepcalling this by µ in order to avoid cluttering of notation) is given by µ = √ s (cid:0) N / (1 − Φ ) − s (cid:1) πN / (15)The difference in the expressions of eq.(11) and (15) arises simply due to thefact that the chemical potential involves a derivative with respect to the num-ber of colours, and hence assumes a different value once the densities are in-volved (as the AdS radius L is related to the number of colours via the AdS-CFT dictionary). From eq.(15), we see that the chemical potential changessign at s = N / (1 − Φ ) and we find that this corresponds to a tempera-ture T µ =0 = √ T HP , i.e ∼ . T HP . This is different from the factor of 0 .
91 thatwe obtained in our previous analysis.In what follows, we will confine ourselves to the case where the chemicalpotential is defined via eq.(14). We now consider the five dimensional RN-AdSblack hole, where the effect of considering the densities is more drastic. Theprocedure adopted here is similar to the one outlined for the four dimensionalexamples, and we simply record the expressions for the temperature, Gibbs freeenergy density and the chemical potential, in terms of the entropy density s , theelectric potential Φ and the number of colours N . These are T = N / (cid:0) − + 12 × / N − / s / (cid:1) × / πs / g = N / s / (cid:0) − − × / N − / s / (cid:1) × / πµ = s / (cid:0) / N / (3 − ) − s / (cid:1) × / πN / (16)As before, the Hawking-Page phase transition temperature T HP can be obtainedfrom the zero of the Gibbs free energy density, and this can be compared withthe temperature at which the chemical potential changes sign, T µ =0 . The corre-sponding entropy densities read s HP = 112 √ N / (cid:0) − (cid:1) / , s µ =0 = 196 √ N / (cid:0) − (cid:1) / (17) It might be argued that we should ideally consider the “density” of the number of degrees offreedom. If we do use such a definition in the first law, it is straightforward to convince oneselfthat the numerical value of the chemical potential might change by an appropriate power of N . This does not affect the temperature at which µ changes sign, which is what we will beinterested in. We will thus continue to define µ via eq.(14). .25 0.30 0.35 0.40T - - Μ , g Figure 1: Gibbs potential (blue, scaledby 0 .
5) and chemical potential µ (red,scaled by 80) for 5-D RN-AdS blackholes as a function of the temperature,for the values N = 10 and Φ = 0 . - - - - Μ , g Figure 2: Gibbs potential (blue, scaledby 5) and chemical potential µ (red,scaled by 8 × ) for 5-D RN-AdS blackholes as a function of temperature, forthe values N = 20 and Φ = 0 . T HP = T µ =0 = 3 (3 − )2 √ πN / (3 − ) / (18)This might seem a little surprising, given that the entropy densities of eq.(17)differ by a factor of 1 /
8. The resolution is as follows. Inverting the first ofeqs.(16), it can be seen that there are two solutions to the entropy as in the fourdimensional example discussed before. These are (apart from a multiplicativefactor of 1 / s ± = N / (cid:16) πN / t ± √ π √ N t + 8Φ − / (cid:17) (19)While the Hawking-Page phase transition always occurs on the branch corre-sponding to s + , the chemical potential goes to zero on the branch s − . We showthis explicitly in figs.(1) and (2). In fig.(1), we have chosen N = 10 and Φ = 0 . . s = s + and s = s − respec-tively, in their expressions of eq.(16). We see that both these quantities changesign at the same temperature, as alluded to above, and that whereas for g , thesign change occurs for the branch s + , it occurs in the branch s − for µ . In fig.(2),the same quantities are plotted, for N = 20 and Φ = 0 .
6. Here, g is scaled by afactor of 5 and µ by a factor of 8 × . The same qualitative behaviour is seenin this case as well. We note that there is a critical value of Φ = √ / g and µ are alwaysnegative.
7s alluded to in the introduction, a vanishing chemical potential signals theonset of quantum effects. What we have established is that for five dimensionalRN-AdS black holes, this occurs precisely at the Hawking-Page phase transition.The same result holds after setting Q = Φ = 0, i.e for the five dimensional AdS-Schwarzschild black hole as well. This result differs from the that of [8] wherethe chemical potential for the five dimensional AdS-Schwarzschild black hole wasdefined via the thermodynamic variables, rather than their densities, and it wasfound that the sign change of µ occured in for temperatures less than T HP . Now we turn to the case of rotating black holes. We will first consider Kerr-AdSblack holes in four dimensions. The metric is standard, and given by [13] ds = − ∆ ρ (cid:16) dt − a Ξ sin θdφ (cid:17) + ρ dr ∆ + ρ dθ ∆ θ + ∆ θ sin θρ (cid:18) adt − r + a Ξ dφ (cid:19) (20)where we have defined∆ = ( r + a ) (cid:18) r L (cid:19) − mr, ∆ θ = 1 − a L cos θρ = r + a cos θ, Ξ = 1 − a L (21)We first record the expressions for the relevant thermodynamic quantities, themass M , entropy S , temperature T and angular velocity Ω in terms of the rotationparameter a and the horizon radius r h as M = L ( a + r h ) ( r h + L )2 G ( a − L ) r h , S = π ( a + r h ) G (cid:0) − a L (cid:1) ,T = a ( r h − L ) + r h (3 r h + L )4 πL r h ( a + r h ) , Ω = a (cid:16) r h L + 1 (cid:17) a + r h (22)We will also need the expression for the angular momentum J and Gibbs freeenergy G = M − T S − Ω J . These are expressed in terms of the same variables a and r h , and read J = aL ( a + r h ) ( r h + L )2 ( a − L ) r h , G = − ( a + r h ) ( L − r h )4 G ( a − L ) r h (23) Note that the angular velocity is measured with respect to a static observer at infinity [13].
8n order to switch to physical variables, we solve for r h and a in terms of S andΩ. The solutions read r h = √ G √ S √ G S − G L S Ω + πL √ πG S + π L , a = G L S Ω G S + πL (24)Similar expressions can obtained for r h and a , solved in terms of S and J . Thiswill be important in calculating the chemical potential in what follows.As before, we use the AdS-CFT dictionary to express the thermodynamicvariables in terms of the eleven dimensional Planck length and the eleven dimen-sional Newton’s constant. Also, we use the mass density ρ , the entropy density s and the angular momentum density j for the analysis that follows. It can beverified that the Hawking-Page phase transition occurs at the temperature T HP = 2 − N / Ω + 2 √ − N / Ω πN / (cid:16) √ − N / Ω + 1 (cid:17) (25)The temperature at which the chemical potential conjugate to the number ofcolours changes sign (denoted by T µ =0 ) can be straightforwardly computed, butthe expression is lengthy and we do not show it here. We will rather presentsome limiting expressions which will serve to illustrate out point. First let usconsider small values of the angular velocity Ω. In this case, we get the followingexpansion for the Hawking-Page phase transition temperature : T HP = 1 πN / − N / Ω π + O (cid:0) Ω (cid:1) (26)On the other hand, in this limit we obtain T µ =0 = 2 √ πN / − (cid:0) √ N / (cid:1) Ω π + O (cid:0) Ω (cid:1) (27)The above expressions indicate that at low values of the angular velocity, T µ =0 isalways less than T HP , indicating that quantum effects become large in the regionwhere pure AdS is preferred over the black hole. The situation however changesfor large Ω. Note that for four dimensional rotating black holes physicality of thesolutions demand that Ω < /L , which translates into Ω < N − / . Hence, welook at the region where the angular velocity is close to N − / , and obtain T HP = 12 πN / + √ α √ πN / − N / α / √ π + O (cid:0) α / (cid:1) T µ =0 = 0 . N / + 0 . α − . N / α + O (cid:0) α (cid:1) (28)where we have defined α = (cid:0) N − / − Ω (cid:1) . With 1 / (2 π ) ∼ .
16, it is clear thatclose to the the maximum value of Ω, T µ =0 is greater than T HP , i.e quantum We are actually considering small values of the dimensionless quantity Ω l . Since l isset to unity, this fact is suppressed in what follows. ds = ∆ ρ (cid:18) dt − a sin θ Ξ dφ (cid:19) + ∆ θ sin θρ (cid:18) adt − r + a Ξ dφ (cid:19) (29)+ ∆ θ r cos θρ + (cid:16) r L (cid:17) a r cos θdψ ρ dψ + ρ dr ∆ + ρ dθ ∆ θ (30)(31)where, ∆ = (cid:0) r + a (cid:1) (cid:18) r L (cid:19) − m, ∆ θ = 1 − a cos θL ρ = r + a cos θ, Ξ = 1 − a L (32)The relevant thermodynamic quantities, i.e the mass M , angular momentum J , entropy S , temperature T and angular velocity Ω in terms of the rotationparameter a and the horizon radius r h read : M = π (3 L − a ) ( a + r h ) ( L + r h )8 G ( a − L ) , J = πa ( a + r h ) (cid:16) r h L + 1 (cid:17) G (cid:0) − a L (cid:1) T = r h ( a + 2 r h + L )2 πL ( a + r h ) , Ω = a (cid:16) r h L + 1 (cid:17) a + r h , S = π r h ( a + r h )2 G (cid:0) − a L (cid:1) (33)In this case however, solving simultaneously for the rotation parameter a andthe horizon radius r h in terms of Ω and J (as in eq.(24) for the four dimensionalexample) is difficult. Thus we first solve for a in terms of r h , Ω and L , andfeed this back in the expression for the temperature. This latter relation is theninverted to obtain r h in terms of Ω, T and L . This information is then used tocompute the Gibbs free energy, which in terms of r h and a has a simple form G = π ( a + r h ) ( r h − L )8 G ( a − L ) (34)Then using the AdS-CFT dictionary, to set L ∼ N / , we obtain the Gibbsfree energy in terms of Ω, T and N . Finally, the chemical potential is obtainedvia µ = ( ∂g/∂N ) T, Ω where g is the Gibbs free energy density (note that thenumber of degrees of freedom scales as N in five dimensions). The expressions10 .25 0.26 0.27 0.28 0.29 0.30T - - Μ , g Figure 3: Gibbs potential density (blue,scaled by 10 ) and chemical potential µ (red, scaled by 4 × ) for 5-D Kerr-AdSblack holes as a function of the tempera-ture, for the values N = 10 and Ω = 0 . - - Μ , g Figure 4: Gibbs potential density (blue,scaled by 10 ) and chemical potential µ (red, scaled by 4 × ) for 5-D Kerr-AdSblack holes as a function of temperature,for the values N = 10 and Φ = 0 . . .
52 respectively. In both cases, we have taken N = 10. For small valuesof Ω, the chemical potential becomes positive very close to the Hawking-Pagetemperature T HP , as expected from the AdS-Schwarzschild analysis. However,as we increase Ω towards its maximum value (= N − / ∼ . T µ =0 is seen tobe greater than T HP . In these figures, the dotted and solid lines correspond tothe higher and lower entropy branches, and as before, the T HP ( T µ =0 ) occurs inthe higher (lower) entropy branch, respectively. We finally come to the case of the Gauss Bonnet black holes in five dimensions.The Einstein-Maxwell Gauss-Bonnet action in five dimensions with a cosmologi-cal constant Λ is given by [14] S = 116 πG Z d x √− g [ R − α R µνγδ R µνγδ − R µν R µν + R ) − πF µν F µν ] (35)where G is Newton’s constant in d = 5 and F µν is the field strength tensor and α is the Gauss-Bonnet parameter. The action has a well known solution givenby the metric ds = − f ( r ) dt + f − ( r ) dr + r d Ω , (36)11
10 12 14T - Μ , g Figure 5: Gibbs potential density (blue,scaled by 10 ) and chemical potential µ (red, scaled by 10 ) for 5-D AdS-GaussBonnet black holes as a function of thetemperature (scaled by 10 ), for the val-ues N = 1, α = 0 .
75 80 85 90T - Μ , g Figure 6: Gibbs potential density (blue,scaled by 10) and chemical potential µ (red, scaled by 10 ) for 5-D AdS-GaussBonnet black holes as a function of thetemperature (scaled by 10 ) for the val-ues N = 1000 and α = 0 . d Ω represents the line element of the 3-sphere and f ( r ) = 1 + r α − r αM G πr − αQ G π r − αL ! M is the thermodynamic mass and Q is the electric charge of the five-dimensionalGauss-Bonnet black hole with spherical event horizon topology. As before, wewill denote r h as the radius of the event horizon. In terms of r h and Q , thethermodynamic mass M , electric potential Φ, temperature T and entropy S aregiven as M = 3 πr h G (cid:18) αr h + r h L (cid:19) + G Q πr h , Φ = G Qπr h T = 3 π r h (2 r h + L ) − G L Q π L r h ( r h + 2 α ) , S = π r h G (cid:18) αr h (cid:19) , (37)The Gibbs free energy density is, g = − π r h ( − L (6 α − αr h + r h ) + 18 αr h + r h ) + 4 G L Q ( r h − α )24 πG L r h (2 α + r h ) (38)We will focus on the simple case Q = 0. In this case, one finds a subtle interplaybetween the two parameters of the theory, namely the Gauss-Bonnet coupling α and the number of colours N . We first fix α to a small value, say α = 0 .
01. Inthis case, one observes a swallow-tail behaviour for the Gibbs free energy density12tarting from N = 1. However, with α = 0 .
1, this swallow tail behaviour sets infor a larger value of N ∼
15. Conversely, for a fixed N , the swallow-tail behaviourfor a given value of α disappears when one increases α .To discuss the nature of the chemical potential vis a vis the Gibbs free energydensity, we fix α = 0 .
5. In figs.(5) and (6), we show the behaviour of the Gibbsfree energy (blue) and the chemical potential (red) for N = 1 and N = 1000respectively. In figs.(5) and (6), the Gibbs free energy density has been scaled by10 and 10, the chemical potential by 10 and 10 and the temperature by 10 and 10 respectively, for better visibility. We see that for this value of α , as oneincreases the number of colours, the chemical potential changes sign at a temper-ature greater than T HP . A qualitatively similar analysis can be straightforwardlydone for non-zero charge. We will however not present the details here. In this paper, we have considered extended phase space thermodynamics of aclass of charged and rotating AdS black holes in four and five dimensions, andthe AdS-Gauss-Bonnet black hole in five dimensions. The analysis was donein the grand canonical ensemble. We defined the chemical potential dual tothe number of colours of the boundary gauge theory via densities of standardthermodynamic variables. Our main conclusion here is that for five dimensionalAdS-Schwarzschild and RN-AdS black holes, this chemical potential changes signprecisely at the location of the Hawking-Page phase transition. This signals theonset of quantum effects, since a vanishing chemical potential conventionallysignals non-conservation of particle number. This is physically reasonable, andmight point to important physics at the Hawking-Page transition, which, in con-ventional thermodynamics, is dual to a confinement-deconfinement transition ofthe boundary gauge theory.For rotating black holes in four and five dimensions, our analysis shows thatfor a sufficiently large value of the angular frequency (close to its maximum value),the chemical potential changes sign in a stable black hole region, i.e above theHawking-Page temperature. The precise angular frequency where the potentialchanges sign at the black hole phase transition should be an important pointto revisit. We further analysed non-rotating Gauss-Bonnet-AdS black holes infive dimensions and saw features of the chemical potential that are similar toKerr-AdS black holes.Understanding the nature of the quantum effects due to a vanishing chemicalpotential, from the boundary field theory perspective, should be an importantissue for future research.
Acknowledgements
The work of RM is supported by the Department of Science and Technology,Govt. of India, by the grant IFA12-PH-34. The swallow-tail is always in the region of positive Gibbs free energy density and hencepossibly not particularly interesting. eferences [1] D. Kastor, S. Ray,J. Traschen, “Enthalpy and the Mechanics of AdS BlackHoles,” Class. Quant. Grav. , 195011 (2009) [arXiv:0904.2765 [hep-th]].[2] D. Kubiznak and R. B. Mann, “Black hole chemistry,” Can. J. Phys. , no.9, 999 (2015) [arXiv:1404.2126 [gr-qc]].[3] B. P. Dolan, “Where is the PdV term in the fist law of black hole thermo-dynamics?,” arXiv:1209.1272 [gr-qc].[4] A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, “Charged AdSblack holes and catastrophic holography,” Phys. Rev. D , 064018 (1999)[hep-th/9902170].[5] N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, “Ther-modynamics of rotating black holes and black rings: phase transitions andthermodynamic volume,” Galaxies , 89 (2014) [arXiv:1401.2586 [hep-th]].[6] D. Kubiznak and R. B. Mann, “P-V criticality of charged AdS black holes,”JHEP , 033 (2012) [arXiv:1205.0559 [hep-th]].[7] M. Cvetic, G. W. Gibbons, D. Kubiznak and C. N. Pope, “Black Hole En-thalpy and an Entropy Inequality for the Thermodynamic Volume,” Phys.Rev. D , 024037 (2011) [arXiv:1012.2888 [hep-th]].[8] B. P. Dolan, “Bose condensation and branes,” JHEP , 179 (2014)[arXiv:1406.7267 [hep-th]][9] J. -L. Zhang, R. G. Cai and H. Yu, “Phase transition and Thermodynam-ical geometry of Reissner-Nordstrom-AdS Black Holes in Extended PhaseSpace,” Phys. Rev. D , 044028 (2015) [arXiv:1502.01428 [hep-th]][10] J. -L. Zhang, R. G. Cai and H. Yu, “Phase transition and thermodynamicalgeometry for Schwarzschild AdS black hole in AdS × S spacetime,” JHEP , 143 (2015) [arXiv:1409.5305v2 [hep-th]][11] A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, “Charged AdSblack holes and catastrophic holography,” Phys. Rev. D , 064018 (1999)[hep-th/9902170][12] M. M. Caldarelli, G. Cognola and D. Klemm, “Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories,” Class. Quant. Grav. , 399 (2000) [hep-th/9908022][13] G. W. Gibbons, M. J. Perry and C. N. Pope, “The First law of thermody-namics for Kerr-anti-de Sitter black holes,” Class. Quant. Grav. , 1503(2005) [hep-th/0408217][14] R. G. Cai, “Gauss-Bonnet black holes in AdS spaces,” Phys. Rev. D65